direct product, metabelian, nilpotent (class 2), monomial, 2-elementary
Aliases: C5×C23.7Q8, (C22×C4)⋊8C20, (C22×C20)⋊30C4, (C2×C20).511D4, C23.7(C5×Q8), (C23×C4).7C10, C23.21(C5×D4), C20⋊12(C22⋊C4), C22.9(Q8×C10), C24.24(C2×C10), C23.25(C2×C20), (C23×C20).22C2, C22.30(D4×C10), (C22×C10).19Q8, C2.C42⋊1C10, (C22×C10).126D4, C10.82(C22⋊Q8), C10.132(C4⋊D4), C22.29(C22×C20), (C23×C10).84C22, C23.53(C22×C10), C10.72(C42⋊C2), (C22×C10).444C23, (C22×C20).573C22, (C2×C4⋊C4)⋊1C10, C2.4(C10×C4⋊C4), C4⋊2(C5×C22⋊C4), C22⋊1(C5×C4⋊C4), (C10×C4⋊C4)⋊28C2, (C2×C10)⋊7(C4⋊C4), C10.82(C2×C4⋊C4), C2.1(C5×C4⋊D4), (C2×C4).55(C2×C20), C2.1(C5×C22⋊Q8), (C2×C4).116(C5×D4), C2.5(C10×C22⋊C4), (C2×C20).503(C2×C4), (C2×C10).597(C2×D4), (C2×C22⋊C4).3C10, (C10×C22⋊C4).9C2, C2.5(C5×C42⋊C2), (C2×C10).101(C2×Q8), C22.15(C5×C4○D4), C10.133(C2×C22⋊C4), (C5×C2.C42)⋊3C2, (C22×C4).84(C2×C10), (C2×C10).205(C4○D4), (C2×C10).317(C22×C4), (C22×C10).179(C2×C4), SmallGroup(320,881)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C5×C23.7Q8
G = < a,b,c,d,e,f | a5=b2=c2=d2=e4=1, f2=ce2, ab=ba, ac=ca, ad=da, ae=ea, af=fa, bc=cb, fbf-1=bd=db, be=eb, cd=dc, ce=ec, cf=fc, de=ed, df=fd, fef-1=e-1 >
Subgroups: 370 in 234 conjugacy classes, 114 normal (26 characteristic)
C1, C2, C2, C2, C4, C4, C22, C22, C22, C5, C2×C4, C2×C4, C23, C23, C23, C10, C10, C10, C22⋊C4, C4⋊C4, C22×C4, C22×C4, C22×C4, C24, C20, C20, C2×C10, C2×C10, C2×C10, C2.C42, C2×C22⋊C4, C2×C4⋊C4, C23×C4, C2×C20, C2×C20, C22×C10, C22×C10, C22×C10, C23.7Q8, C5×C22⋊C4, C5×C4⋊C4, C22×C20, C22×C20, C22×C20, C23×C10, C5×C2.C42, C10×C22⋊C4, C10×C4⋊C4, C23×C20, C5×C23.7Q8
Quotients: C1, C2, C4, C22, C5, C2×C4, D4, Q8, C23, C10, C22⋊C4, C4⋊C4, C22×C4, C2×D4, C2×Q8, C4○D4, C20, C2×C10, C2×C22⋊C4, C2×C4⋊C4, C42⋊C2, C4⋊D4, C22⋊Q8, C2×C20, C5×D4, C5×Q8, C22×C10, C23.7Q8, C5×C22⋊C4, C5×C4⋊C4, C22×C20, D4×C10, Q8×C10, C5×C4○D4, C10×C22⋊C4, C10×C4⋊C4, C5×C42⋊C2, C5×C4⋊D4, C5×C22⋊Q8, C5×C23.7Q8
►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 54)(2 55)(3 51)(4 52)(5 53)(6 31)(7 32)(8 33)(9 34)(10 35)(11 28)(12 29)(13 30)(14 26)(15 27)(16 23)(17 24)(18 25)(19 21)(20 22)(36 41)(37 42)(38 43)(39 44)(40 45)(46 70)(47 66)(48 67)(49 68)(50 69)(56 63)(57 64)(58 65)(59 61)(60 62)(71 80)(72 76)(73 77)(74 78)(75 79)(81 94)(82 95)(83 91)(84 92)(85 93)(86 110)(87 106)(88 107)(89 108)(90 109)(96 112)(97 113)(98 114)(99 115)(100 111)(101 119)(102 120)(103 116)(104 117)(105 118)(121 127)(122 128)(123 129)(124 130)(125 126)(131 148)(132 149)(133 150)(134 146)(135 147)(136 152)(137 153)(138 154)(139 155)(140 151)(141 159)(142 160)(143 156)(144 157)(145 158)
(1 54)(2 55)(3 51)(4 52)(5 53)(6 22)(7 23)(8 24)(9 25)(10 21)(11 28)(12 29)(13 30)(14 26)(15 27)(16 32)(17 33)(18 34)(19 35)(20 31)(36 41)(37 42)(38 43)(39 44)(40 45)(46 70)(47 66)(48 67)(49 68)(50 69)(56 63)(57 64)(58 65)(59 61)(60 62)(71 80)(72 76)(73 77)(74 78)(75 79)(81 94)(82 95)(83 91)(84 92)(85 93)(86 110)(87 106)(88 107)(89 108)(90 109)(96 103)(97 104)(98 105)(99 101)(100 102)(111 120)(112 116)(113 117)(114 118)(115 119)(121 134)(122 135)(123 131)(124 132)(125 133)(126 150)(127 146)(128 147)(129 148)(130 149)(136 143)(137 144)(138 145)(139 141)(140 142)(151 160)(152 156)(153 157)(154 158)(155 159)
(1 66)(2 67)(3 68)(4 69)(5 70)(6 20)(7 16)(8 17)(9 18)(10 19)(11 41)(12 42)(13 43)(14 44)(15 45)(21 35)(22 31)(23 32)(24 33)(25 34)(26 39)(27 40)(28 36)(29 37)(30 38)(46 53)(47 54)(48 55)(49 51)(50 52)(56 76)(57 77)(58 78)(59 79)(60 80)(61 75)(62 71)(63 72)(64 73)(65 74)(81 106)(82 107)(83 108)(84 109)(85 110)(86 93)(87 94)(88 95)(89 91)(90 92)(96 116)(97 117)(98 118)(99 119)(100 120)(101 115)(102 111)(103 112)(104 113)(105 114)(121 146)(122 147)(123 148)(124 149)(125 150)(126 133)(127 134)(128 135)(129 131)(130 132)(136 156)(137 157)(138 158)(139 159)(140 160)(141 155)(142 151)(143 152)(144 153)(145 154)
(1 64 12 81)(2 65 13 82)(3 61 14 83)(4 62 15 84)(5 63 11 85)(6 103 143 125)(7 104 144 121)(8 105 145 122)(9 101 141 123)(10 102 142 124)(16 113 153 146)(17 114 154 147)(18 115 155 148)(19 111 151 149)(20 112 152 150)(21 100 140 132)(22 96 136 133)(23 97 137 134)(24 98 138 135)(25 99 139 131)(26 91 51 59)(27 92 52 60)(28 93 53 56)(29 94 54 57)(30 95 55 58)(31 116 156 126)(32 117 157 127)(33 118 158 128)(34 119 159 129)(35 120 160 130)(36 86 46 76)(37 87 47 77)(38 88 48 78)(39 89 49 79)(40 90 50 80)(41 110 70 72)(42 106 66 73)(43 107 67 74)(44 108 68 75)(45 109 69 71)
(1 97 29 121)(2 98 30 122)(3 99 26 123)(4 100 27 124)(5 96 28 125)(6 85 136 56)(7 81 137 57)(8 82 138 58)(9 83 139 59)(10 84 140 60)(11 133 53 103)(12 134 54 104)(13 135 55 105)(14 131 51 101)(15 132 52 102)(16 106 157 77)(17 107 158 78)(18 108 159 79)(19 109 160 80)(20 110 156 76)(21 92 142 62)(22 93 143 63)(23 94 144 64)(24 95 145 65)(25 91 141 61)(31 86 152 72)(32 87 153 73)(33 88 154 74)(34 89 155 75)(35 90 151 71)(36 150 70 116)(37 146 66 117)(38 147 67 118)(39 148 68 119)(40 149 69 120)(41 126 46 112)(42 127 47 113)(43 128 48 114)(44 129 49 115)(45 130 50 111)
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), (1,54)(2,55)(3,51)(4,52)(5,53)(6,31)(7,32)(8,33)(9,34)(10,35)(11,28)(12,29)(13,30)(14,26)(15,27)(16,23)(17,24)(18,25)(19,21)(20,22)(36,41)(37,42)(38,43)(39,44)(40,45)(46,70)(47,66)(48,67)(49,68)(50,69)(56,63)(57,64)(58,65)(59,61)(60,62)(71,80)(72,76)(73,77)(74,78)(75,79)(81,94)(82,95)(83,91)(84,92)(85,93)(86,110)(87,106)(88,107)(89,108)(90,109)(96,112)(97,113)(98,114)(99,115)(100,111)(101,119)(102,120)(103,116)(104,117)(105,118)(121,127)(122,128)(123,129)(124,130)(125,126)(131,148)(132,149)(133,150)(134,146)(135,147)(136,152)(137,153)(138,154)(139,155)(140,151)(141,159)(142,160)(143,156)(144,157)(145,158), (1,54)(2,55)(3,51)(4,52)(5,53)(6,22)(7,23)(8,24)(9,25)(10,21)(11,28)(12,29)(13,30)(14,26)(15,27)(16,32)(17,33)(18,34)(19,35)(20,31)(36,41)(37,42)(38,43)(39,44)(40,45)(46,70)(47,66)(48,67)(49,68)(50,69)(56,63)(57,64)(58,65)(59,61)(60,62)(71,80)(72,76)(73,77)(74,78)(75,79)(81,94)(82,95)(83,91)(84,92)(85,93)(86,110)(87,106)(88,107)(89,108)(90,109)(96,103)(97,104)(98,105)(99,101)(100,102)(111,120)(112,116)(113,117)(114,118)(115,119)(121,134)(122,135)(123,131)(124,132)(125,133)(126,150)(127,146)(128,147)(129,148)(130,149)(136,143)(137,144)(138,145)(139,141)(140,142)(151,160)(152,156)(153,157)(154,158)(155,159), (1,66)(2,67)(3,68)(4,69)(5,70)(6,20)(7,16)(8,17)(9,18)(10,19)(11,41)(12,42)(13,43)(14,44)(15,45)(21,35)(22,31)(23,32)(24,33)(25,34)(26,39)(27,40)(28,36)(29,37)(30,38)(46,53)(47,54)(48,55)(49,51)(50,52)(56,76)(57,77)(58,78)(59,79)(60,80)(61,75)(62,71)(63,72)(64,73)(65,74)(81,106)(82,107)(83,108)(84,109)(85,110)(86,93)(87,94)(88,95)(89,91)(90,92)(96,116)(97,117)(98,118)(99,119)(100,120)(101,115)(102,111)(103,112)(104,113)(105,114)(121,146)(122,147)(123,148)(124,149)(125,150)(126,133)(127,134)(128,135)(129,131)(130,132)(136,156)(137,157)(138,158)(139,159)(140,160)(141,155)(142,151)(143,152)(144,153)(145,154), (1,64,12,81)(2,65,13,82)(3,61,14,83)(4,62,15,84)(5,63,11,85)(6,103,143,125)(7,104,144,121)(8,105,145,122)(9,101,141,123)(10,102,142,124)(16,113,153,146)(17,114,154,147)(18,115,155,148)(19,111,151,149)(20,112,152,150)(21,100,140,132)(22,96,136,133)(23,97,137,134)(24,98,138,135)(25,99,139,131)(26,91,51,59)(27,92,52,60)(28,93,53,56)(29,94,54,57)(30,95,55,58)(31,116,156,126)(32,117,157,127)(33,118,158,128)(34,119,159,129)(35,120,160,130)(36,86,46,76)(37,87,47,77)(38,88,48,78)(39,89,49,79)(40,90,50,80)(41,110,70,72)(42,106,66,73)(43,107,67,74)(44,108,68,75)(45,109,69,71), (1,97,29,121)(2,98,30,122)(3,99,26,123)(4,100,27,124)(5,96,28,125)(6,85,136,56)(7,81,137,57)(8,82,138,58)(9,83,139,59)(10,84,140,60)(11,133,53,103)(12,134,54,104)(13,135,55,105)(14,131,51,101)(15,132,52,102)(16,106,157,77)(17,107,158,78)(18,108,159,79)(19,109,160,80)(20,110,156,76)(21,92,142,62)(22,93,143,63)(23,94,144,64)(24,95,145,65)(25,91,141,61)(31,86,152,72)(32,87,153,73)(33,88,154,74)(34,89,155,75)(35,90,151,71)(36,150,70,116)(37,146,66,117)(38,147,67,118)(39,148,68,119)(40,149,69,120)(41,126,46,112)(42,127,47,113)(43,128,48,114)(44,129,49,115)(45,130,50,111)>;
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), (1,54)(2,55)(3,51)(4,52)(5,53)(6,31)(7,32)(8,33)(9,34)(10,35)(11,28)(12,29)(13,30)(14,26)(15,27)(16,23)(17,24)(18,25)(19,21)(20,22)(36,41)(37,42)(38,43)(39,44)(40,45)(46,70)(47,66)(48,67)(49,68)(50,69)(56,63)(57,64)(58,65)(59,61)(60,62)(71,80)(72,76)(73,77)(74,78)(75,79)(81,94)(82,95)(83,91)(84,92)(85,93)(86,110)(87,106)(88,107)(89,108)(90,109)(96,112)(97,113)(98,114)(99,115)(100,111)(101,119)(102,120)(103,116)(104,117)(105,118)(121,127)(122,128)(123,129)(124,130)(125,126)(131,148)(132,149)(133,150)(134,146)(135,147)(136,152)(137,153)(138,154)(139,155)(140,151)(141,159)(142,160)(143,156)(144,157)(145,158), (1,54)(2,55)(3,51)(4,52)(5,53)(6,22)(7,23)(8,24)(9,25)(10,21)(11,28)(12,29)(13,30)(14,26)(15,27)(16,32)(17,33)(18,34)(19,35)(20,31)(36,41)(37,42)(38,43)(39,44)(40,45)(46,70)(47,66)(48,67)(49,68)(50,69)(56,63)(57,64)(58,65)(59,61)(60,62)(71,80)(72,76)(73,77)(74,78)(75,79)(81,94)(82,95)(83,91)(84,92)(85,93)(86,110)(87,106)(88,107)(89,108)(90,109)(96,103)(97,104)(98,105)(99,101)(100,102)(111,120)(112,116)(113,117)(114,118)(115,119)(121,134)(122,135)(123,131)(124,132)(125,133)(126,150)(127,146)(128,147)(129,148)(130,149)(136,143)(137,144)(138,145)(139,141)(140,142)(151,160)(152,156)(153,157)(154,158)(155,159), (1,66)(2,67)(3,68)(4,69)(5,70)(6,20)(7,16)(8,17)(9,18)(10,19)(11,41)(12,42)(13,43)(14,44)(15,45)(21,35)(22,31)(23,32)(24,33)(25,34)(26,39)(27,40)(28,36)(29,37)(30,38)(46,53)(47,54)(48,55)(49,51)(50,52)(56,76)(57,77)(58,78)(59,79)(60,80)(61,75)(62,71)(63,72)(64,73)(65,74)(81,106)(82,107)(83,108)(84,109)(85,110)(86,93)(87,94)(88,95)(89,91)(90,92)(96,116)(97,117)(98,118)(99,119)(100,120)(101,115)(102,111)(103,112)(104,113)(105,114)(121,146)(122,147)(123,148)(124,149)(125,150)(126,133)(127,134)(128,135)(129,131)(130,132)(136,156)(137,157)(138,158)(139,159)(140,160)(141,155)(142,151)(143,152)(144,153)(145,154), (1,64,12,81)(2,65,13,82)(3,61,14,83)(4,62,15,84)(5,63,11,85)(6,103,143,125)(7,104,144,121)(8,105,145,122)(9,101,141,123)(10,102,142,124)(16,113,153,146)(17,114,154,147)(18,115,155,148)(19,111,151,149)(20,112,152,150)(21,100,140,132)(22,96,136,133)(23,97,137,134)(24,98,138,135)(25,99,139,131)(26,91,51,59)(27,92,52,60)(28,93,53,56)(29,94,54,57)(30,95,55,58)(31,116,156,126)(32,117,157,127)(33,118,158,128)(34,119,159,129)(35,120,160,130)(36,86,46,76)(37,87,47,77)(38,88,48,78)(39,89,49,79)(40,90,50,80)(41,110,70,72)(42,106,66,73)(43,107,67,74)(44,108,68,75)(45,109,69,71), (1,97,29,121)(2,98,30,122)(3,99,26,123)(4,100,27,124)(5,96,28,125)(6,85,136,56)(7,81,137,57)(8,82,138,58)(9,83,139,59)(10,84,140,60)(11,133,53,103)(12,134,54,104)(13,135,55,105)(14,131,51,101)(15,132,52,102)(16,106,157,77)(17,107,158,78)(18,108,159,79)(19,109,160,80)(20,110,156,76)(21,92,142,62)(22,93,143,63)(23,94,144,64)(24,95,145,65)(25,91,141,61)(31,86,152,72)(32,87,153,73)(33,88,154,74)(34,89,155,75)(35,90,151,71)(36,150,70,116)(37,146,66,117)(38,147,67,118)(39,148,68,119)(40,149,69,120)(41,126,46,112)(42,127,47,113)(43,128,48,114)(44,129,49,115)(45,130,50,111) );
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)], [(1,54),(2,55),(3,51),(4,52),(5,53),(6,31),(7,32),(8,33),(9,34),(10,35),(11,28),(12,29),(13,30),(14,26),(15,27),(16,23),(17,24),(18,25),(19,21),(20,22),(36,41),(37,42),(38,43),(39,44),(40,45),(46,70),(47,66),(48,67),(49,68),(50,69),(56,63),(57,64),(58,65),(59,61),(60,62),(71,80),(72,76),(73,77),(74,78),(75,79),(81,94),(82,95),(83,91),(84,92),(85,93),(86,110),(87,106),(88,107),(89,108),(90,109),(96,112),(97,113),(98,114),(99,115),(100,111),(101,119),(102,120),(103,116),(104,117),(105,118),(121,127),(122,128),(123,129),(124,130),(125,126),(131,148),(132,149),(133,150),(134,146),(135,147),(136,152),(137,153),(138,154),(139,155),(140,151),(141,159),(142,160),(143,156),(144,157),(145,158)], [(1,54),(2,55),(3,51),(4,52),(5,53),(6,22),(7,23),(8,24),(9,25),(10,21),(11,28),(12,29),(13,30),(14,26),(15,27),(16,32),(17,33),(18,34),(19,35),(20,31),(36,41),(37,42),(38,43),(39,44),(40,45),(46,70),(47,66),(48,67),(49,68),(50,69),(56,63),(57,64),(58,65),(59,61),(60,62),(71,80),(72,76),(73,77),(74,78),(75,79),(81,94),(82,95),(83,91),(84,92),(85,93),(86,110),(87,106),(88,107),(89,108),(90,109),(96,103),(97,104),(98,105),(99,101),(100,102),(111,120),(112,116),(113,117),(114,118),(115,119),(121,134),(122,135),(123,131),(124,132),(125,133),(126,150),(127,146),(128,147),(129,148),(130,149),(136,143),(137,144),(138,145),(139,141),(140,142),(151,160),(152,156),(153,157),(154,158),(155,159)], [(1,66),(2,67),(3,68),(4,69),(5,70),(6,20),(7,16),(8,17),(9,18),(10,19),(11,41),(12,42),(13,43),(14,44),(15,45),(21,35),(22,31),(23,32),(24,33),(25,34),(26,39),(27,40),(28,36),(29,37),(30,38),(46,53),(47,54),(48,55),(49,51),(50,52),(56,76),(57,77),(58,78),(59,79),(60,80),(61,75),(62,71),(63,72),(64,73),(65,74),(81,106),(82,107),(83,108),(84,109),(85,110),(86,93),(87,94),(88,95),(89,91),(90,92),(96,116),(97,117),(98,118),(99,119),(100,120),(101,115),(102,111),(103,112),(104,113),(105,114),(121,146),(122,147),(123,148),(124,149),(125,150),(126,133),(127,134),(128,135),(129,131),(130,132),(136,156),(137,157),(138,158),(139,159),(140,160),(141,155),(142,151),(143,152),(144,153),(145,154)], [(1,64,12,81),(2,65,13,82),(3,61,14,83),(4,62,15,84),(5,63,11,85),(6,103,143,125),(7,104,144,121),(8,105,145,122),(9,101,141,123),(10,102,142,124),(16,113,153,146),(17,114,154,147),(18,115,155,148),(19,111,151,149),(20,112,152,150),(21,100,140,132),(22,96,136,133),(23,97,137,134),(24,98,138,135),(25,99,139,131),(26,91,51,59),(27,92,52,60),(28,93,53,56),(29,94,54,57),(30,95,55,58),(31,116,156,126),(32,117,157,127),(33,118,158,128),(34,119,159,129),(35,120,160,130),(36,86,46,76),(37,87,47,77),(38,88,48,78),(39,89,49,79),(40,90,50,80),(41,110,70,72),(42,106,66,73),(43,107,67,74),(44,108,68,75),(45,109,69,71)], [(1,97,29,121),(2,98,30,122),(3,99,26,123),(4,100,27,124),(5,96,28,125),(6,85,136,56),(7,81,137,57),(8,82,138,58),(9,83,139,59),(10,84,140,60),(11,133,53,103),(12,134,54,104),(13,135,55,105),(14,131,51,101),(15,132,52,102),(16,106,157,77),(17,107,158,78),(18,108,159,79),(19,109,160,80),(20,110,156,76),(21,92,142,62),(22,93,143,63),(23,94,144,64),(24,95,145,65),(25,91,141,61),(31,86,152,72),(32,87,153,73),(33,88,154,74),(34,89,155,75),(35,90,151,71),(36,150,70,116),(37,146,66,117),(38,147,67,118),(39,148,68,119),(40,149,69,120),(41,126,46,112),(42,127,47,113),(43,128,48,114),(44,129,49,115),(45,130,50,111)]])
140 conjugacy classes
| class | 1 | 2A | ··· | 2G | 2H | 2I | 2J | 2K | 4A | ··· | 4H | 4I | ··· | 4P | 5A | 5B | 5C | 5D | 10A | ··· | 10AB | 10AC | ··· | 10AR | 20A | ··· | 20AF | 20AG | ··· | 20BL |
| order | 1 | 2 | ··· | 2 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 4 | ··· | 4 | 5 | 5 | 5 | 5 | 10 | ··· | 10 | 10 | ··· | 10 | 20 | ··· | 20 | 20 | ··· | 20 |
| size | 1 | 1 | ··· | 1 | 2 | 2 | 2 | 2 | 2 | ··· | 2 | 4 | ··· | 4 | 1 | 1 | 1 | 1 | 1 | ··· | 1 | 2 | ··· | 2 | 2 | ··· | 2 | 4 | ··· | 4 |
140 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | - | ||||||||||||
| image | C1 | C2 | C2 | C2 | C2 | C4 | C5 | C10 | C10 | C10 | C10 | C20 | D4 | D4 | Q8 | C4○D4 | C5×D4 | C5×D4 | C5×Q8 | C5×C4○D4 |
| kernel | C5×C23.7Q8 | C5×C2.C42 | C10×C22⋊C4 | C10×C4⋊C4 | C23×C20 | C22×C20 | C23.7Q8 | C2.C42 | C2×C22⋊C4 | C2×C4⋊C4 | C23×C4 | C22×C4 | C2×C20 | C22×C10 | C22×C10 | C2×C10 | C2×C4 | C23 | C23 | C22 |
| # reps | 1 | 2 | 2 | 2 | 1 | 8 | 4 | 8 | 8 | 8 | 4 | 32 | 4 | 2 | 2 | 4 | 16 | 8 | 8 | 16 |
Matrix representation of C5×C23.7Q8 ►in GL5(𝔽41)
| 1 | 0 | 0 | 0 | 0 |
| 0 | 10 | 0 | 0 | 0 |
| 0 | 0 | 10 | 0 | 0 |
| 0 | 0 | 0 | 16 | 0 |
| 0 | 0 | 0 | 0 | 16 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 40 | 0 | 0 |
| 0 | 0 | 0 | 40 | 0 |
| 0 | 0 | 0 | 0 | 40 |
| 40 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 40 | 0 |
| 0 | 0 | 0 | 0 | 40 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 40 | 0 | 0 | 0 |
| 0 | 0 | 40 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 1 |
| 40 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 40 | 39 |
| 0 | 0 | 0 | 1 | 1 |
| 32 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 16 | 38 |
| 0 | 0 | 0 | 3 | 25 |
G:=sub<GL(5,GF(41))| [1,0,0,0,0,0,10,0,0,0,0,0,10,0,0,0,0,0,16,0,0,0,0,0,16],[1,0,0,0,0,0,1,0,0,0,0,0,40,0,0,0,0,0,40,0,0,0,0,0,40],[40,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,40,0,0,0,0,0,40],[1,0,0,0,0,0,40,0,0,0,0,0,40,0,0,0,0,0,1,0,0,0,0,0,1],[40,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,40,1,0,0,0,39,1],[32,0,0,0,0,0,0,1,0,0,0,1,0,0,0,0,0,0,16,3,0,0,0,38,25] >;
C5×C23.7Q8 in GAP, Magma, Sage, TeX
C_5\times C_2^3._7Q_8
% in TeX
G:=Group("C5xC2^3.7Q8"); // GroupNames label
G:=SmallGroup(320,881);
// by ID
G=gap.SmallGroup(320,881);
# by ID
G:=PCGroup([7,-2,-2,-2,-5,-2,-2,-2,1120,589,288,1766]);
// Polycyclic
G:=Group<a,b,c,d,e,f|a^5=b^2=c^2=d^2=e^4=1,f^2=c*e^2,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,a*f=f*a,b*c=c*b,f*b*f^-1=b*d=d*b,b*e=e*b,c*d=d*c,c*e=e*c,c*f=f*c,d*e=e*d,d*f=f*d,f*e*f^-1=e^-1>;
// generators/relations