Copied to
clipboard

G = C5×C22.56C24  order 320 = 26·5

Direct product of C5 and C22.56C24

direct product, metabelian, nilpotent (class 2), monomial, 2-elementary

Aliases: C5×C22.56C24, C10.1712+ 1+4, C10.1232- 1+4, C4⋊D4⋊18C10, C22⋊Q8⋊19C10, C4.4D4⋊16C10, C42.54(C2×C10), C42.C2⋊11C10, (C4×C20).295C22, (C2×C20).683C23, (C2×C10).382C24, (D4×C10).224C22, C22.D4⋊14C10, C22.56(C23×C10), C23.25(C22×C10), (Q8×C10).187C22, C2.23(C5×2+ 1+4), C2.15(C5×2- 1+4), (C22×C20).462C22, (C22×C10).108C23, (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

Derived series
C1 — C22 — C5×C22.56C24
Chief series
C1 — C2 — C22 — C2×C10 — C22×C10 — D4×C10 — C5×C4.4D4 — C5×C22.56C24
Lower central
C1 — C22 — C5×C22.56C24
Upper central
C1 — C2×C10 — C5×C22.56C24

Generators and relations for C5×C22.56C24
 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 >

Subgroups: 362 in 220 conjugacy classes, 142 normal (14 characteristic)
C1, C2, C2, C2, C4, C22, C22, C5, C2×C4, C2×C4, C2×C4, D4, Q8, C23, C10, C10, C10, C42, C22⋊C4, C4⋊C4, C22×C4, C2×D4, C2×Q8, C20, C2×C10, C2×C10, C4⋊D4, C22⋊Q8, C22.D4, C4.4D4, C42.C2, C2×C20, C2×C20, C2×C20, C5×D4, C5×Q8, C22×C10, C22.56C24, C4×C20, C5×C22⋊C4, C5×C4⋊C4, C22×C20, D4×C10, Q8×C10, C5×C4⋊D4, C5×C22⋊Q8, C5×C22.D4, C5×C4.4D4, C5×C42.C2, C5×C22.56C24
Quotients: C1, C2, C22, C5, C23, C10, C24, C2×C10, 2+ 1+4, 2- 1+4, C22×C10, C22.56C24, C23×C10, C5×2+ 1+4, C5×2- 1+4, C5×C22.56C24

Smallest permutation representation of C5×C22.56C24
►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 35)(2 31)(3 32)(4 33)(5 34)(6 160)(7 156)(8 157)(9 158)(10 159)(11 17)(12 18)(13 19)(14 20)(15 16)(21 27)(22 28)(23 29)(24 30)(25 26)(36 55)(37 51)(38 52)(39 53)(40 54)(41 47)(42 48)(43 49)(44 50)(45 46)(56 75)(57 71)(58 72)(59 73)(60 74)(61 67)(62 68)(63 69)(64 70)(65 66)(76 95)(77 91)(78 92)(79 93)(80 94)(81 87)(82 88)(83 89)(84 90)(85 86)(96 115)(97 111)(98 112)(99 113)(100 114)(101 107)(102 108)(103 109)(104 110)(105 106)(116 135)(117 131)(118 132)(119 133)(120 134)(121 127)(122 128)(123 129)(124 130)(125 126)(136 155)(137 151)(138 152)(139 153)(140 154)(141 147)(142 148)(143 149)(144 150)(145 146)

(1 21)(2 22)(3 23)(4 24)(5 25)(6 20)(7 16)(8 17)(9 18)(10 19)(11 157)(12 158)(13 159)(14 160)(15 156)(26 34)(27 35)(28 31)(29 32)(30 33)(36 41)(37 42)(38 43)(39 44)(40 45)(46 54)(47 55)(48 51)(49 52)(50 53)(56 61)(57 62)(58 63)(59 64)(60 65)(66 74)(67 75)(68 71)(69 72)(70 73)(76 81)(77 82)(78 83)(79 84)(80 85)(86 94)(87 95)(88 91)(89 92)(90 93)(96 101)(97 102)(98 103)(99 104)(100 105)(106 114)(107 115)(108 111)(109 112)(110 113)(116 121)(117 122)(118 123)(119 124)(120 125)(126 134)(127 135)(128 131)(129 132)(130 133)(136 141)(137 142)(138 143)(139 144)(140 145)(146 154)(147 155)(148 151)(149 152)(150 153)

(1 96)(2 97)(3 98)(4 99)(5 100)(6 94)(7 95)(8 91)(9 92)(10 93)(11 82)(12 83)(13 84)(14 85)(15 81)(16 87)(17 88)(18 89)(19 90)(20 86)(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)(6 120)(7 116)(8 117)(9 118)(10 119)(11 128)(12 129)(13 130)(14 126)(15 127)(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)(96 155)(97 151)(98 152)(99 153)(100 154)(101 147)(102 148)(103 149)(104 150)(105 146)(106 145)(107 141)(108 142)(109 143)(110 144)(111 137)(112 138)(113 139)(114 140)(115 136)(131 157)(132 158)(133 159)(134 160)(135 156)

(1 55 35 36)(2 51 31 37)(3 52 32 38)(4 53 33 39)(5 54 34 40)(6 140 160 154)(7 136 156 155)(8 137 157 151)(9 138 158 152)(10 139 159 153)(11 148 17 142)(12 149 18 143)(13 150 19 144)(14 146 20 145)(15 147 16 141)(21 47 27 41)(22 48 28 42)(23 49 29 43)(24 50 30 44)(25 46 26 45)(56 81 75 87)(57 82 71 88)(58 83 72 89)(59 84 73 90)(60 85 74 86)(61 76 67 95)(62 77 68 91)(63 78 69 92)(64 79 70 93)(65 80 66 94)(96 127 115 121)(97 128 111 122)(98 129 112 123)(99 130 113 124)(100 126 114 125)(101 135 107 116)(102 131 108 117)(103 132 109 118)(104 133 110 119)(105 134 106 120)

(6 20)(7 16)(8 17)(9 18)(10 19)(11 157)(12 158)(13 159)(14 160)(15 156)(56 75)(57 71)(58 72)(59 73)(60 74)(61 67)(62 68)(63 69)(64 70)(65 66)(76 95)(77 91)(78 92)(79 93)(80 94)(81 87)(82 88)(83 89)(84 90)(85 86)(96 107)(97 108)(98 109)(99 110)(100 106)(101 115)(102 111)(103 112)(104 113)(105 114)(116 127)(117 128)(118 129)(119 130)(120 126)(121 135)(122 131)(123 132)(124 133)(125 134)(136 141)(137 142)(138 143)(139 144)(140 145)(146 154)(147 155)(148 151)(149 152)(150 153)
 
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), (1,35)(2,31)(3,32)(4,33)(5,34)(6,160)(7,156)(8,157)(9,158)(10,159)(11,17)(12,18)(13,19)(14,20)(15,16)(21,27)(22,28)(23,29)(24,30)(25,26)(36,55)(37,51)(38,52)(39,53)(40,54)(41,47)(42,48)(43,49)(44,50)(45,46)(56,75)(57,71)(58,72)(59,73)(60,74)(61,67)(62,68)(63,69)(64,70)(65,66)(76,95)(77,91)(78,92)(79,93)(80,94)(81,87)(82,88)(83,89)(84,90)(85,86)(96,115)(97,111)(98,112)(99,113)(100,114)(101,107)(102,108)(103,109)(104,110)(105,106)(116,135)(117,131)(118,132)(119,133)(120,134)(121,127)(122,128)(123,129)(124,130)(125,126)(136,155)(137,151)(138,152)(139,153)(140,154)(141,147)(142,148)(143,149)(144,150)(145,146), (1,21)(2,22)(3,23)(4,24)(5,25)(6,20)(7,16)(8,17)(9,18)(10,19)(11,157)(12,158)(13,159)(14,160)(15,156)(26,34)(27,35)(28,31)(29,32)(30,33)(36,41)(37,42)(38,43)(39,44)(40,45)(46,54)(47,55)(48,51)(49,52)(50,53)(56,61)(57,62)(58,63)(59,64)(60,65)(66,74)(67,75)(68,71)(69,72)(70,73)(76,81)(77,82)(78,83)(79,84)(80,85)(86,94)(87,95)(88,91)(89,92)(90,93)(96,101)(97,102)(98,103)(99,104)(100,105)(106,114)(107,115)(108,111)(109,112)(110,113)(116,121)(117,122)(118,123)(119,124)(120,125)(126,134)(127,135)(128,131)(129,132)(130,133)(136,141)(137,142)(138,143)(139,144)(140,145)(146,154)(147,155)(148,151)(149,152)(150,153), (1,96)(2,97)(3,98)(4,99)(5,100)(6,94)(7,95)(8,91)(9,92)(10,93)(11,82)(12,83)(13,84)(14,85)(15,81)(16,87)(17,88)(18,89)(19,90)(20,86)(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)(6,120)(7,116)(8,117)(9,118)(10,119)(11,128)(12,129)(13,130)(14,126)(15,127)(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)(96,155)(97,151)(98,152)(99,153)(100,154)(101,147)(102,148)(103,149)(104,150)(105,146)(106,145)(107,141)(108,142)(109,143)(110,144)(111,137)(112,138)(113,139)(114,140)(115,136)(131,157)(132,158)(133,159)(134,160)(135,156), (1,55,35,36)(2,51,31,37)(3,52,32,38)(4,53,33,39)(5,54,34,40)(6,140,160,154)(7,136,156,155)(8,137,157,151)(9,138,158,152)(10,139,159,153)(11,148,17,142)(12,149,18,143)(13,150,19,144)(14,146,20,145)(15,147,16,141)(21,47,27,41)(22,48,28,42)(23,49,29,43)(24,50,30,44)(25,46,26,45)(56,81,75,87)(57,82,71,88)(58,83,72,89)(59,84,73,90)(60,85,74,86)(61,76,67,95)(62,77,68,91)(63,78,69,92)(64,79,70,93)(65,80,66,94)(96,127,115,121)(97,128,111,122)(98,129,112,123)(99,130,113,124)(100,126,114,125)(101,135,107,116)(102,131,108,117)(103,132,109,118)(104,133,110,119)(105,134,106,120), (6,20)(7,16)(8,17)(9,18)(10,19)(11,157)(12,158)(13,159)(14,160)(15,156)(56,75)(57,71)(58,72)(59,73)(60,74)(61,67)(62,68)(63,69)(64,70)(65,66)(76,95)(77,91)(78,92)(79,93)(80,94)(81,87)(82,88)(83,89)(84,90)(85,86)(96,107)(97,108)(98,109)(99,110)(100,106)(101,115)(102,111)(103,112)(104,113)(105,114)(116,127)(117,128)(118,129)(119,130)(120,126)(121,135)(122,131)(123,132)(124,133)(125,134)(136,141)(137,142)(138,143)(139,144)(140,145)(146,154)(147,155)(148,151)(149,152)(150,153)>;
 
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), (1,35)(2,31)(3,32)(4,33)(5,34)(6,160)(7,156)(8,157)(9,158)(10,159)(11,17)(12,18)(13,19)(14,20)(15,16)(21,27)(22,28)(23,29)(24,30)(25,26)(36,55)(37,51)(38,52)(39,53)(40,54)(41,47)(42,48)(43,49)(44,50)(45,46)(56,75)(57,71)(58,72)(59,73)(60,74)(61,67)(62,68)(63,69)(64,70)(65,66)(76,95)(77,91)(78,92)(79,93)(80,94)(81,87)(82,88)(83,89)(84,90)(85,86)(96,115)(97,111)(98,112)(99,113)(100,114)(101,107)(102,108)(103,109)(104,110)(105,106)(116,135)(117,131)(118,132)(119,133)(120,134)(121,127)(122,128)(123,129)(124,130)(125,126)(136,155)(137,151)(138,152)(139,153)(140,154)(141,147)(142,148)(143,149)(144,150)(145,146), (1,21)(2,22)(3,23)(4,24)(5,25)(6,20)(7,16)(8,17)(9,18)(10,19)(11,157)(12,158)(13,159)(14,160)(15,156)(26,34)(27,35)(28,31)(29,32)(30,33)(36,41)(37,42)(38,43)(39,44)(40,45)(46,54)(47,55)(48,51)(49,52)(50,53)(56,61)(57,62)(58,63)(59,64)(60,65)(66,74)(67,75)(68,71)(69,72)(70,73)(76,81)(77,82)(78,83)(79,84)(80,85)(86,94)(87,95)(88,91)(89,92)(90,93)(96,101)(97,102)(98,103)(99,104)(100,105)(106,114)(107,115)(108,111)(109,112)(110,113)(116,121)(117,122)(118,123)(119,124)(120,125)(126,134)(127,135)(128,131)(129,132)(130,133)(136,141)(137,142)(138,143)(139,144)(140,145)(146,154)(147,155)(148,151)(149,152)(150,153), (1,96)(2,97)(3,98)(4,99)(5,100)(6,94)(7,95)(8,91)(9,92)(10,93)(11,82)(12,83)(13,84)(14,85)(15,81)(16,87)(17,88)(18,89)(19,90)(20,86)(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)(6,120)(7,116)(8,117)(9,118)(10,119)(11,128)(12,129)(13,130)(14,126)(15,127)(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)(96,155)(97,151)(98,152)(99,153)(100,154)(101,147)(102,148)(103,149)(104,150)(105,146)(106,145)(107,141)(108,142)(109,143)(110,144)(111,137)(112,138)(113,139)(114,140)(115,136)(131,157)(132,158)(133,159)(134,160)(135,156), (1,55,35,36)(2,51,31,37)(3,52,32,38)(4,53,33,39)(5,54,34,40)(6,140,160,154)(7,136,156,155)(8,137,157,151)(9,138,158,152)(10,139,159,153)(11,148,17,142)(12,149,18,143)(13,150,19,144)(14,146,20,145)(15,147,16,141)(21,47,27,41)(22,48,28,42)(23,49,29,43)(24,50,30,44)(25,46,26,45)(56,81,75,87)(57,82,71,88)(58,83,72,89)(59,84,73,90)(60,85,74,86)(61,76,67,95)(62,77,68,91)(63,78,69,92)(64,79,70,93)(65,80,66,94)(96,127,115,121)(97,128,111,122)(98,129,112,123)(99,130,113,124)(100,126,114,125)(101,135,107,116)(102,131,108,117)(103,132,109,118)(104,133,110,119)(105,134,106,120), (6,20)(7,16)(8,17)(9,18)(10,19)(11,157)(12,158)(13,159)(14,160)(15,156)(56,75)(57,71)(58,72)(59,73)(60,74)(61,67)(62,68)(63,69)(64,70)(65,66)(76,95)(77,91)(78,92)(79,93)(80,94)(81,87)(82,88)(83,89)(84,90)(85,86)(96,107)(97,108)(98,109)(99,110)(100,106)(101,115)(102,111)(103,112)(104,113)(105,114)(116,127)(117,128)(118,129)(119,130)(120,126)(121,135)(122,131)(123,132)(124,133)(125,134)(136,141)(137,142)(138,143)(139,144)(140,145)(146,154)(147,155)(148,151)(149,152)(150,153) );
 
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)], [(1,35),(2,31),(3,32),(4,33),(5,34),(6,160),(7,156),(8,157),(9,158),(10,159),(11,17),(12,18),(13,19),(14,20),(15,16),(21,27),(22,28),(23,29),(24,30),(25,26),(36,55),(37,51),(38,52),(39,53),(40,54),(41,47),(42,48),(43,49),(44,50),(45,46),(56,75),(57,71),(58,72),(59,73),(60,74),(61,67),(62,68),(63,69),(64,70),(65,66),(76,95),(77,91),(78,92),(79,93),(80,94),(81,87),(82,88),(83,89),(84,90),(85,86),(96,115),(97,111),(98,112),(99,113),(100,114),(101,107),(102,108),(103,109),(104,110),(105,106),(116,135),(117,131),(118,132),(119,133),(120,134),(121,127),(122,128),(123,129),(124,130),(125,126),(136,155),(137,151),(138,152),(139,153),(140,154),(141,147),(142,148),(143,149),(144,150),(145,146)], [(1,21),(2,22),(3,23),(4,24),(5,25),(6,20),(7,16),(8,17),(9,18),(10,19),(11,157),(12,158),(13,159),(14,160),(15,156),(26,34),(27,35),(28,31),(29,32),(30,33),(36,41),(37,42),(38,43),(39,44),(40,45),(46,54),(47,55),(48,51),(49,52),(50,53),(56,61),(57,62),(58,63),(59,64),(60,65),(66,74),(67,75),(68,71),(69,72),(70,73),(76,81),(77,82),(78,83),(79,84),(80,85),(86,94),(87,95),(88,91),(89,92),(90,93),(96,101),(97,102),(98,103),(99,104),(100,105),(106,114),(107,115),(108,111),(109,112),(110,113),(116,121),(117,122),(118,123),(119,124),(120,125),(126,134),(127,135),(128,131),(129,132),(130,133),(136,141),(137,142),(138,143),(139,144),(140,145),(146,154),(147,155),(148,151),(149,152),(150,153)], [(1,96),(2,97),(3,98),(4,99),(5,100),(6,94),(7,95),(8,91),(9,92),(10,93),(11,82),(12,83),(13,84),(14,85),(15,81),(16,87),(17,88),(18,89),(19,90),(20,86),(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),(6,120),(7,116),(8,117),(9,118),(10,119),(11,128),(12,129),(13,130),(14,126),(15,127),(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),(96,155),(97,151),(98,152),(99,153),(100,154),(101,147),(102,148),(103,149),(104,150),(105,146),(106,145),(107,141),(108,142),(109,143),(110,144),(111,137),(112,138),(113,139),(114,140),(115,136),(131,157),(132,158),(133,159),(134,160),(135,156)], [(1,55,35,36),(2,51,31,37),(3,52,32,38),(4,53,33,39),(5,54,34,40),(6,140,160,154),(7,136,156,155),(8,137,157,151),(9,138,158,152),(10,139,159,153),(11,148,17,142),(12,149,18,143),(13,150,19,144),(14,146,20,145),(15,147,16,141),(21,47,27,41),(22,48,28,42),(23,49,29,43),(24,50,30,44),(25,46,26,45),(56,81,75,87),(57,82,71,88),(58,83,72,89),(59,84,73,90),(60,85,74,86),(61,76,67,95),(62,77,68,91),(63,78,69,92),(64,79,70,93),(65,80,66,94),(96,127,115,121),(97,128,111,122),(98,129,112,123),(99,130,113,124),(100,126,114,125),(101,135,107,116),(102,131,108,117),(103,132,109,118),(104,133,110,119),(105,134,106,120)], [(6,20),(7,16),(8,17),(9,18),(10,19),(11,157),(12,158),(13,159),(14,160),(15,156),(56,75),(57,71),(58,72),(59,73),(60,74),(61,67),(62,68),(63,69),(64,70),(65,66),(76,95),(77,91),(78,92),(79,93),(80,94),(81,87),(82,88),(83,89),(84,90),(85,86),(96,107),(97,108),(98,109),(99,110),(100,106),(101,115),(102,111),(103,112),(104,113),(105,114),(116,127),(117,128),(118,129),(119,130),(120,126),(121,135),(122,131),(123,132),(124,133),(125,134),(136,141),(137,142),(138,143),(139,144),(140,145),(146,154),(147,155),(148,151),(149,152),(150,153)]])
 

95 conjugacy classes

class 1 2A2B2C2D2E2F2G4A···4K5A5B5C5D10A···10L10M···10AB20A···20AR
order122222224···4555510···1010···1020···20
size111144444···411111···14···44···4

95 irreducible representations

dim1111111111114444
type+++++++-
imageC1C2C2C2C2C2C5C10C10C10C10C102+ 1+42- 1+4C5×2+ 1+4C5×2- 1+4
kernelC5×C22.56C24C5×C4⋊D4C5×C22⋊Q8C5×C22.D4C5×C4.4D4C5×C42.C2C22.56C24C4⋊D4C22⋊Q8C22.D4C4.4D4C42.C2C10C10C2C2
# reps1444214161616842184

Matrix representation of C5×C22.56C24 ►in GL8(𝔽41)

160000000
016000000
001600000
000160000
000010000
000001000
000000100
000000010
,
400000000
040000000
004000000
000400000
000040000
000004000
000000400
000000040
,
400000000
040000000
004000000
000400000
00001000
00000100
00000010
00000001
,
2823000000
2313000000
23013180000
02318280000
00002038321
00003202138
00000402138
0000400321
,
013900000
100390000
000400000
004000000
000001400
000010040
000000040
000000400
,
01000000
400000000
400010000
014000000
00000100
000040000
000000040
00000010
,
10000000
01000000
014000000
100400000
00001000
00000100
000002400
000020040

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] >;
 

C5×C22.56C24 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
○
≀
ℚ
◁