direct product, metabelian, nilpotent (class 2), monomial, 2-elementary
Aliases: C5×C22.35C24, C10.1142- 1+4, C4⋊Q8⋊10C10, (C4×Q8)⋊9C10, (Q8×C20)⋊29C2, C42⋊2C2.C10, C42.C2⋊6C10, C22⋊Q8.8C10, C42.39(C2×C10), C20.277(C4○D4), (C2×C10).361C24, (C2×C20).670C23, (C4×C20).280C22, C42⋊C2.12C10, C2.6(C5×2- 1+4), (C22×C10).96C23, C23.13(C22×C10), C22.35(C23×C10), (Q8×C10).272C22, (C22×C20).449C22, (C5×C4⋊Q8)⋊31C2, C4.21(C5×C4○D4), C4⋊C4.69(C2×C10), C2.18(C10×C4○D4), C10.237(C2×C4○D4), C22⋊C4.3(C2×C10), (C2×Q8).59(C2×C10), (C5×C42.C2)⋊23C2, (C5×C22⋊Q8).18C2, (C5×C4⋊C4).248C22, (C22×C4).61(C2×C10), (C2×C4).28(C22×C10), (C5×C42⋊2C2).2C2, (C5×C42⋊C2).26C2, (C5×C22⋊C4).149C22, SmallGroup(320,1543)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C5×C22.35C24
G = < a,b,c,d,e,f,g | a5=b2=c2=f2=1, d2=g2=b, e2=c, ab=ba, ac=ca, ad=da, ae=ea, af=fa, ag=ga, bc=cb, ede-1=gdg-1=bd=db, fef=be=eb, bf=fb, bg=gb, fdf=cd=dc, ce=ec, cf=fc, cg=gc, eg=ge, fg=gf >
Subgroups: 242 in 192 conjugacy classes, 146 normal (26 characteristic)
C1, C2, C2, C2, C4, C4, C22, C22, C5, C2×C4, C2×C4, C2×C4, Q8, C23, C10, C10, C10, C42, C42, C22⋊C4, C4⋊C4, C22×C4, C2×Q8, C20, C20, C2×C10, C2×C10, C42⋊C2, C4×Q8, C22⋊Q8, C42.C2, C42.C2, C42⋊2C2, C4⋊Q8, C2×C20, C2×C20, C2×C20, C5×Q8, C22×C10, C22.35C24, C4×C20, C4×C20, C5×C22⋊C4, C5×C4⋊C4, C22×C20, Q8×C10, C5×C42⋊C2, Q8×C20, C5×C22⋊Q8, C5×C42.C2, C5×C42.C2, C5×C42⋊2C2, C5×C4⋊Q8, C5×C22.35C24
Quotients: C1, C2, C22, C5, C23, C10, C4○D4, C24, C2×C10, C2×C4○D4, 2- 1+4, C22×C10, C22.35C24, C5×C4○D4, C23×C10, C10×C4○D4, C5×2- 1+4, C5×C22.35C24
►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)
(1 106 26 96)(2 107 27 97)(3 108 28 98)(4 109 29 99)(5 110 30 100)(6 81 16 91)(7 82 17 92)(8 83 18 93)(9 84 19 94)(10 85 20 95)(11 86 156 76)(12 87 157 77)(13 88 158 78)(14 89 159 79)(15 90 160 80)(21 111 31 101)(22 112 32 102)(23 113 33 103)(24 114 34 104)(25 115 35 105)(36 126 46 116)(37 127 47 117)(38 128 48 118)(39 129 49 119)(40 130 50 120)(41 131 51 121)(42 132 52 122)(43 133 53 123)(44 134 54 124)(45 135 55 125)(56 146 66 136)(57 147 67 137)(58 148 68 138)(59 149 69 139)(60 150 70 140)(61 151 71 141)(62 152 72 142)(63 153 73 143)(64 154 74 144)(65 155 75 145)
(1 61 21 56)(2 62 22 57)(3 63 23 58)(4 64 24 59)(5 65 25 60)(6 131 156 126)(7 132 157 127)(8 133 158 128)(9 134 159 129)(10 135 160 130)(11 116 16 121)(12 117 17 122)(13 118 18 123)(14 119 19 124)(15 120 20 125)(26 71 31 66)(27 72 32 67)(28 73 33 68)(29 74 34 69)(30 75 35 70)(36 81 41 76)(37 82 42 77)(38 83 43 78)(39 84 44 79)(40 85 45 80)(46 91 51 86)(47 92 52 87)(48 93 53 88)(49 94 54 89)(50 95 55 90)(96 151 101 146)(97 152 102 147)(98 153 103 148)(99 154 104 149)(100 155 105 150)(106 141 111 136)(107 142 112 137)(108 143 113 138)(109 144 114 139)(110 145 115 140)
(6 11)(7 12)(8 13)(9 14)(10 15)(16 156)(17 157)(18 158)(19 159)(20 160)(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)(96 101)(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)(136 151)(137 152)(138 153)(139 154)(140 155)(141 146)(142 147)(143 148)(144 149)(145 150)
(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 86 66 76)(57 87 67 77)(58 88 68 78)(59 89 69 79)(60 90 70 80)(61 91 71 81)(62 92 72 82)(63 93 73 83)(64 94 74 84)(65 95 75 85)(96 116 106 126)(97 117 107 127)(98 118 108 128)(99 119 109 129)(100 120 110 130)(101 121 111 131)(102 122 112 132)(103 123 113 133)(104 124 114 134)(105 125 115 135)
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), (1,106,26,96)(2,107,27,97)(3,108,28,98)(4,109,29,99)(5,110,30,100)(6,81,16,91)(7,82,17,92)(8,83,18,93)(9,84,19,94)(10,85,20,95)(11,86,156,76)(12,87,157,77)(13,88,158,78)(14,89,159,79)(15,90,160,80)(21,111,31,101)(22,112,32,102)(23,113,33,103)(24,114,34,104)(25,115,35,105)(36,126,46,116)(37,127,47,117)(38,128,48,118)(39,129,49,119)(40,130,50,120)(41,131,51,121)(42,132,52,122)(43,133,53,123)(44,134,54,124)(45,135,55,125)(56,146,66,136)(57,147,67,137)(58,148,68,138)(59,149,69,139)(60,150,70,140)(61,151,71,141)(62,152,72,142)(63,153,73,143)(64,154,74,144)(65,155,75,145), (1,61,21,56)(2,62,22,57)(3,63,23,58)(4,64,24,59)(5,65,25,60)(6,131,156,126)(7,132,157,127)(8,133,158,128)(9,134,159,129)(10,135,160,130)(11,116,16,121)(12,117,17,122)(13,118,18,123)(14,119,19,124)(15,120,20,125)(26,71,31,66)(27,72,32,67)(28,73,33,68)(29,74,34,69)(30,75,35,70)(36,81,41,76)(37,82,42,77)(38,83,43,78)(39,84,44,79)(40,85,45,80)(46,91,51,86)(47,92,52,87)(48,93,53,88)(49,94,54,89)(50,95,55,90)(96,151,101,146)(97,152,102,147)(98,153,103,148)(99,154,104,149)(100,155,105,150)(106,141,111,136)(107,142,112,137)(108,143,113,138)(109,144,114,139)(110,145,115,140), (6,11)(7,12)(8,13)(9,14)(10,15)(16,156)(17,157)(18,158)(19,159)(20,160)(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)(96,101)(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)(136,151)(137,152)(138,153)(139,154)(140,155)(141,146)(142,147)(143,148)(144,149)(145,150), (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)(56,86,66,76)(57,87,67,77)(58,88,68,78)(59,89,69,79)(60,90,70,80)(61,91,71,81)(62,92,72,82)(63,93,73,83)(64,94,74,84)(65,95,75,85)(96,116,106,126)(97,117,107,127)(98,118,108,128)(99,119,109,129)(100,120,110,130)(101,121,111,131)(102,122,112,132)(103,123,113,133)(104,124,114,134)(105,125,115,135)>;
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), (1,106,26,96)(2,107,27,97)(3,108,28,98)(4,109,29,99)(5,110,30,100)(6,81,16,91)(7,82,17,92)(8,83,18,93)(9,84,19,94)(10,85,20,95)(11,86,156,76)(12,87,157,77)(13,88,158,78)(14,89,159,79)(15,90,160,80)(21,111,31,101)(22,112,32,102)(23,113,33,103)(24,114,34,104)(25,115,35,105)(36,126,46,116)(37,127,47,117)(38,128,48,118)(39,129,49,119)(40,130,50,120)(41,131,51,121)(42,132,52,122)(43,133,53,123)(44,134,54,124)(45,135,55,125)(56,146,66,136)(57,147,67,137)(58,148,68,138)(59,149,69,139)(60,150,70,140)(61,151,71,141)(62,152,72,142)(63,153,73,143)(64,154,74,144)(65,155,75,145), (1,61,21,56)(2,62,22,57)(3,63,23,58)(4,64,24,59)(5,65,25,60)(6,131,156,126)(7,132,157,127)(8,133,158,128)(9,134,159,129)(10,135,160,130)(11,116,16,121)(12,117,17,122)(13,118,18,123)(14,119,19,124)(15,120,20,125)(26,71,31,66)(27,72,32,67)(28,73,33,68)(29,74,34,69)(30,75,35,70)(36,81,41,76)(37,82,42,77)(38,83,43,78)(39,84,44,79)(40,85,45,80)(46,91,51,86)(47,92,52,87)(48,93,53,88)(49,94,54,89)(50,95,55,90)(96,151,101,146)(97,152,102,147)(98,153,103,148)(99,154,104,149)(100,155,105,150)(106,141,111,136)(107,142,112,137)(108,143,113,138)(109,144,114,139)(110,145,115,140), (6,11)(7,12)(8,13)(9,14)(10,15)(16,156)(17,157)(18,158)(19,159)(20,160)(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)(96,101)(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)(136,151)(137,152)(138,153)(139,154)(140,155)(141,146)(142,147)(143,148)(144,149)(145,150), (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)(56,86,66,76)(57,87,67,77)(58,88,68,78)(59,89,69,79)(60,90,70,80)(61,91,71,81)(62,92,72,82)(63,93,73,83)(64,94,74,84)(65,95,75,85)(96,116,106,126)(97,117,107,127)(98,118,108,128)(99,119,109,129)(100,120,110,130)(101,121,111,131)(102,122,112,132)(103,123,113,133)(104,124,114,134)(105,125,115,135) );
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)], [(1,106,26,96),(2,107,27,97),(3,108,28,98),(4,109,29,99),(5,110,30,100),(6,81,16,91),(7,82,17,92),(8,83,18,93),(9,84,19,94),(10,85,20,95),(11,86,156,76),(12,87,157,77),(13,88,158,78),(14,89,159,79),(15,90,160,80),(21,111,31,101),(22,112,32,102),(23,113,33,103),(24,114,34,104),(25,115,35,105),(36,126,46,116),(37,127,47,117),(38,128,48,118),(39,129,49,119),(40,130,50,120),(41,131,51,121),(42,132,52,122),(43,133,53,123),(44,134,54,124),(45,135,55,125),(56,146,66,136),(57,147,67,137),(58,148,68,138),(59,149,69,139),(60,150,70,140),(61,151,71,141),(62,152,72,142),(63,153,73,143),(64,154,74,144),(65,155,75,145)], [(1,61,21,56),(2,62,22,57),(3,63,23,58),(4,64,24,59),(5,65,25,60),(6,131,156,126),(7,132,157,127),(8,133,158,128),(9,134,159,129),(10,135,160,130),(11,116,16,121),(12,117,17,122),(13,118,18,123),(14,119,19,124),(15,120,20,125),(26,71,31,66),(27,72,32,67),(28,73,33,68),(29,74,34,69),(30,75,35,70),(36,81,41,76),(37,82,42,77),(38,83,43,78),(39,84,44,79),(40,85,45,80),(46,91,51,86),(47,92,52,87),(48,93,53,88),(49,94,54,89),(50,95,55,90),(96,151,101,146),(97,152,102,147),(98,153,103,148),(99,154,104,149),(100,155,105,150),(106,141,111,136),(107,142,112,137),(108,143,113,138),(109,144,114,139),(110,145,115,140)], [(6,11),(7,12),(8,13),(9,14),(10,15),(16,156),(17,157),(18,158),(19,159),(20,160),(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),(96,101),(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),(136,151),(137,152),(138,153),(139,154),(140,155),(141,146),(142,147),(143,148),(144,149),(145,150)], [(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),(56,86,66,76),(57,87,67,77),(58,88,68,78),(59,89,69,79),(60,90,70,80),(61,91,71,81),(62,92,72,82),(63,93,73,83),(64,94,74,84),(65,95,75,85),(96,116,106,126),(97,117,107,127),(98,118,108,128),(99,119,109,129),(100,120,110,130),(101,121,111,131),(102,122,112,132),(103,123,113,133),(104,124,114,134),(105,125,115,135)]])
110 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 4A | ··· | 4F | 4G | ··· | 4Q | 5A | 5B | 5C | 5D | 10A | ··· | 10L | 10M | 10N | 10O | 10P | 20A | ··· | 20X | 20Y | ··· | 20BP |
| order | 1 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 4 | ··· | 4 | 5 | 5 | 5 | 5 | 10 | ··· | 10 | 10 | 10 | 10 | 10 | 20 | ··· | 20 | 20 | ··· | 20 |
| size | 1 | 1 | 1 | 1 | 4 | 2 | ··· | 2 | 4 | ··· | 4 | 1 | 1 | 1 | 1 | 1 | ··· | 1 | 4 | 4 | 4 | 4 | 2 | ··· | 2 | 4 | ··· | 4 |
110 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | - | ||||||||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C5 | C10 | C10 | C10 | C10 | C10 | C10 | C4○D4 | C5×C4○D4 | 2- 1+4 | C5×2- 1+4 |
| kernel | C5×C22.35C24 | C5×C42⋊C2 | Q8×C20 | C5×C22⋊Q8 | C5×C42.C2 | C5×C42⋊2C2 | C5×C4⋊Q8 | C22.35C24 | C42⋊C2 | C4×Q8 | C22⋊Q8 | C42.C2 | C42⋊2C2 | C4⋊Q8 | C20 | C4 | C10 | C2 |
| # reps | 1 | 1 | 2 | 2 | 5 | 4 | 1 | 4 | 4 | 8 | 8 | 20 | 16 | 4 | 4 | 16 | 2 | 8 |
Matrix representation of C5×C22.35C24 ►in GL6(𝔽41)
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 10 | 0 | 0 | 0 |
| 0 | 0 | 0 | 10 | 0 | 0 |
| 0 | 0 | 0 | 0 | 10 | 0 |
| 0 | 0 | 0 | 0 | 0 | 10 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 40 | 0 | 0 | 0 |
| 0 | 0 | 0 | 40 | 0 | 0 |
| 0 | 0 | 0 | 0 | 40 | 0 |
| 0 | 0 | 0 | 0 | 0 | 40 |
| 40 | 0 | 0 | 0 | 0 | 0 |
| 0 | 40 | 0 | 0 | 0 | 0 |
| 0 | 0 | 40 | 0 | 0 | 0 |
| 0 | 0 | 0 | 40 | 0 | 0 |
| 0 | 0 | 0 | 0 | 40 | 0 |
| 0 | 0 | 0 | 0 | 0 | 40 |
| 28 | 39 | 0 | 0 | 0 | 0 |
| 2 | 13 | 0 | 0 | 0 | 0 |
| 0 | 0 | 31 | 0 | 29 | 37 |
| 0 | 0 | 13 | 0 | 37 | 33 |
| 0 | 0 | 33 | 2 | 28 | 23 |
| 0 | 0 | 39 | 35 | 28 | 23 |
| 32 | 0 | 0 | 0 | 0 | 0 |
| 0 | 32 | 0 | 0 | 0 | 0 |
| 0 | 0 | 9 | 0 | 39 | 0 |
| 0 | 0 | 0 | 0 | 40 | 1 |
| 0 | 0 | 0 | 0 | 32 | 0 |
| 0 | 0 | 0 | 40 | 32 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 28 | 40 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 9 | 0 | 40 | 0 |
| 0 | 0 | 9 | 0 | 0 | 40 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 39 | 0 | 0 |
| 0 | 0 | 1 | 40 | 0 | 0 |
| 0 | 0 | 0 | 32 | 0 | 1 |
| 0 | 0 | 9 | 32 | 40 | 0 |
G:=sub<GL(6,GF(41))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,10,0,0,0,0,0,0,10,0,0,0,0,0,0,10,0,0,0,0,0,0,10],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40],[40,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40],[28,2,0,0,0,0,39,13,0,0,0,0,0,0,31,13,33,39,0,0,0,0,2,35,0,0,29,37,28,28,0,0,37,33,23,23],[32,0,0,0,0,0,0,32,0,0,0,0,0,0,9,0,0,0,0,0,0,0,0,40,0,0,39,40,32,32,0,0,0,1,0,0],[1,28,0,0,0,0,0,40,0,0,0,0,0,0,1,0,9,9,0,0,0,1,0,0,0,0,0,0,40,0,0,0,0,0,0,40],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,1,0,9,0,0,39,40,32,32,0,0,0,0,0,40,0,0,0,0,1,0] >;
C5×C22.35C24 in GAP, Magma, Sage, TeX
C_5\times C_2^2._{35}C_2^4 % in TeX
G:=Group("C5xC2^2.35C2^4"); // GroupNames label
G:=SmallGroup(320,1543);
// by ID
G=gap.SmallGroup(320,1543);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-5,-2,-2,560,1149,1128,3446,891,2467,304]);
// Polycyclic
G:=Group<a,b,c,d,e,f,g|a^5=b^2=c^2=f^2=1,d^2=g^2=b,e^2=c,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^-1=g*d*g^-1=b*d=d*b,f*e*f=b*e=e*b,b*f=f*b,b*g=g*b,f*d*f=c*d=d*c,c*e=e*c,c*f=f*c,c*g=g*c,e*g=g*e,f*g=g*f>;
// generators/relations