metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C24.44D10, C23.9Dic10, C22.94(D4×D5), C23.49(C4×D5), (C22×C4).22D10, (C22×C10).60D4, (C22×C10).10Q8, C10.81(C4⋊D4), C5⋊3(C23.7Q8), Dic5⋊4(C22⋊C4), (C2×Dic5).227D4, (C22×Dic5)⋊11C4, C23.35(C5⋊D4), C10.15(C22⋊Q8), (C23×Dic5).2C2, C10.10C42⋊9C2, C2.1(Dic5⋊D4), C22⋊1(C10.D4), (C22×C20).21C22, (C23×C10).25C22, C22.23(C2×Dic10), C23.275(C22×D5), C10.43(C42⋊C2), C22.41(D4⋊2D5), (C22×C10).317C23, C2.5(Dic5.14D4), (C22×Dic5).34C22, C2.11(C23.11D10), (C2×C10)⋊4(C4⋊C4), C10.52(C2×C4⋊C4), (C2×C22⋊C4).4D5, (C2×C10).30(C2×Q8), C2.27(D5×C22⋊C4), C22.121(C2×C4×D5), (C2×C10).429(C2×D4), (C2×C10.D4)⋊6C2, (C10×C22⋊C4).5C2, C10.68(C2×C22⋊C4), C2.5(C2×C10.D4), C22.45(C2×C5⋊D4), (C2×C23.D5).4C2, (C2×C10).138(C4○D4), (C2×C10).204(C22×C4), (C22×C10).113(C2×C4), (C2×Dic5).147(C2×C4), SmallGroup(320,569)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C24.44D10
G = < a,b,c,d,e,f | a2=b2=c2=d2=1, e10=f2=b, ab=ba, ac=ca, eae-1=faf-1=ad=da, bc=cb, bd=db, be=eb, bf=fb, cd=dc, ce=ec, cf=fc, de=ed, df=fd, fef-1=ce9 >
Subgroups: 734 in 234 conjugacy classes, 87 normal (25 characteristic)
C1, C2, C2, C2, C4, C22, C22, C22, C5, C2×C4, C23, C23, C23, C10, C10, C10, C22⋊C4, C4⋊C4, C22×C4, C22×C4, C24, Dic5, Dic5, C20, C2×C10, C2×C10, C2×C10, C2.C42, C2×C22⋊C4, C2×C22⋊C4, C2×C4⋊C4, C23×C4, C2×Dic5, C2×Dic5, C2×C20, C22×C10, C22×C10, C22×C10, C23.7Q8, C10.D4, C23.D5, C5×C22⋊C4, C22×Dic5, C22×Dic5, C22×Dic5, C22×C20, C23×C10, C10.10C42, C2×C10.D4, C2×C23.D5, C10×C22⋊C4, C23×Dic5, C24.44D10
Quotients: C1, C2, C4, C22, C2×C4, D4, Q8, C23, D5, C22⋊C4, C4⋊C4, C22×C4, C2×D4, C2×Q8, C4○D4, D10, C2×C22⋊C4, C2×C4⋊C4, C42⋊C2, C4⋊D4, C22⋊Q8, Dic10, C4×D5, C5⋊D4, C22×D5, C23.7Q8, C10.D4, C2×Dic10, C2×C4×D5, D4×D5, D4⋊2D5, C2×C5⋊D4, C23.11D10, Dic5.14D4, D5×C22⋊C4, C2×C10.D4, Dic5⋊D4, C24.44D10
►On 160 points
Generators in S160
(1 11)(2 30)(3 13)(4 32)(5 15)(6 34)(7 17)(8 36)(9 19)(10 38)(12 40)(14 22)(16 24)(18 26)(20 28)(21 31)(23 33)(25 35)(27 37)(29 39)(41 51)(42 62)(43 53)(44 64)(45 55)(46 66)(47 57)(48 68)(49 59)(50 70)(52 72)(54 74)(56 76)(58 78)(60 80)(61 71)(63 73)(65 75)(67 77)(69 79)(81 112)(82 92)(83 114)(84 94)(85 116)(86 96)(87 118)(88 98)(89 120)(90 100)(91 102)(93 104)(95 106)(97 108)(99 110)(101 111)(103 113)(105 115)(107 117)(109 119)(121 131)(122 147)(123 133)(124 149)(125 135)(126 151)(127 137)(128 153)(129 139)(130 155)(132 157)(134 159)(136 141)(138 143)(140 145)(142 152)(144 154)(146 156)(148 158)(150 160)
(1 11)(2 12)(3 13)(4 14)(5 15)(6 16)(7 17)(8 18)(9 19)(10 20)(21 31)(22 32)(23 33)(24 34)(25 35)(26 36)(27 37)(28 38)(29 39)(30 40)(41 51)(42 52)(43 53)(44 54)(45 55)(46 56)(47 57)(48 58)(49 59)(50 60)(61 71)(62 72)(63 73)(64 74)(65 75)(66 76)(67 77)(68 78)(69 79)(70 80)(81 91)(82 92)(83 93)(84 94)(85 95)(86 96)(87 97)(88 98)(89 99)(90 100)(101 111)(102 112)(103 113)(104 114)(105 115)(106 116)(107 117)(108 118)(109 119)(110 120)(121 131)(122 132)(123 133)(124 134)(125 135)(126 136)(127 137)(128 138)(129 139)(130 140)(141 151)(142 152)(143 153)(144 154)(145 155)(146 156)(147 157)(148 158)(149 159)(150 160)
(1 59)(2 60)(3 41)(4 42)(5 43)(6 44)(7 45)(8 46)(9 47)(10 48)(11 49)(12 50)(13 51)(14 52)(15 53)(16 54)(17 55)(18 56)(19 57)(20 58)(21 71)(22 72)(23 73)(24 74)(25 75)(26 76)(27 77)(28 78)(29 79)(30 80)(31 61)(32 62)(33 63)(34 64)(35 65)(36 66)(37 67)(38 68)(39 69)(40 70)(81 141)(82 142)(83 143)(84 144)(85 145)(86 146)(87 147)(88 148)(89 149)(90 150)(91 151)(92 152)(93 153)(94 154)(95 155)(96 156)(97 157)(98 158)(99 159)(100 160)(101 125)(102 126)(103 127)(104 128)(105 129)(106 130)(107 131)(108 132)(109 133)(110 134)(111 135)(112 136)(113 137)(114 138)(115 139)(116 140)(117 121)(118 122)(119 123)(120 124)
(1 39)(2 40)(3 21)(4 22)(5 23)(6 24)(7 25)(8 26)(9 27)(10 28)(11 29)(12 30)(13 31)(14 32)(15 33)(16 34)(17 35)(18 36)(19 37)(20 38)(41 71)(42 72)(43 73)(44 74)(45 75)(46 76)(47 77)(48 78)(49 79)(50 80)(51 61)(52 62)(53 63)(54 64)(55 65)(56 66)(57 67)(58 68)(59 69)(60 70)(81 102)(82 103)(83 104)(84 105)(85 106)(86 107)(87 108)(88 109)(89 110)(90 111)(91 112)(92 113)(93 114)(94 115)(95 116)(96 117)(97 118)(98 119)(99 120)(100 101)(121 156)(122 157)(123 158)(124 159)(125 160)(126 141)(127 142)(128 143)(129 144)(130 145)(131 146)(132 147)(133 148)(134 149)(135 150)(136 151)(137 152)(138 153)(139 154)(140 155)
(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 81 11 91)(2 150 12 160)(3 99 13 89)(4 148 14 158)(5 97 15 87)(6 146 16 156)(7 95 17 85)(8 144 18 154)(9 93 19 83)(10 142 20 152)(21 120 31 110)(22 133 32 123)(23 118 33 108)(24 131 34 121)(25 116 35 106)(26 129 36 139)(27 114 37 104)(28 127 38 137)(29 112 39 102)(30 125 40 135)(41 159 51 149)(42 88 52 98)(43 157 53 147)(44 86 54 96)(45 155 55 145)(46 84 56 94)(47 153 57 143)(48 82 58 92)(49 151 59 141)(50 100 60 90)(61 134 71 124)(62 119 72 109)(63 132 73 122)(64 117 74 107)(65 130 75 140)(66 115 76 105)(67 128 77 138)(68 113 78 103)(69 126 79 136)(70 111 80 101)
G:=sub<Sym(160)| (1,11)(2,30)(3,13)(4,32)(5,15)(6,34)(7,17)(8,36)(9,19)(10,38)(12,40)(14,22)(16,24)(18,26)(20,28)(21,31)(23,33)(25,35)(27,37)(29,39)(41,51)(42,62)(43,53)(44,64)(45,55)(46,66)(47,57)(48,68)(49,59)(50,70)(52,72)(54,74)(56,76)(58,78)(60,80)(61,71)(63,73)(65,75)(67,77)(69,79)(81,112)(82,92)(83,114)(84,94)(85,116)(86,96)(87,118)(88,98)(89,120)(90,100)(91,102)(93,104)(95,106)(97,108)(99,110)(101,111)(103,113)(105,115)(107,117)(109,119)(121,131)(122,147)(123,133)(124,149)(125,135)(126,151)(127,137)(128,153)(129,139)(130,155)(132,157)(134,159)(136,141)(138,143)(140,145)(142,152)(144,154)(146,156)(148,158)(150,160), (1,11)(2,12)(3,13)(4,14)(5,15)(6,16)(7,17)(8,18)(9,19)(10,20)(21,31)(22,32)(23,33)(24,34)(25,35)(26,36)(27,37)(28,38)(29,39)(30,40)(41,51)(42,52)(43,53)(44,54)(45,55)(46,56)(47,57)(48,58)(49,59)(50,60)(61,71)(62,72)(63,73)(64,74)(65,75)(66,76)(67,77)(68,78)(69,79)(70,80)(81,91)(82,92)(83,93)(84,94)(85,95)(86,96)(87,97)(88,98)(89,99)(90,100)(101,111)(102,112)(103,113)(104,114)(105,115)(106,116)(107,117)(108,118)(109,119)(110,120)(121,131)(122,132)(123,133)(124,134)(125,135)(126,136)(127,137)(128,138)(129,139)(130,140)(141,151)(142,152)(143,153)(144,154)(145,155)(146,156)(147,157)(148,158)(149,159)(150,160), (1,59)(2,60)(3,41)(4,42)(5,43)(6,44)(7,45)(8,46)(9,47)(10,48)(11,49)(12,50)(13,51)(14,52)(15,53)(16,54)(17,55)(18,56)(19,57)(20,58)(21,71)(22,72)(23,73)(24,74)(25,75)(26,76)(27,77)(28,78)(29,79)(30,80)(31,61)(32,62)(33,63)(34,64)(35,65)(36,66)(37,67)(38,68)(39,69)(40,70)(81,141)(82,142)(83,143)(84,144)(85,145)(86,146)(87,147)(88,148)(89,149)(90,150)(91,151)(92,152)(93,153)(94,154)(95,155)(96,156)(97,157)(98,158)(99,159)(100,160)(101,125)(102,126)(103,127)(104,128)(105,129)(106,130)(107,131)(108,132)(109,133)(110,134)(111,135)(112,136)(113,137)(114,138)(115,139)(116,140)(117,121)(118,122)(119,123)(120,124), (1,39)(2,40)(3,21)(4,22)(5,23)(6,24)(7,25)(8,26)(9,27)(10,28)(11,29)(12,30)(13,31)(14,32)(15,33)(16,34)(17,35)(18,36)(19,37)(20,38)(41,71)(42,72)(43,73)(44,74)(45,75)(46,76)(47,77)(48,78)(49,79)(50,80)(51,61)(52,62)(53,63)(54,64)(55,65)(56,66)(57,67)(58,68)(59,69)(60,70)(81,102)(82,103)(83,104)(84,105)(85,106)(86,107)(87,108)(88,109)(89,110)(90,111)(91,112)(92,113)(93,114)(94,115)(95,116)(96,117)(97,118)(98,119)(99,120)(100,101)(121,156)(122,157)(123,158)(124,159)(125,160)(126,141)(127,142)(128,143)(129,144)(130,145)(131,146)(132,147)(133,148)(134,149)(135,150)(136,151)(137,152)(138,153)(139,154)(140,155), (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), (1,81,11,91)(2,150,12,160)(3,99,13,89)(4,148,14,158)(5,97,15,87)(6,146,16,156)(7,95,17,85)(8,144,18,154)(9,93,19,83)(10,142,20,152)(21,120,31,110)(22,133,32,123)(23,118,33,108)(24,131,34,121)(25,116,35,106)(26,129,36,139)(27,114,37,104)(28,127,38,137)(29,112,39,102)(30,125,40,135)(41,159,51,149)(42,88,52,98)(43,157,53,147)(44,86,54,96)(45,155,55,145)(46,84,56,94)(47,153,57,143)(48,82,58,92)(49,151,59,141)(50,100,60,90)(61,134,71,124)(62,119,72,109)(63,132,73,122)(64,117,74,107)(65,130,75,140)(66,115,76,105)(67,128,77,138)(68,113,78,103)(69,126,79,136)(70,111,80,101)>;
G:=Group( (1,11)(2,30)(3,13)(4,32)(5,15)(6,34)(7,17)(8,36)(9,19)(10,38)(12,40)(14,22)(16,24)(18,26)(20,28)(21,31)(23,33)(25,35)(27,37)(29,39)(41,51)(42,62)(43,53)(44,64)(45,55)(46,66)(47,57)(48,68)(49,59)(50,70)(52,72)(54,74)(56,76)(58,78)(60,80)(61,71)(63,73)(65,75)(67,77)(69,79)(81,112)(82,92)(83,114)(84,94)(85,116)(86,96)(87,118)(88,98)(89,120)(90,100)(91,102)(93,104)(95,106)(97,108)(99,110)(101,111)(103,113)(105,115)(107,117)(109,119)(121,131)(122,147)(123,133)(124,149)(125,135)(126,151)(127,137)(128,153)(129,139)(130,155)(132,157)(134,159)(136,141)(138,143)(140,145)(142,152)(144,154)(146,156)(148,158)(150,160), (1,11)(2,12)(3,13)(4,14)(5,15)(6,16)(7,17)(8,18)(9,19)(10,20)(21,31)(22,32)(23,33)(24,34)(25,35)(26,36)(27,37)(28,38)(29,39)(30,40)(41,51)(42,52)(43,53)(44,54)(45,55)(46,56)(47,57)(48,58)(49,59)(50,60)(61,71)(62,72)(63,73)(64,74)(65,75)(66,76)(67,77)(68,78)(69,79)(70,80)(81,91)(82,92)(83,93)(84,94)(85,95)(86,96)(87,97)(88,98)(89,99)(90,100)(101,111)(102,112)(103,113)(104,114)(105,115)(106,116)(107,117)(108,118)(109,119)(110,120)(121,131)(122,132)(123,133)(124,134)(125,135)(126,136)(127,137)(128,138)(129,139)(130,140)(141,151)(142,152)(143,153)(144,154)(145,155)(146,156)(147,157)(148,158)(149,159)(150,160), (1,59)(2,60)(3,41)(4,42)(5,43)(6,44)(7,45)(8,46)(9,47)(10,48)(11,49)(12,50)(13,51)(14,52)(15,53)(16,54)(17,55)(18,56)(19,57)(20,58)(21,71)(22,72)(23,73)(24,74)(25,75)(26,76)(27,77)(28,78)(29,79)(30,80)(31,61)(32,62)(33,63)(34,64)(35,65)(36,66)(37,67)(38,68)(39,69)(40,70)(81,141)(82,142)(83,143)(84,144)(85,145)(86,146)(87,147)(88,148)(89,149)(90,150)(91,151)(92,152)(93,153)(94,154)(95,155)(96,156)(97,157)(98,158)(99,159)(100,160)(101,125)(102,126)(103,127)(104,128)(105,129)(106,130)(107,131)(108,132)(109,133)(110,134)(111,135)(112,136)(113,137)(114,138)(115,139)(116,140)(117,121)(118,122)(119,123)(120,124), (1,39)(2,40)(3,21)(4,22)(5,23)(6,24)(7,25)(8,26)(9,27)(10,28)(11,29)(12,30)(13,31)(14,32)(15,33)(16,34)(17,35)(18,36)(19,37)(20,38)(41,71)(42,72)(43,73)(44,74)(45,75)(46,76)(47,77)(48,78)(49,79)(50,80)(51,61)(52,62)(53,63)(54,64)(55,65)(56,66)(57,67)(58,68)(59,69)(60,70)(81,102)(82,103)(83,104)(84,105)(85,106)(86,107)(87,108)(88,109)(89,110)(90,111)(91,112)(92,113)(93,114)(94,115)(95,116)(96,117)(97,118)(98,119)(99,120)(100,101)(121,156)(122,157)(123,158)(124,159)(125,160)(126,141)(127,142)(128,143)(129,144)(130,145)(131,146)(132,147)(133,148)(134,149)(135,150)(136,151)(137,152)(138,153)(139,154)(140,155), (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), (1,81,11,91)(2,150,12,160)(3,99,13,89)(4,148,14,158)(5,97,15,87)(6,146,16,156)(7,95,17,85)(8,144,18,154)(9,93,19,83)(10,142,20,152)(21,120,31,110)(22,133,32,123)(23,118,33,108)(24,131,34,121)(25,116,35,106)(26,129,36,139)(27,114,37,104)(28,127,38,137)(29,112,39,102)(30,125,40,135)(41,159,51,149)(42,88,52,98)(43,157,53,147)(44,86,54,96)(45,155,55,145)(46,84,56,94)(47,153,57,143)(48,82,58,92)(49,151,59,141)(50,100,60,90)(61,134,71,124)(62,119,72,109)(63,132,73,122)(64,117,74,107)(65,130,75,140)(66,115,76,105)(67,128,77,138)(68,113,78,103)(69,126,79,136)(70,111,80,101) );
G=PermutationGroup([[(1,11),(2,30),(3,13),(4,32),(5,15),(6,34),(7,17),(8,36),(9,19),(10,38),(12,40),(14,22),(16,24),(18,26),(20,28),(21,31),(23,33),(25,35),(27,37),(29,39),(41,51),(42,62),(43,53),(44,64),(45,55),(46,66),(47,57),(48,68),(49,59),(50,70),(52,72),(54,74),(56,76),(58,78),(60,80),(61,71),(63,73),(65,75),(67,77),(69,79),(81,112),(82,92),(83,114),(84,94),(85,116),(86,96),(87,118),(88,98),(89,120),(90,100),(91,102),(93,104),(95,106),(97,108),(99,110),(101,111),(103,113),(105,115),(107,117),(109,119),(121,131),(122,147),(123,133),(124,149),(125,135),(126,151),(127,137),(128,153),(129,139),(130,155),(132,157),(134,159),(136,141),(138,143),(140,145),(142,152),(144,154),(146,156),(148,158),(150,160)], [(1,11),(2,12),(3,13),(4,14),(5,15),(6,16),(7,17),(8,18),(9,19),(10,20),(21,31),(22,32),(23,33),(24,34),(25,35),(26,36),(27,37),(28,38),(29,39),(30,40),(41,51),(42,52),(43,53),(44,54),(45,55),(46,56),(47,57),(48,58),(49,59),(50,60),(61,71),(62,72),(63,73),(64,74),(65,75),(66,76),(67,77),(68,78),(69,79),(70,80),(81,91),(82,92),(83,93),(84,94),(85,95),(86,96),(87,97),(88,98),(89,99),(90,100),(101,111),(102,112),(103,113),(104,114),(105,115),(106,116),(107,117),(108,118),(109,119),(110,120),(121,131),(122,132),(123,133),(124,134),(125,135),(126,136),(127,137),(128,138),(129,139),(130,140),(141,151),(142,152),(143,153),(144,154),(145,155),(146,156),(147,157),(148,158),(149,159),(150,160)], [(1,59),(2,60),(3,41),(4,42),(5,43),(6,44),(7,45),(8,46),(9,47),(10,48),(11,49),(12,50),(13,51),(14,52),(15,53),(16,54),(17,55),(18,56),(19,57),(20,58),(21,71),(22,72),(23,73),(24,74),(25,75),(26,76),(27,77),(28,78),(29,79),(30,80),(31,61),(32,62),(33,63),(34,64),(35,65),(36,66),(37,67),(38,68),(39,69),(40,70),(81,141),(82,142),(83,143),(84,144),(85,145),(86,146),(87,147),(88,148),(89,149),(90,150),(91,151),(92,152),(93,153),(94,154),(95,155),(96,156),(97,157),(98,158),(99,159),(100,160),(101,125),(102,126),(103,127),(104,128),(105,129),(106,130),(107,131),(108,132),(109,133),(110,134),(111,135),(112,136),(113,137),(114,138),(115,139),(116,140),(117,121),(118,122),(119,123),(120,124)], [(1,39),(2,40),(3,21),(4,22),(5,23),(6,24),(7,25),(8,26),(9,27),(10,28),(11,29),(12,30),(13,31),(14,32),(15,33),(16,34),(17,35),(18,36),(19,37),(20,38),(41,71),(42,72),(43,73),(44,74),(45,75),(46,76),(47,77),(48,78),(49,79),(50,80),(51,61),(52,62),(53,63),(54,64),(55,65),(56,66),(57,67),(58,68),(59,69),(60,70),(81,102),(82,103),(83,104),(84,105),(85,106),(86,107),(87,108),(88,109),(89,110),(90,111),(91,112),(92,113),(93,114),(94,115),(95,116),(96,117),(97,118),(98,119),(99,120),(100,101),(121,156),(122,157),(123,158),(124,159),(125,160),(126,141),(127,142),(128,143),(129,144),(130,145),(131,146),(132,147),(133,148),(134,149),(135,150),(136,151),(137,152),(138,153),(139,154),(140,155)], [(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)], [(1,81,11,91),(2,150,12,160),(3,99,13,89),(4,148,14,158),(5,97,15,87),(6,146,16,156),(7,95,17,85),(8,144,18,154),(9,93,19,83),(10,142,20,152),(21,120,31,110),(22,133,32,123),(23,118,33,108),(24,131,34,121),(25,116,35,106),(26,129,36,139),(27,114,37,104),(28,127,38,137),(29,112,39,102),(30,125,40,135),(41,159,51,149),(42,88,52,98),(43,157,53,147),(44,86,54,96),(45,155,55,145),(46,84,56,94),(47,153,57,143),(48,82,58,92),(49,151,59,141),(50,100,60,90),(61,134,71,124),(62,119,72,109),(63,132,73,122),(64,117,74,107),(65,130,75,140),(66,115,76,105),(67,128,77,138),(68,113,78,103),(69,126,79,136),(70,111,80,101)]])
68 conjugacy classes
| class | 1 | 2A | ··· | 2G | 2H | 2I | 2J | 2K | 4A | 4B | 4C | 4D | 4E | ··· | 4L | 4M | 4N | 4O | 4P | 5A | 5B | 10A | ··· | 10N | 10O | ··· | 10V | 20A | ··· | 20P |
| order | 1 | 2 | ··· | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 10 | ··· | 10 | 10 | ··· | 10 | 20 | ··· | 20 |
| size | 1 | 1 | ··· | 1 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 10 | ··· | 10 | 20 | 20 | 20 | 20 | 2 | 2 | 2 | ··· | 2 | 4 | ··· | 4 | 4 | ··· | 4 |
68 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | - | + | + | + | - | + | - | ||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C4 | D4 | D4 | Q8 | D5 | C4○D4 | D10 | D10 | Dic10 | C4×D5 | C5⋊D4 | D4×D5 | D4⋊2D5 |
| kernel | C24.44D10 | C10.10C42 | C2×C10.D4 | C2×C23.D5 | C10×C22⋊C4 | C23×Dic5 | C22×Dic5 | C2×Dic5 | C22×C10 | C22×C10 | C2×C22⋊C4 | C2×C10 | C22×C4 | C24 | C23 | C23 | C23 | C22 | C22 |
| # reps | 1 | 2 | 2 | 1 | 1 | 1 | 8 | 4 | 2 | 2 | 2 | 4 | 4 | 2 | 8 | 8 | 8 | 4 | 4 |
Matrix representation of C24.44D10 ►in GL5(𝔽41)
| 1 | 0 | 0 | 0 | 0 |
| 0 | 1 | 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 | 1 | 0 |
| 0 | 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 40 | 0 | 0 | 0 |
| 0 | 0 | 40 | 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 |
| 32 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 3 | 4 |
| 0 | 0 | 0 | 20 | 40 |
| 9 | 0 | 0 | 0 | 0 |
| 0 | 0 | 32 | 0 | 0 |
| 0 | 9 | 0 | 0 | 0 |
| 0 | 0 | 0 | 7 | 8 |
| 0 | 0 | 0 | 35 | 34 |
G:=sub<GL(5,GF(41))| [1,0,0,0,0,0,1,0,0,0,0,0,40,0,0,0,0,0,1,0,0,0,0,0,1],[40,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1],[1,0,0,0,0,0,40,0,0,0,0,0,40,0,0,0,0,0,40,0,0,0,0,0,40],[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],[32,0,0,0,0,0,0,1,0,0,0,1,0,0,0,0,0,0,3,20,0,0,0,4,40],[9,0,0,0,0,0,0,9,0,0,0,32,0,0,0,0,0,0,7,35,0,0,0,8,34] >;
C24.44D10 in GAP, Magma, Sage, TeX
C_2^4._{44}D_{10} % in TeX
G:=Group("C2^4.44D10"); // GroupNames label
G:=SmallGroup(320,569);
// by ID
G=gap.SmallGroup(320,569);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,112,422,387,58,12550]);
// Polycyclic
G:=Group<a,b,c,d,e,f|a^2=b^2=c^2=d^2=1,e^10=f^2=b,a*b=b*a,a*c=c*a,e*a*e^-1=f*a*f^-1=a*d=d*a,b*c=c*b,b*d=d*b,b*e=e*b,b*f=f*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^9>;
// generators/relations