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G = C24.4D10order 320 = 26·5

4th non-split extension by C24 of D10 acting via D10/C5=C22

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C24.4D10, C10.72(C4×D4), (C2×C20).248D4, C23.D515C4, C22.96(D4×D5), C23.16(C4×D5), (C22×C4).25D10, C10.83(C4⋊D4), (C2×Dic5).228D4, C2.2(Dic5⋊D4), C2.1(C20.17D4), C10.37(C4.4D4), C22.51(C4○D20), (C23×C10).28C22, C56(C24.C22), C23.278(C22×D5), C10.10C4211C2, C10.46(C42⋊C2), C10.12(C422C2), C2.25(Dic54D4), C2.5(D10.12D4), C22.44(D42D5), (C22×C10).320C23, (C22×C20).342C22, C2.6(C23.D10), C10.28(C22.D4), (C22×Dic5).36C22, C2.14(C23.11D10), C2.8(C4×C5⋊D4), (C2×C4×Dic5)⋊22C2, (C2×C22⋊C4).7D5, C22.124(C2×C4×D5), (C2×C10).430(C2×D4), (C2×C4).98(C5⋊D4), C22.48(C2×C5⋊D4), (C2×C23.D5).7C2, (C2×C10).75(C4○D4), (C2×C10.D4)⋊32C2, (C10×C22⋊C4).24C2, (C2×C10).207(C22×C4), (C22×C10).116(C2×C4), (C2×Dic5).104(C2×C4), SmallGroup(320,572)

Series: Derived Chief Lower central Upper central

C1C2×C10 — C24.4D10
C1C5C10C2×C10C22×C10C22×Dic5C2×C4×Dic5 — C24.4D10
C5C2×C10 — C24.4D10
C1C23C2×C22⋊C4

Generators and relations for C24.4D10
 G = < a,b,c,d,e,f | a2=b2=c2=d2=1, e10=b, f2=db=bd, ab=ba, eae-1=ac=ca, ad=da, faf-1=abcd, bc=cb, be=eb, bf=fb, cd=dc, ce=ec, cf=fc, de=ed, df=fd, fef-1=e9 >

Subgroups: 590 in 190 conjugacy classes, 69 normal (51 characteristic)
C1, C2 [×7], C2 [×2], C4 [×10], C22 [×7], C22 [×10], C5, C2×C4 [×2], C2×C4 [×20], C23, C23 [×2], C23 [×6], C10 [×7], C10 [×2], C42 [×2], C22⋊C4 [×8], C4⋊C4 [×2], C22×C4 [×2], C22×C4 [×4], C24, Dic5 [×7], C20 [×3], C2×C10 [×7], C2×C10 [×10], C2.C42 [×2], C2×C42, C2×C22⋊C4, C2×C22⋊C4 [×2], C2×C4⋊C4, C2×Dic5 [×6], C2×Dic5 [×9], C2×C20 [×2], C2×C20 [×5], C22×C10, C22×C10 [×2], C22×C10 [×6], C24.C22, C4×Dic5 [×2], C10.D4 [×2], C23.D5 [×4], C23.D5 [×2], C5×C22⋊C4 [×2], C22×Dic5 [×4], C22×C20 [×2], C23×C10, C10.10C42 [×2], C2×C4×Dic5, C2×C10.D4, C2×C23.D5 [×2], C10×C22⋊C4, C24.4D10
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C2×C4 [×6], D4 [×4], C23, D5, C22×C4, C2×D4 [×2], C4○D4 [×4], D10 [×3], C42⋊C2, C4×D4 [×2], C4⋊D4, C22.D4, C4.4D4, C422C2, C4×D5 [×2], C5⋊D4 [×2], C22×D5, C24.C22, C2×C4×D5, C4○D20, D4×D5, D42D5 [×3], C2×C5⋊D4, C23.11D10, C23.D10, Dic54D4, D10.12D4, C4×C5⋊D4, C20.17D4, Dic5⋊D4, C24.4D10

Smallest permutation representation of C24.4D10
On 160 points
Generators in S160
(2 90)(4 92)(6 94)(8 96)(10 98)(12 100)(14 82)(16 84)(18 86)(20 88)(22 158)(24 160)(26 142)(28 144)(30 146)(32 148)(34 150)(36 152)(38 154)(40 156)(41 65)(42 124)(43 67)(44 126)(45 69)(46 128)(47 71)(48 130)(49 73)(50 132)(51 75)(52 134)(53 77)(54 136)(55 79)(56 138)(57 61)(58 140)(59 63)(60 122)(62 103)(64 105)(66 107)(68 109)(70 111)(72 113)(74 115)(76 117)(78 119)(80 101)(102 139)(104 121)(106 123)(108 125)(110 127)(112 129)(114 131)(116 133)(118 135)(120 137)
(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 89)(2 90)(3 91)(4 92)(5 93)(6 94)(7 95)(8 96)(9 97)(10 98)(11 99)(12 100)(13 81)(14 82)(15 83)(16 84)(17 85)(18 86)(19 87)(20 88)(21 157)(22 158)(23 159)(24 160)(25 141)(26 142)(27 143)(28 144)(29 145)(30 146)(31 147)(32 148)(33 149)(34 150)(35 151)(36 152)(37 153)(38 154)(39 155)(40 156)(41 106)(42 107)(43 108)(44 109)(45 110)(46 111)(47 112)(48 113)(49 114)(50 115)(51 116)(52 117)(53 118)(54 119)(55 120)(56 101)(57 102)(58 103)(59 104)(60 105)(61 139)(62 140)(63 121)(64 122)(65 123)(66 124)(67 125)(68 126)(69 127)(70 128)(71 129)(72 130)(73 131)(74 132)(75 133)(76 134)(77 135)(78 136)(79 137)(80 138)
(1 33)(2 34)(3 35)(4 36)(5 37)(6 38)(7 39)(8 40)(9 21)(10 22)(11 23)(12 24)(13 25)(14 26)(15 27)(16 28)(17 29)(18 30)(19 31)(20 32)(41 133)(42 134)(43 135)(44 136)(45 137)(46 138)(47 139)(48 140)(49 121)(50 122)(51 123)(52 124)(53 125)(54 126)(55 127)(56 128)(57 129)(58 130)(59 131)(60 132)(61 112)(62 113)(63 114)(64 115)(65 116)(66 117)(67 118)(68 119)(69 120)(70 101)(71 102)(72 103)(73 104)(74 105)(75 106)(76 107)(77 108)(78 109)(79 110)(80 111)(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)
(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 73 23 114)(2 62 24 103)(3 71 25 112)(4 80 26 101)(5 69 27 110)(6 78 28 119)(7 67 29 108)(8 76 30 117)(9 65 31 106)(10 74 32 115)(11 63 33 104)(12 72 34 113)(13 61 35 102)(14 70 36 111)(15 79 37 120)(16 68 38 109)(17 77 39 118)(18 66 40 107)(19 75 21 116)(20 64 22 105)(41 97 123 147)(42 86 124 156)(43 95 125 145)(44 84 126 154)(45 93 127 143)(46 82 128 152)(47 91 129 141)(48 100 130 150)(49 89 131 159)(50 98 132 148)(51 87 133 157)(52 96 134 146)(53 85 135 155)(54 94 136 144)(55 83 137 153)(56 92 138 142)(57 81 139 151)(58 90 140 160)(59 99 121 149)(60 88 122 158)

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

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

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

68 conjugacy classes

class 1 2A···2G2H2I4A4B4C4D4E4F4G···4N4O4P4Q4R5A5B10A···10N10O···10V20A···20P
order12···2224444444···444445510···1010···1020···20
size11···14422224410···1020202020222···24···44···4

68 irreducible representations

dim111111122222222244
type++++++++++++-
imageC1C2C2C2C2C2C4D4D4D5C4○D4D10D10C5⋊D4C4×D5C4○D20D4×D5D42D5
kernelC24.4D10C10.10C42C2×C4×Dic5C2×C10.D4C2×C23.D5C10×C22⋊C4C23.D5C2×Dic5C2×C20C2×C22⋊C4C2×C10C22×C4C24C2×C4C23C22C22C22
# reps121121822284288826

Matrix representation of C24.4D10 in GL6(𝔽41)

100000
0400000
001000
0004000
000010
00002140
,
4000000
0400000
0040000
0004000
0000400
0000040
,
4000000
0400000
0040000
0004000
000010
000001
,
100000
010000
0040000
0004000
000010
000001
,
010000
4000000
000100
0040000
0000360
00001133
,
900000
090000
0003200
009000
0000196
0000822

G:=sub<GL(6,GF(41))| [1,0,0,0,0,0,0,40,0,0,0,0,0,0,1,0,0,0,0,0,0,40,0,0,0,0,0,0,1,21,0,0,0,0,0,40],[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],[40,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[0,40,0,0,0,0,1,0,0,0,0,0,0,0,0,40,0,0,0,0,1,0,0,0,0,0,0,0,36,11,0,0,0,0,0,33],[9,0,0,0,0,0,0,9,0,0,0,0,0,0,0,9,0,0,0,0,32,0,0,0,0,0,0,0,19,8,0,0,0,0,6,22] >;

C24.4D10 in GAP, Magma, Sage, TeX

C_2^4._4D_{10}
% in TeX

G:=Group("C2^4.4D10");
// GroupNames label

G:=SmallGroup(320,572);
// by ID

G=gap.SmallGroup(320,572);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,224,64,926,219,12550]);
// Polycyclic

G:=Group<a,b,c,d,e,f|a^2=b^2=c^2=d^2=1,e^10=b,f^2=d*b=b*d,a*b=b*a,e*a*e^-1=a*c=c*a,a*d=d*a,f*a*f^-1=a*b*c*d,b*c=c*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=e^9>;
// generators/relations

׿
×
𝔽