direct product, metabelian, nilpotent (class 2), monomial, 2-elementary
Aliases: C5×C23.8Q8, C2.4(D4×C20), C22⋊C4⋊4C20, (C2×C20).454D4, C10.135(C4×D4), C23.8(C5×Q8), (C23×C4).4C10, (C23×C20).7C2, C23.35(C5×D4), C10.88C22≀C2, C23.15(C2×C20), C24.26(C2×C10), C22.34(D4×C10), (C22×C10).20Q8, C22.12(Q8×C10), C2.C42⋊7C10, (C22×C10).155D4, C10.83(C22⋊Q8), C23.58(C22×C10), (C23×C10).86C22, C22.34(C22×C20), (C22×C20).493C22, (C22×C10).449C23, C10.86(C22.D4), (C2×C4⋊C4)⋊2C10, (C2×C4)⋊2(C2×C20), C2.7(C10×C4⋊C4), C22⋊2(C5×C4⋊C4), (C10×C4⋊C4)⋊29C2, (C2×C10)⋊8(C4⋊C4), (C2×C20)⋊36(C2×C4), C10.85(C2×C4⋊C4), (C2×C4).99(C5×D4), (C5×C22⋊C4)⋊16C4, C2.2(C5×C22⋊Q8), C2.2(C5×C22≀C2), (C2×C10).601(C2×D4), (C2×C22⋊C4).5C10, (C2×C10).104(C2×Q8), C22.19(C5×C4○D4), (C10×C22⋊C4).11C2, (C22×C4).86(C2×C10), (C2×C10).209(C4○D4), C2.2(C5×C22.D4), (C2×C10).322(C22×C4), (C5×C2.C42)⋊23C2, (C22×C10).150(C2×C4), SmallGroup(320,886)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C5×C23.8Q8
G = < a,b,c,d,e,f | a5=b2=c2=d2=e4=1, f2=e2, 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=ce-1 >
Subgroups: 370 in 234 conjugacy classes, 106 normal (30 characteristic)
C1, C2, C2, C2, C4, C22, C22, C22, C5, C2×C4, C2×C4, C23, C23, C23, C10, C10, C10, C22⋊C4, C22⋊C4, C4⋊C4, C22×C4, C22×C4, C22×C4, C24, 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, C23.8Q8, C5×C22⋊C4, 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×C23.8Q8
Quotients: C1, C2, C4, C22, C5, C2×C4, D4, Q8, C23, C10, C4⋊C4, C22×C4, C2×D4, C2×Q8, C4○D4, C20, C2×C10, C2×C4⋊C4, C4×D4, C22≀C2, C22⋊Q8, C22.D4, C2×C20, C5×D4, C5×Q8, C22×C10, C23.8Q8, C5×C4⋊C4, C22×C20, D4×C10, Q8×C10, C5×C4○D4, C10×C4⋊C4, D4×C20, C5×C22≀C2, C5×C22⋊Q8, C5×C22.D4, C5×C23.8Q8
►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 44)(12 45)(13 41)(14 42)(15 43)(16 25)(17 21)(18 22)(19 23)(20 24)(26 40)(27 36)(28 37)(29 38)(30 39)(46 70)(47 66)(48 67)(49 68)(50 69)(56 65)(57 61)(58 62)(59 63)(60 64)(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)(96 117)(97 118)(98 119)(99 120)(100 116)(101 114)(102 115)(103 111)(104 112)(105 113)(121 127)(122 128)(123 129)(124 130)(125 126)(131 148)(132 149)(133 150)(134 146)(135 147)(136 157)(137 158)(138 159)(139 160)(140 156)(141 154)(142 155)(143 151)(144 152)(145 153)
(1 54)(2 55)(3 51)(4 52)(5 53)(6 19)(7 20)(8 16)(9 17)(10 18)(11 44)(12 45)(13 41)(14 42)(15 43)(21 34)(22 35)(23 31)(24 32)(25 33)(26 40)(27 36)(28 37)(29 38)(30 39)(46 70)(47 66)(48 67)(49 68)(50 69)(56 65)(57 61)(58 62)(59 63)(60 64)(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)(96 105)(97 101)(98 102)(99 103)(100 104)(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)(136 145)(137 141)(138 142)(139 143)(140 144)(151 160)(152 156)(153 157)(154 158)(155 159)
(1 66)(2 67)(3 68)(4 69)(5 70)(6 23)(7 24)(8 25)(9 21)(10 22)(11 36)(12 37)(13 38)(14 39)(15 40)(16 33)(17 34)(18 35)(19 31)(20 32)(26 43)(27 44)(28 45)(29 41)(30 42)(46 53)(47 54)(48 55)(49 51)(50 52)(56 73)(57 74)(58 75)(59 71)(60 72)(61 78)(62 79)(63 80)(64 76)(65 77)(81 106)(82 107)(83 108)(84 109)(85 110)(86 93)(87 94)(88 95)(89 91)(90 92)(96 113)(97 114)(98 115)(99 111)(100 112)(101 118)(102 119)(103 120)(104 116)(105 117)(121 146)(122 147)(123 148)(124 149)(125 150)(126 133)(127 134)(128 135)(129 131)(130 132)(136 153)(137 154)(138 155)(139 151)(140 152)(141 158)(142 159)(143 160)(144 156)(145 157)
(1 65 12 81)(2 61 13 82)(3 62 14 83)(4 63 15 84)(5 64 11 85)(6 100 144 133)(7 96 145 134)(8 97 141 135)(9 98 142 131)(10 99 143 132)(16 101 137 122)(17 102 138 123)(18 103 139 124)(19 104 140 125)(20 105 136 121)(21 115 159 129)(22 111 160 130)(23 112 156 126)(24 113 157 127)(25 114 158 128)(26 90 50 71)(27 86 46 72)(28 87 47 73)(29 88 48 74)(30 89 49 75)(31 116 152 150)(32 117 153 146)(33 118 154 147)(34 119 155 148)(35 120 151 149)(36 110 70 76)(37 106 66 77)(38 107 67 78)(39 108 68 79)(40 109 69 80)(41 95 55 57)(42 91 51 58)(43 92 52 59)(44 93 53 60)(45 94 54 56)
(1 105 12 121)(2 101 13 122)(3 102 14 123)(4 103 15 124)(5 104 11 125)(6 85 144 64)(7 81 145 65)(8 82 141 61)(9 83 142 62)(10 84 143 63)(16 95 137 57)(17 91 138 58)(18 92 139 59)(19 93 140 60)(20 94 136 56)(21 108 159 79)(22 109 160 80)(23 110 156 76)(24 106 157 77)(25 107 158 78)(26 130 50 111)(27 126 46 112)(28 127 47 113)(29 128 48 114)(30 129 49 115)(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)(41 135 55 97)(42 131 51 98)(43 132 52 99)(44 133 53 100)(45 134 54 96)
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)(11,44)(12,45)(13,41)(14,42)(15,43)(16,25)(17,21)(18,22)(19,23)(20,24)(26,40)(27,36)(28,37)(29,38)(30,39)(46,70)(47,66)(48,67)(49,68)(50,69)(56,65)(57,61)(58,62)(59,63)(60,64)(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)(96,117)(97,118)(98,119)(99,120)(100,116)(101,114)(102,115)(103,111)(104,112)(105,113)(121,127)(122,128)(123,129)(124,130)(125,126)(131,148)(132,149)(133,150)(134,146)(135,147)(136,157)(137,158)(138,159)(139,160)(140,156)(141,154)(142,155)(143,151)(144,152)(145,153), (1,54)(2,55)(3,51)(4,52)(5,53)(6,19)(7,20)(8,16)(9,17)(10,18)(11,44)(12,45)(13,41)(14,42)(15,43)(21,34)(22,35)(23,31)(24,32)(25,33)(26,40)(27,36)(28,37)(29,38)(30,39)(46,70)(47,66)(48,67)(49,68)(50,69)(56,65)(57,61)(58,62)(59,63)(60,64)(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)(96,105)(97,101)(98,102)(99,103)(100,104)(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)(136,145)(137,141)(138,142)(139,143)(140,144)(151,160)(152,156)(153,157)(154,158)(155,159), (1,66)(2,67)(3,68)(4,69)(5,70)(6,23)(7,24)(8,25)(9,21)(10,22)(11,36)(12,37)(13,38)(14,39)(15,40)(16,33)(17,34)(18,35)(19,31)(20,32)(26,43)(27,44)(28,45)(29,41)(30,42)(46,53)(47,54)(48,55)(49,51)(50,52)(56,73)(57,74)(58,75)(59,71)(60,72)(61,78)(62,79)(63,80)(64,76)(65,77)(81,106)(82,107)(83,108)(84,109)(85,110)(86,93)(87,94)(88,95)(89,91)(90,92)(96,113)(97,114)(98,115)(99,111)(100,112)(101,118)(102,119)(103,120)(104,116)(105,117)(121,146)(122,147)(123,148)(124,149)(125,150)(126,133)(127,134)(128,135)(129,131)(130,132)(136,153)(137,154)(138,155)(139,151)(140,152)(141,158)(142,159)(143,160)(144,156)(145,157), (1,65,12,81)(2,61,13,82)(3,62,14,83)(4,63,15,84)(5,64,11,85)(6,100,144,133)(7,96,145,134)(8,97,141,135)(9,98,142,131)(10,99,143,132)(16,101,137,122)(17,102,138,123)(18,103,139,124)(19,104,140,125)(20,105,136,121)(21,115,159,129)(22,111,160,130)(23,112,156,126)(24,113,157,127)(25,114,158,128)(26,90,50,71)(27,86,46,72)(28,87,47,73)(29,88,48,74)(30,89,49,75)(31,116,152,150)(32,117,153,146)(33,118,154,147)(34,119,155,148)(35,120,151,149)(36,110,70,76)(37,106,66,77)(38,107,67,78)(39,108,68,79)(40,109,69,80)(41,95,55,57)(42,91,51,58)(43,92,52,59)(44,93,53,60)(45,94,54,56), (1,105,12,121)(2,101,13,122)(3,102,14,123)(4,103,15,124)(5,104,11,125)(6,85,144,64)(7,81,145,65)(8,82,141,61)(9,83,142,62)(10,84,143,63)(16,95,137,57)(17,91,138,58)(18,92,139,59)(19,93,140,60)(20,94,136,56)(21,108,159,79)(22,109,160,80)(23,110,156,76)(24,106,157,77)(25,107,158,78)(26,130,50,111)(27,126,46,112)(28,127,47,113)(29,128,48,114)(30,129,49,115)(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)(41,135,55,97)(42,131,51,98)(43,132,52,99)(44,133,53,100)(45,134,54,96)>;
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)(11,44)(12,45)(13,41)(14,42)(15,43)(16,25)(17,21)(18,22)(19,23)(20,24)(26,40)(27,36)(28,37)(29,38)(30,39)(46,70)(47,66)(48,67)(49,68)(50,69)(56,65)(57,61)(58,62)(59,63)(60,64)(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)(96,117)(97,118)(98,119)(99,120)(100,116)(101,114)(102,115)(103,111)(104,112)(105,113)(121,127)(122,128)(123,129)(124,130)(125,126)(131,148)(132,149)(133,150)(134,146)(135,147)(136,157)(137,158)(138,159)(139,160)(140,156)(141,154)(142,155)(143,151)(144,152)(145,153), (1,54)(2,55)(3,51)(4,52)(5,53)(6,19)(7,20)(8,16)(9,17)(10,18)(11,44)(12,45)(13,41)(14,42)(15,43)(21,34)(22,35)(23,31)(24,32)(25,33)(26,40)(27,36)(28,37)(29,38)(30,39)(46,70)(47,66)(48,67)(49,68)(50,69)(56,65)(57,61)(58,62)(59,63)(60,64)(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)(96,105)(97,101)(98,102)(99,103)(100,104)(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)(136,145)(137,141)(138,142)(139,143)(140,144)(151,160)(152,156)(153,157)(154,158)(155,159), (1,66)(2,67)(3,68)(4,69)(5,70)(6,23)(7,24)(8,25)(9,21)(10,22)(11,36)(12,37)(13,38)(14,39)(15,40)(16,33)(17,34)(18,35)(19,31)(20,32)(26,43)(27,44)(28,45)(29,41)(30,42)(46,53)(47,54)(48,55)(49,51)(50,52)(56,73)(57,74)(58,75)(59,71)(60,72)(61,78)(62,79)(63,80)(64,76)(65,77)(81,106)(82,107)(83,108)(84,109)(85,110)(86,93)(87,94)(88,95)(89,91)(90,92)(96,113)(97,114)(98,115)(99,111)(100,112)(101,118)(102,119)(103,120)(104,116)(105,117)(121,146)(122,147)(123,148)(124,149)(125,150)(126,133)(127,134)(128,135)(129,131)(130,132)(136,153)(137,154)(138,155)(139,151)(140,152)(141,158)(142,159)(143,160)(144,156)(145,157), (1,65,12,81)(2,61,13,82)(3,62,14,83)(4,63,15,84)(5,64,11,85)(6,100,144,133)(7,96,145,134)(8,97,141,135)(9,98,142,131)(10,99,143,132)(16,101,137,122)(17,102,138,123)(18,103,139,124)(19,104,140,125)(20,105,136,121)(21,115,159,129)(22,111,160,130)(23,112,156,126)(24,113,157,127)(25,114,158,128)(26,90,50,71)(27,86,46,72)(28,87,47,73)(29,88,48,74)(30,89,49,75)(31,116,152,150)(32,117,153,146)(33,118,154,147)(34,119,155,148)(35,120,151,149)(36,110,70,76)(37,106,66,77)(38,107,67,78)(39,108,68,79)(40,109,69,80)(41,95,55,57)(42,91,51,58)(43,92,52,59)(44,93,53,60)(45,94,54,56), (1,105,12,121)(2,101,13,122)(3,102,14,123)(4,103,15,124)(5,104,11,125)(6,85,144,64)(7,81,145,65)(8,82,141,61)(9,83,142,62)(10,84,143,63)(16,95,137,57)(17,91,138,58)(18,92,139,59)(19,93,140,60)(20,94,136,56)(21,108,159,79)(22,109,160,80)(23,110,156,76)(24,106,157,77)(25,107,158,78)(26,130,50,111)(27,126,46,112)(28,127,47,113)(29,128,48,114)(30,129,49,115)(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)(41,135,55,97)(42,131,51,98)(43,132,52,99)(44,133,53,100)(45,134,54,96) );
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),(11,44),(12,45),(13,41),(14,42),(15,43),(16,25),(17,21),(18,22),(19,23),(20,24),(26,40),(27,36),(28,37),(29,38),(30,39),(46,70),(47,66),(48,67),(49,68),(50,69),(56,65),(57,61),(58,62),(59,63),(60,64),(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),(96,117),(97,118),(98,119),(99,120),(100,116),(101,114),(102,115),(103,111),(104,112),(105,113),(121,127),(122,128),(123,129),(124,130),(125,126),(131,148),(132,149),(133,150),(134,146),(135,147),(136,157),(137,158),(138,159),(139,160),(140,156),(141,154),(142,155),(143,151),(144,152),(145,153)], [(1,54),(2,55),(3,51),(4,52),(5,53),(6,19),(7,20),(8,16),(9,17),(10,18),(11,44),(12,45),(13,41),(14,42),(15,43),(21,34),(22,35),(23,31),(24,32),(25,33),(26,40),(27,36),(28,37),(29,38),(30,39),(46,70),(47,66),(48,67),(49,68),(50,69),(56,65),(57,61),(58,62),(59,63),(60,64),(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),(96,105),(97,101),(98,102),(99,103),(100,104),(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),(136,145),(137,141),(138,142),(139,143),(140,144),(151,160),(152,156),(153,157),(154,158),(155,159)], [(1,66),(2,67),(3,68),(4,69),(5,70),(6,23),(7,24),(8,25),(9,21),(10,22),(11,36),(12,37),(13,38),(14,39),(15,40),(16,33),(17,34),(18,35),(19,31),(20,32),(26,43),(27,44),(28,45),(29,41),(30,42),(46,53),(47,54),(48,55),(49,51),(50,52),(56,73),(57,74),(58,75),(59,71),(60,72),(61,78),(62,79),(63,80),(64,76),(65,77),(81,106),(82,107),(83,108),(84,109),(85,110),(86,93),(87,94),(88,95),(89,91),(90,92),(96,113),(97,114),(98,115),(99,111),(100,112),(101,118),(102,119),(103,120),(104,116),(105,117),(121,146),(122,147),(123,148),(124,149),(125,150),(126,133),(127,134),(128,135),(129,131),(130,132),(136,153),(137,154),(138,155),(139,151),(140,152),(141,158),(142,159),(143,160),(144,156),(145,157)], [(1,65,12,81),(2,61,13,82),(3,62,14,83),(4,63,15,84),(5,64,11,85),(6,100,144,133),(7,96,145,134),(8,97,141,135),(9,98,142,131),(10,99,143,132),(16,101,137,122),(17,102,138,123),(18,103,139,124),(19,104,140,125),(20,105,136,121),(21,115,159,129),(22,111,160,130),(23,112,156,126),(24,113,157,127),(25,114,158,128),(26,90,50,71),(27,86,46,72),(28,87,47,73),(29,88,48,74),(30,89,49,75),(31,116,152,150),(32,117,153,146),(33,118,154,147),(34,119,155,148),(35,120,151,149),(36,110,70,76),(37,106,66,77),(38,107,67,78),(39,108,68,79),(40,109,69,80),(41,95,55,57),(42,91,51,58),(43,92,52,59),(44,93,53,60),(45,94,54,56)], [(1,105,12,121),(2,101,13,122),(3,102,14,123),(4,103,15,124),(5,104,11,125),(6,85,144,64),(7,81,145,65),(8,82,141,61),(9,83,142,62),(10,84,143,63),(16,95,137,57),(17,91,138,58),(18,92,139,59),(19,93,140,60),(20,94,136,56),(21,108,159,79),(22,109,160,80),(23,110,156,76),(24,106,157,77),(25,107,158,78),(26,130,50,111),(27,126,46,112),(28,127,47,113),(29,128,48,114),(30,129,49,115),(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),(41,135,55,97),(42,131,51,98),(43,132,52,99),(44,133,53,100),(45,134,54,96)]])
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×C23.8Q8 | C5×C2.C42 | C10×C22⋊C4 | C10×C4⋊C4 | C23×C20 | C5×C22⋊C4 | C23.8Q8 | 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×C23.8Q8 ►in GL6(𝔽41)
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 16 | 0 |
| 0 | 0 | 0 | 0 | 0 | 16 |
| 40 | 0 | 0 | 0 | 0 | 0 |
| 0 | 40 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 40 | 0 |
| 0 | 0 | 0 | 0 | 4 | 1 |
| 40 | 0 | 0 | 0 | 0 | 0 |
| 0 | 40 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 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 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 40 | 0 |
| 0 | 0 | 0 | 0 | 0 | 40 |
| 1 | 39 | 0 | 0 | 0 | 0 |
| 0 | 40 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 40 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 32 | 0 |
| 0 | 0 | 0 | 0 | 0 | 32 |
| 17 | 12 | 0 | 0 | 0 | 0 |
| 17 | 24 | 0 | 0 | 0 | 0 |
| 0 | 0 | 15 | 15 | 0 | 0 |
| 0 | 0 | 15 | 26 | 0 | 0 |
| 0 | 0 | 0 | 0 | 37 | 39 |
| 0 | 0 | 0 | 0 | 29 | 4 |
G:=sub<GL(6,GF(41))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,16,0,0,0,0,0,0,16],[40,0,0,0,0,0,0,40,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,40,4,0,0,0,0,0,1],[40,0,0,0,0,0,0,40,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,40,0,0,0,0,0,0,40],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,40,0,0,0,0,0,0,40],[1,0,0,0,0,0,39,40,0,0,0,0,0,0,0,40,0,0,0,0,1,0,0,0,0,0,0,0,32,0,0,0,0,0,0,32],[17,17,0,0,0,0,12,24,0,0,0,0,0,0,15,15,0,0,0,0,15,26,0,0,0,0,0,0,37,29,0,0,0,0,39,4] >;
C5×C23.8Q8 in GAP, Magma, Sage, TeX
C_5\times C_2^3._8Q_8
% in TeX
G:=Group("C5xC2^3.8Q8"); // GroupNames label
G:=SmallGroup(320,886);
// by ID
G=gap.SmallGroup(320,886);
# by ID
G:=PCGroup([7,-2,-2,-2,-5,-2,-2,-2,1120,589,1128,1766]);
// Polycyclic
G:=Group<a,b,c,d,e,f|a^5=b^2=c^2=d^2=e^4=1,f^2=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=c*e^-1>;
// generators/relations