direct product, metabelian, nilpotent (class 2), monomial, 2-elementary
Aliases: C5×C22.56C24, C10.1232- (1+4), C10.1712+ (1+4), C4⋊D4⋊18C10, C22⋊Q8⋊19C10, C4.4D4⋊16C10, C42.54(C2×C10), C42.C2⋊11C10, (C2×C20).683C23, (C4×C20).295C22, (C2×C10).382C24, (D4×C10).224C22, C22.D4⋊14C10, C23.25(C22×C10), C22.56(C23×C10), (Q8×C10).187C22, C2.15(C5×2- (1+4)), C2.23(C5×2+ (1+4)), (C22×C10).108C23, (C22×C20).462C22, (C5×C4⋊D4)⋊45C2, C4⋊C4.34(C2×C10), (C5×C22⋊Q8)⋊46C2, (C2×D4).37(C2×C10), (C5×C4.4D4)⋊36C2, C22⋊C4.7(C2×C10), (C2×Q8).30(C2×C10), (C5×C42.C2)⋊28C2, (C5×C4⋊C4).251C22, (C22×C4).73(C2×C10), (C2×C4).42(C22×C10), (C5×C22.D4)⋊33C2, (C5×C22⋊C4).92C22, SmallGroup(320,1564)
Series: Derived ►Chief ►Lower central ►Upper central
Subgroups: 362 in 220 conjugacy classes, 142 normal (14 characteristic)
C1, C2, C2 [×2], C2 [×4], C4 [×11], C22, C22 [×12], C5, C2×C4, C2×C4 [×10], C2×C4 [×4], D4 [×6], Q8 [×2], C23 [×4], C10, C10 [×2], C10 [×4], C42, C22⋊C4 [×12], C4⋊C4 [×10], C22×C4 [×4], C2×D4 [×6], C2×Q8 [×2], C20 [×11], C2×C10, C2×C10 [×12], C4⋊D4 [×4], C22⋊Q8 [×4], C22.D4 [×4], C4.4D4 [×2], C42.C2, C2×C20, C2×C20 [×10], C2×C20 [×4], C5×D4 [×6], C5×Q8 [×2], C22×C10 [×4], C22.56C24, C4×C20, C5×C22⋊C4 [×12], C5×C4⋊C4 [×10], C22×C20 [×4], D4×C10 [×6], Q8×C10 [×2], C5×C4⋊D4 [×4], C5×C22⋊Q8 [×4], C5×C22.D4 [×4], C5×C4.4D4 [×2], C5×C42.C2, C5×C22.56C24
Quotients:
C1, C2 [×15], C22 [×35], C5, C23 [×15], C10 [×15], C24, C2×C10 [×35], 2+ (1+4) [×2], 2- (1+4), C22×C10 [×15], C22.56C24, C23×C10, C5×2+ (1+4) [×2], C5×2- (1+4), C5×C22.56C24
Generators and relations
G = < a,b,c,d,e,f,g | a5=b2=c2=d2=e2=g2=1, f2=b, ab=ba, ac=ca, ad=da, ae=ea, af=fa, ag=ga, bc=cb, ede=bd=db, geg=be=eb, bf=fb, bg=gb, fdf-1=cd=dc, ce=ec, cf=fc, cg=gc, gdg=bcd, fef-1=bce, fg=gf >
►On 160 points
Generators in S160
(1 2 3 4 5)(6 7 8 9 10)(11 12 13 14 15)(16 17 18 19 20)(21 22 23 24 25)(26 27 28 29 30)(31 32 33 34 35)(36 37 38 39 40)(41 42 43 44 45)(46 47 48 49 50)(51 52 53 54 55)(56 57 58 59 60)(61 62 63 64 65)(66 67 68 69 70)(71 72 73 74 75)(76 77 78 79 80)(81 82 83 84 85)(86 87 88 89 90)(91 92 93 94 95)(96 97 98 99 100)(101 102 103 104 105)(106 107 108 109 110)(111 112 113 114 115)(116 117 118 119 120)(121 122 123 124 125)(126 127 128 129 130)(131 132 133 134 135)(136 137 138 139 140)(141 142 143 144 145)(146 147 148 149 150)(151 152 153 154 155)(156 157 158 159 160)
(1 26)(2 27)(3 28)(4 29)(5 30)(6 16)(7 17)(8 18)(9 19)(10 20)(11 156)(12 157)(13 158)(14 159)(15 160)(21 31)(22 32)(23 33)(24 34)(25 35)(36 46)(37 47)(38 48)(39 49)(40 50)(41 51)(42 52)(43 53)(44 54)(45 55)(56 66)(57 67)(58 68)(59 69)(60 70)(61 71)(62 72)(63 73)(64 74)(65 75)(76 86)(77 87)(78 88)(79 89)(80 90)(81 91)(82 92)(83 93)(84 94)(85 95)(96 106)(97 107)(98 108)(99 109)(100 110)(101 111)(102 112)(103 113)(104 114)(105 115)(116 126)(117 127)(118 128)(119 129)(120 130)(121 131)(122 132)(123 133)(124 134)(125 135)(136 146)(137 147)(138 148)(139 149)(140 150)(141 151)(142 152)(143 153)(144 154)(145 155)
(1 21)(2 22)(3 23)(4 24)(5 25)(6 156)(7 157)(8 158)(9 159)(10 160)(11 16)(12 17)(13 18)(14 19)(15 20)(26 31)(27 32)(28 33)(29 34)(30 35)(36 41)(37 42)(38 43)(39 44)(40 45)(46 51)(47 52)(48 53)(49 54)(50 55)(56 61)(57 62)(58 63)(59 64)(60 65)(66 71)(67 72)(68 73)(69 74)(70 75)(76 81)(77 82)(78 83)(79 84)(80 85)(86 91)(87 92)(88 93)(89 94)(90 95)(96 101)(97 102)(98 103)(99 104)(100 105)(106 111)(107 112)(108 113)(109 114)(110 115)(116 121)(117 122)(118 123)(119 124)(120 125)(126 131)(127 132)(128 133)(129 134)(130 135)(136 141)(137 142)(138 143)(139 144)(140 145)(146 151)(147 152)(148 153)(149 154)(150 155)
(1 96)(2 97)(3 98)(4 99)(5 100)(6 81)(7 82)(8 83)(9 84)(10 85)(11 86)(12 87)(13 88)(14 89)(15 90)(16 91)(17 92)(18 93)(19 94)(20 95)(21 101)(22 102)(23 103)(24 104)(25 105)(26 106)(27 107)(28 108)(29 109)(30 110)(31 111)(32 112)(33 113)(34 114)(35 115)(36 116)(37 117)(38 118)(39 119)(40 120)(41 121)(42 122)(43 123)(44 124)(45 125)(46 126)(47 127)(48 128)(49 129)(50 130)(51 131)(52 132)(53 133)(54 134)(55 135)(56 136)(57 137)(58 138)(59 139)(60 140)(61 141)(62 142)(63 143)(64 144)(65 145)(66 146)(67 147)(68 148)(69 149)(70 150)(71 151)(72 152)(73 153)(74 154)(75 155)(76 156)(77 157)(78 158)(79 159)(80 160)
(1 56)(2 57)(3 58)(4 59)(5 60)(6 131)(7 132)(8 133)(9 134)(10 135)(11 116)(12 117)(13 118)(14 119)(15 120)(16 121)(17 122)(18 123)(19 124)(20 125)(21 61)(22 62)(23 63)(24 64)(25 65)(26 66)(27 67)(28 68)(29 69)(30 70)(31 71)(32 72)(33 73)(34 74)(35 75)(36 76)(37 77)(38 78)(39 79)(40 80)(41 81)(42 82)(43 83)(44 84)(45 85)(46 86)(47 87)(48 88)(49 89)(50 90)(51 91)(52 92)(53 93)(54 94)(55 95)(96 146)(97 147)(98 148)(99 149)(100 150)(101 151)(102 152)(103 153)(104 154)(105 155)(106 136)(107 137)(108 138)(109 139)(110 140)(111 141)(112 142)(113 143)(114 144)(115 145)(126 156)(127 157)(128 158)(129 159)(130 160)
(1 46 26 36)(2 47 27 37)(3 48 28 38)(4 49 29 39)(5 50 30 40)(6 151 16 141)(7 152 17 142)(8 153 18 143)(9 154 19 144)(10 155 20 145)(11 136 156 146)(12 137 157 147)(13 138 158 148)(14 139 159 149)(15 140 160 150)(21 51 31 41)(22 52 32 42)(23 53 33 43)(24 54 34 44)(25 55 35 45)(56 81 66 91)(57 82 67 92)(58 83 68 93)(59 84 69 94)(60 85 70 95)(61 76 71 86)(62 77 72 87)(63 78 73 88)(64 79 74 89)(65 80 75 90)(96 131 106 121)(97 132 107 122)(98 133 108 123)(99 134 109 124)(100 135 110 125)(101 126 111 116)(102 127 112 117)(103 128 113 118)(104 129 114 119)(105 130 115 120)
(6 156)(7 157)(8 158)(9 159)(10 160)(11 16)(12 17)(13 18)(14 19)(15 20)(56 66)(57 67)(58 68)(59 69)(60 70)(61 71)(62 72)(63 73)(64 74)(65 75)(76 86)(77 87)(78 88)(79 89)(80 90)(81 91)(82 92)(83 93)(84 94)(85 95)(96 111)(97 112)(98 113)(99 114)(100 115)(101 106)(102 107)(103 108)(104 109)(105 110)(116 131)(117 132)(118 133)(119 134)(120 135)(121 126)(122 127)(123 128)(124 129)(125 130)(136 141)(137 142)(138 143)(139 144)(140 145)(146 151)(147 152)(148 153)(149 154)(150 155)
G:=sub<Sym(160)| (1,2,3,4,5)(6,7,8,9,10)(11,12,13,14,15)(16,17,18,19,20)(21,22,23,24,25)(26,27,28,29,30)(31,32,33,34,35)(36,37,38,39,40)(41,42,43,44,45)(46,47,48,49,50)(51,52,53,54,55)(56,57,58,59,60)(61,62,63,64,65)(66,67,68,69,70)(71,72,73,74,75)(76,77,78,79,80)(81,82,83,84,85)(86,87,88,89,90)(91,92,93,94,95)(96,97,98,99,100)(101,102,103,104,105)(106,107,108,109,110)(111,112,113,114,115)(116,117,118,119,120)(121,122,123,124,125)(126,127,128,129,130)(131,132,133,134,135)(136,137,138,139,140)(141,142,143,144,145)(146,147,148,149,150)(151,152,153,154,155)(156,157,158,159,160), (1,26)(2,27)(3,28)(4,29)(5,30)(6,16)(7,17)(8,18)(9,19)(10,20)(11,156)(12,157)(13,158)(14,159)(15,160)(21,31)(22,32)(23,33)(24,34)(25,35)(36,46)(37,47)(38,48)(39,49)(40,50)(41,51)(42,52)(43,53)(44,54)(45,55)(56,66)(57,67)(58,68)(59,69)(60,70)(61,71)(62,72)(63,73)(64,74)(65,75)(76,86)(77,87)(78,88)(79,89)(80,90)(81,91)(82,92)(83,93)(84,94)(85,95)(96,106)(97,107)(98,108)(99,109)(100,110)(101,111)(102,112)(103,113)(104,114)(105,115)(116,126)(117,127)(118,128)(119,129)(120,130)(121,131)(122,132)(123,133)(124,134)(125,135)(136,146)(137,147)(138,148)(139,149)(140,150)(141,151)(142,152)(143,153)(144,154)(145,155), (1,21)(2,22)(3,23)(4,24)(5,25)(6,156)(7,157)(8,158)(9,159)(10,160)(11,16)(12,17)(13,18)(14,19)(15,20)(26,31)(27,32)(28,33)(29,34)(30,35)(36,41)(37,42)(38,43)(39,44)(40,45)(46,51)(47,52)(48,53)(49,54)(50,55)(56,61)(57,62)(58,63)(59,64)(60,65)(66,71)(67,72)(68,73)(69,74)(70,75)(76,81)(77,82)(78,83)(79,84)(80,85)(86,91)(87,92)(88,93)(89,94)(90,95)(96,101)(97,102)(98,103)(99,104)(100,105)(106,111)(107,112)(108,113)(109,114)(110,115)(116,121)(117,122)(118,123)(119,124)(120,125)(126,131)(127,132)(128,133)(129,134)(130,135)(136,141)(137,142)(138,143)(139,144)(140,145)(146,151)(147,152)(148,153)(149,154)(150,155), (1,96)(2,97)(3,98)(4,99)(5,100)(6,81)(7,82)(8,83)(9,84)(10,85)(11,86)(12,87)(13,88)(14,89)(15,90)(16,91)(17,92)(18,93)(19,94)(20,95)(21,101)(22,102)(23,103)(24,104)(25,105)(26,106)(27,107)(28,108)(29,109)(30,110)(31,111)(32,112)(33,113)(34,114)(35,115)(36,116)(37,117)(38,118)(39,119)(40,120)(41,121)(42,122)(43,123)(44,124)(45,125)(46,126)(47,127)(48,128)(49,129)(50,130)(51,131)(52,132)(53,133)(54,134)(55,135)(56,136)(57,137)(58,138)(59,139)(60,140)(61,141)(62,142)(63,143)(64,144)(65,145)(66,146)(67,147)(68,148)(69,149)(70,150)(71,151)(72,152)(73,153)(74,154)(75,155)(76,156)(77,157)(78,158)(79,159)(80,160), (1,56)(2,57)(3,58)(4,59)(5,60)(6,131)(7,132)(8,133)(9,134)(10,135)(11,116)(12,117)(13,118)(14,119)(15,120)(16,121)(17,122)(18,123)(19,124)(20,125)(21,61)(22,62)(23,63)(24,64)(25,65)(26,66)(27,67)(28,68)(29,69)(30,70)(31,71)(32,72)(33,73)(34,74)(35,75)(36,76)(37,77)(38,78)(39,79)(40,80)(41,81)(42,82)(43,83)(44,84)(45,85)(46,86)(47,87)(48,88)(49,89)(50,90)(51,91)(52,92)(53,93)(54,94)(55,95)(96,146)(97,147)(98,148)(99,149)(100,150)(101,151)(102,152)(103,153)(104,154)(105,155)(106,136)(107,137)(108,138)(109,139)(110,140)(111,141)(112,142)(113,143)(114,144)(115,145)(126,156)(127,157)(128,158)(129,159)(130,160), (1,46,26,36)(2,47,27,37)(3,48,28,38)(4,49,29,39)(5,50,30,40)(6,151,16,141)(7,152,17,142)(8,153,18,143)(9,154,19,144)(10,155,20,145)(11,136,156,146)(12,137,157,147)(13,138,158,148)(14,139,159,149)(15,140,160,150)(21,51,31,41)(22,52,32,42)(23,53,33,43)(24,54,34,44)(25,55,35,45)(56,81,66,91)(57,82,67,92)(58,83,68,93)(59,84,69,94)(60,85,70,95)(61,76,71,86)(62,77,72,87)(63,78,73,88)(64,79,74,89)(65,80,75,90)(96,131,106,121)(97,132,107,122)(98,133,108,123)(99,134,109,124)(100,135,110,125)(101,126,111,116)(102,127,112,117)(103,128,113,118)(104,129,114,119)(105,130,115,120), (6,156)(7,157)(8,158)(9,159)(10,160)(11,16)(12,17)(13,18)(14,19)(15,20)(56,66)(57,67)(58,68)(59,69)(60,70)(61,71)(62,72)(63,73)(64,74)(65,75)(76,86)(77,87)(78,88)(79,89)(80,90)(81,91)(82,92)(83,93)(84,94)(85,95)(96,111)(97,112)(98,113)(99,114)(100,115)(101,106)(102,107)(103,108)(104,109)(105,110)(116,131)(117,132)(118,133)(119,134)(120,135)(121,126)(122,127)(123,128)(124,129)(125,130)(136,141)(137,142)(138,143)(139,144)(140,145)(146,151)(147,152)(148,153)(149,154)(150,155)>;
G:=Group( (1,2,3,4,5)(6,7,8,9,10)(11,12,13,14,15)(16,17,18,19,20)(21,22,23,24,25)(26,27,28,29,30)(31,32,33,34,35)(36,37,38,39,40)(41,42,43,44,45)(46,47,48,49,50)(51,52,53,54,55)(56,57,58,59,60)(61,62,63,64,65)(66,67,68,69,70)(71,72,73,74,75)(76,77,78,79,80)(81,82,83,84,85)(86,87,88,89,90)(91,92,93,94,95)(96,97,98,99,100)(101,102,103,104,105)(106,107,108,109,110)(111,112,113,114,115)(116,117,118,119,120)(121,122,123,124,125)(126,127,128,129,130)(131,132,133,134,135)(136,137,138,139,140)(141,142,143,144,145)(146,147,148,149,150)(151,152,153,154,155)(156,157,158,159,160), (1,26)(2,27)(3,28)(4,29)(5,30)(6,16)(7,17)(8,18)(9,19)(10,20)(11,156)(12,157)(13,158)(14,159)(15,160)(21,31)(22,32)(23,33)(24,34)(25,35)(36,46)(37,47)(38,48)(39,49)(40,50)(41,51)(42,52)(43,53)(44,54)(45,55)(56,66)(57,67)(58,68)(59,69)(60,70)(61,71)(62,72)(63,73)(64,74)(65,75)(76,86)(77,87)(78,88)(79,89)(80,90)(81,91)(82,92)(83,93)(84,94)(85,95)(96,106)(97,107)(98,108)(99,109)(100,110)(101,111)(102,112)(103,113)(104,114)(105,115)(116,126)(117,127)(118,128)(119,129)(120,130)(121,131)(122,132)(123,133)(124,134)(125,135)(136,146)(137,147)(138,148)(139,149)(140,150)(141,151)(142,152)(143,153)(144,154)(145,155), (1,21)(2,22)(3,23)(4,24)(5,25)(6,156)(7,157)(8,158)(9,159)(10,160)(11,16)(12,17)(13,18)(14,19)(15,20)(26,31)(27,32)(28,33)(29,34)(30,35)(36,41)(37,42)(38,43)(39,44)(40,45)(46,51)(47,52)(48,53)(49,54)(50,55)(56,61)(57,62)(58,63)(59,64)(60,65)(66,71)(67,72)(68,73)(69,74)(70,75)(76,81)(77,82)(78,83)(79,84)(80,85)(86,91)(87,92)(88,93)(89,94)(90,95)(96,101)(97,102)(98,103)(99,104)(100,105)(106,111)(107,112)(108,113)(109,114)(110,115)(116,121)(117,122)(118,123)(119,124)(120,125)(126,131)(127,132)(128,133)(129,134)(130,135)(136,141)(137,142)(138,143)(139,144)(140,145)(146,151)(147,152)(148,153)(149,154)(150,155), (1,96)(2,97)(3,98)(4,99)(5,100)(6,81)(7,82)(8,83)(9,84)(10,85)(11,86)(12,87)(13,88)(14,89)(15,90)(16,91)(17,92)(18,93)(19,94)(20,95)(21,101)(22,102)(23,103)(24,104)(25,105)(26,106)(27,107)(28,108)(29,109)(30,110)(31,111)(32,112)(33,113)(34,114)(35,115)(36,116)(37,117)(38,118)(39,119)(40,120)(41,121)(42,122)(43,123)(44,124)(45,125)(46,126)(47,127)(48,128)(49,129)(50,130)(51,131)(52,132)(53,133)(54,134)(55,135)(56,136)(57,137)(58,138)(59,139)(60,140)(61,141)(62,142)(63,143)(64,144)(65,145)(66,146)(67,147)(68,148)(69,149)(70,150)(71,151)(72,152)(73,153)(74,154)(75,155)(76,156)(77,157)(78,158)(79,159)(80,160), (1,56)(2,57)(3,58)(4,59)(5,60)(6,131)(7,132)(8,133)(9,134)(10,135)(11,116)(12,117)(13,118)(14,119)(15,120)(16,121)(17,122)(18,123)(19,124)(20,125)(21,61)(22,62)(23,63)(24,64)(25,65)(26,66)(27,67)(28,68)(29,69)(30,70)(31,71)(32,72)(33,73)(34,74)(35,75)(36,76)(37,77)(38,78)(39,79)(40,80)(41,81)(42,82)(43,83)(44,84)(45,85)(46,86)(47,87)(48,88)(49,89)(50,90)(51,91)(52,92)(53,93)(54,94)(55,95)(96,146)(97,147)(98,148)(99,149)(100,150)(101,151)(102,152)(103,153)(104,154)(105,155)(106,136)(107,137)(108,138)(109,139)(110,140)(111,141)(112,142)(113,143)(114,144)(115,145)(126,156)(127,157)(128,158)(129,159)(130,160), (1,46,26,36)(2,47,27,37)(3,48,28,38)(4,49,29,39)(5,50,30,40)(6,151,16,141)(7,152,17,142)(8,153,18,143)(9,154,19,144)(10,155,20,145)(11,136,156,146)(12,137,157,147)(13,138,158,148)(14,139,159,149)(15,140,160,150)(21,51,31,41)(22,52,32,42)(23,53,33,43)(24,54,34,44)(25,55,35,45)(56,81,66,91)(57,82,67,92)(58,83,68,93)(59,84,69,94)(60,85,70,95)(61,76,71,86)(62,77,72,87)(63,78,73,88)(64,79,74,89)(65,80,75,90)(96,131,106,121)(97,132,107,122)(98,133,108,123)(99,134,109,124)(100,135,110,125)(101,126,111,116)(102,127,112,117)(103,128,113,118)(104,129,114,119)(105,130,115,120), (6,156)(7,157)(8,158)(9,159)(10,160)(11,16)(12,17)(13,18)(14,19)(15,20)(56,66)(57,67)(58,68)(59,69)(60,70)(61,71)(62,72)(63,73)(64,74)(65,75)(76,86)(77,87)(78,88)(79,89)(80,90)(81,91)(82,92)(83,93)(84,94)(85,95)(96,111)(97,112)(98,113)(99,114)(100,115)(101,106)(102,107)(103,108)(104,109)(105,110)(116,131)(117,132)(118,133)(119,134)(120,135)(121,126)(122,127)(123,128)(124,129)(125,130)(136,141)(137,142)(138,143)(139,144)(140,145)(146,151)(147,152)(148,153)(149,154)(150,155) );
G=PermutationGroup([(1,2,3,4,5),(6,7,8,9,10),(11,12,13,14,15),(16,17,18,19,20),(21,22,23,24,25),(26,27,28,29,30),(31,32,33,34,35),(36,37,38,39,40),(41,42,43,44,45),(46,47,48,49,50),(51,52,53,54,55),(56,57,58,59,60),(61,62,63,64,65),(66,67,68,69,70),(71,72,73,74,75),(76,77,78,79,80),(81,82,83,84,85),(86,87,88,89,90),(91,92,93,94,95),(96,97,98,99,100),(101,102,103,104,105),(106,107,108,109,110),(111,112,113,114,115),(116,117,118,119,120),(121,122,123,124,125),(126,127,128,129,130),(131,132,133,134,135),(136,137,138,139,140),(141,142,143,144,145),(146,147,148,149,150),(151,152,153,154,155),(156,157,158,159,160)], [(1,26),(2,27),(3,28),(4,29),(5,30),(6,16),(7,17),(8,18),(9,19),(10,20),(11,156),(12,157),(13,158),(14,159),(15,160),(21,31),(22,32),(23,33),(24,34),(25,35),(36,46),(37,47),(38,48),(39,49),(40,50),(41,51),(42,52),(43,53),(44,54),(45,55),(56,66),(57,67),(58,68),(59,69),(60,70),(61,71),(62,72),(63,73),(64,74),(65,75),(76,86),(77,87),(78,88),(79,89),(80,90),(81,91),(82,92),(83,93),(84,94),(85,95),(96,106),(97,107),(98,108),(99,109),(100,110),(101,111),(102,112),(103,113),(104,114),(105,115),(116,126),(117,127),(118,128),(119,129),(120,130),(121,131),(122,132),(123,133),(124,134),(125,135),(136,146),(137,147),(138,148),(139,149),(140,150),(141,151),(142,152),(143,153),(144,154),(145,155)], [(1,21),(2,22),(3,23),(4,24),(5,25),(6,156),(7,157),(8,158),(9,159),(10,160),(11,16),(12,17),(13,18),(14,19),(15,20),(26,31),(27,32),(28,33),(29,34),(30,35),(36,41),(37,42),(38,43),(39,44),(40,45),(46,51),(47,52),(48,53),(49,54),(50,55),(56,61),(57,62),(58,63),(59,64),(60,65),(66,71),(67,72),(68,73),(69,74),(70,75),(76,81),(77,82),(78,83),(79,84),(80,85),(86,91),(87,92),(88,93),(89,94),(90,95),(96,101),(97,102),(98,103),(99,104),(100,105),(106,111),(107,112),(108,113),(109,114),(110,115),(116,121),(117,122),(118,123),(119,124),(120,125),(126,131),(127,132),(128,133),(129,134),(130,135),(136,141),(137,142),(138,143),(139,144),(140,145),(146,151),(147,152),(148,153),(149,154),(150,155)], [(1,96),(2,97),(3,98),(4,99),(5,100),(6,81),(7,82),(8,83),(9,84),(10,85),(11,86),(12,87),(13,88),(14,89),(15,90),(16,91),(17,92),(18,93),(19,94),(20,95),(21,101),(22,102),(23,103),(24,104),(25,105),(26,106),(27,107),(28,108),(29,109),(30,110),(31,111),(32,112),(33,113),(34,114),(35,115),(36,116),(37,117),(38,118),(39,119),(40,120),(41,121),(42,122),(43,123),(44,124),(45,125),(46,126),(47,127),(48,128),(49,129),(50,130),(51,131),(52,132),(53,133),(54,134),(55,135),(56,136),(57,137),(58,138),(59,139),(60,140),(61,141),(62,142),(63,143),(64,144),(65,145),(66,146),(67,147),(68,148),(69,149),(70,150),(71,151),(72,152),(73,153),(74,154),(75,155),(76,156),(77,157),(78,158),(79,159),(80,160)], [(1,56),(2,57),(3,58),(4,59),(5,60),(6,131),(7,132),(8,133),(9,134),(10,135),(11,116),(12,117),(13,118),(14,119),(15,120),(16,121),(17,122),(18,123),(19,124),(20,125),(21,61),(22,62),(23,63),(24,64),(25,65),(26,66),(27,67),(28,68),(29,69),(30,70),(31,71),(32,72),(33,73),(34,74),(35,75),(36,76),(37,77),(38,78),(39,79),(40,80),(41,81),(42,82),(43,83),(44,84),(45,85),(46,86),(47,87),(48,88),(49,89),(50,90),(51,91),(52,92),(53,93),(54,94),(55,95),(96,146),(97,147),(98,148),(99,149),(100,150),(101,151),(102,152),(103,153),(104,154),(105,155),(106,136),(107,137),(108,138),(109,139),(110,140),(111,141),(112,142),(113,143),(114,144),(115,145),(126,156),(127,157),(128,158),(129,159),(130,160)], [(1,46,26,36),(2,47,27,37),(3,48,28,38),(4,49,29,39),(5,50,30,40),(6,151,16,141),(7,152,17,142),(8,153,18,143),(9,154,19,144),(10,155,20,145),(11,136,156,146),(12,137,157,147),(13,138,158,148),(14,139,159,149),(15,140,160,150),(21,51,31,41),(22,52,32,42),(23,53,33,43),(24,54,34,44),(25,55,35,45),(56,81,66,91),(57,82,67,92),(58,83,68,93),(59,84,69,94),(60,85,70,95),(61,76,71,86),(62,77,72,87),(63,78,73,88),(64,79,74,89),(65,80,75,90),(96,131,106,121),(97,132,107,122),(98,133,108,123),(99,134,109,124),(100,135,110,125),(101,126,111,116),(102,127,112,117),(103,128,113,118),(104,129,114,119),(105,130,115,120)], [(6,156),(7,157),(8,158),(9,159),(10,160),(11,16),(12,17),(13,18),(14,19),(15,20),(56,66),(57,67),(58,68),(59,69),(60,70),(61,71),(62,72),(63,73),(64,74),(65,75),(76,86),(77,87),(78,88),(79,89),(80,90),(81,91),(82,92),(83,93),(84,94),(85,95),(96,111),(97,112),(98,113),(99,114),(100,115),(101,106),(102,107),(103,108),(104,109),(105,110),(116,131),(117,132),(118,133),(119,134),(120,135),(121,126),(122,127),(123,128),(124,129),(125,130),(136,141),(137,142),(138,143),(139,144),(140,145),(146,151),(147,152),(148,153),(149,154),(150,155)])
Matrix representation ►G ⊆ GL8(𝔽41)
| 16 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 16 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 16 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 16 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 10 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 10 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 10 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 10 |
| 40 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 40 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 40 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 40 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 40 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 40 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 40 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 40 |
| 40 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 40 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 40 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 40 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 |
| 28 | 23 | 0 | 0 | 0 | 0 | 0 | 0 |
| 23 | 13 | 0 | 0 | 0 | 0 | 0 | 0 |
| 23 | 0 | 13 | 18 | 0 | 0 | 0 | 0 |
| 0 | 23 | 18 | 28 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 20 | 38 | 3 | 21 |
| 0 | 0 | 0 | 0 | 3 | 20 | 21 | 38 |
| 0 | 0 | 0 | 0 | 0 | 40 | 21 | 38 |
| 0 | 0 | 0 | 0 | 40 | 0 | 3 | 21 |
| 0 | 1 | 39 | 0 | 0 | 0 | 0 | 0 |
| 1 | 0 | 0 | 39 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 40 | 0 | 0 | 0 | 0 |
| 0 | 0 | 40 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 | 40 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 | 0 | 40 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 40 |
| 0 | 0 | 0 | 0 | 0 | 0 | 40 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 40 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 40 | 0 | 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 1 | 40 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 40 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 40 |
| 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 40 | 0 | 0 | 0 | 0 | 0 |
| 1 | 0 | 0 | 40 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 2 | 40 | 0 |
| 0 | 0 | 0 | 0 | 2 | 0 | 0 | 40 |
G:=sub<GL(8,GF(41))| [16,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,10,0,0,0,0,0,0,0,0,10,0,0,0,0,0,0,0,0,10,0,0,0,0,0,0,0,0,10],[40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40],[40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[28,23,23,0,0,0,0,0,23,13,0,23,0,0,0,0,0,0,13,18,0,0,0,0,0,0,18,28,0,0,0,0,0,0,0,0,20,3,0,40,0,0,0,0,38,20,40,0,0,0,0,0,3,21,21,3,0,0,0,0,21,38,38,21],[0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,39,0,0,40,0,0,0,0,0,39,40,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,40,0,0,40,0,0,0,0,0,40,40,0],[0,40,40,0,0,0,0,0,1,0,0,1,0,0,0,0,0,0,0,40,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,40,0],[1,0,0,1,0,0,0,0,0,1,1,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,1,0,0,2,0,0,0,0,0,1,2,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40] >;
95 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 4A | ··· | 4K | 5A | 5B | 5C | 5D | 10A | ··· | 10L | 10M | ··· | 10AB | 20A | ··· | 20AR |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 5 | 5 | 5 | 5 | 10 | ··· | 10 | 10 | ··· | 10 | 20 | ··· | 20 |
| size | 1 | 1 | 1 | 1 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 1 | 1 | 1 | 1 | 1 | ··· | 1 | 4 | ··· | 4 | 4 | ··· | 4 |
95 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | - | ||||||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C5 | C10 | C10 | C10 | C10 | C10 | 2+ (1+4) | 2- (1+4) | C5×2+ (1+4) | C5×2- (1+4) |
| kernel | C5×C22.56C24 | C5×C4⋊D4 | C5×C22⋊Q8 | C5×C22.D4 | C5×C4.4D4 | C5×C42.C2 | C22.56C24 | C4⋊D4 | C22⋊Q8 | C22.D4 | C4.4D4 | C42.C2 | C10 | C10 | C2 | C2 |
| # reps | 1 | 4 | 4 | 4 | 2 | 1 | 4 | 16 | 16 | 16 | 8 | 4 | 2 | 1 | 8 | 4 |
In GAP, Magma, Sage, TeX
C_5\times C_2^2._{56}C_2^4 % in TeX
G:=Group("C5xC2^2.56C2^4"); // GroupNames label
G:=SmallGroup(320,1564);
// by ID
G=gap.SmallGroup(320,1564);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-5,-2,-2,1149,3446,2571,436,6947,1242]);
// Polycyclic
G:=Group<a,b,c,d,e,f,g|a^5=b^2=c^2=d^2=e^2=g^2=1,f^2=b,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,a*f=f*a,a*g=g*a,b*c=c*b,e*d*e=b*d=d*b,g*e*g=b*e=e*b,b*f=f*b,b*g=g*b,f*d*f^-1=c*d=d*c,c*e=e*c,c*f=f*c,c*g=g*c,g*d*g=b*c*d,f*e*f^-1=b*c*e,f*g=g*f>;
// generators/relations
×
⋊
ℤ
𝔽
○
≀