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G = C24.6D10order 320 = 26·5

6th non-split extension by C24 of D10 acting via D10/C5=C22

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C24.6D10, C23.4Dic10, (C2×C20).49D4, C2.6(C202D4), (C2×Dic5).62D4, C22.238(D4×D5), (C22×C4).28D10, (C22×C10).12Q8, C10.29(C4⋊D4), C52(C23.Q8), C10.56(C22⋊Q8), C2.32(D10⋊D4), C2.8(Dic5⋊D4), C2.7(C20.48D4), C22.95(C4○D20), (C22×C20).57C22, (C23×C10).31C22, C22.45(C2×Dic10), C23.367(C22×D5), C10.10C4213C2, C10.13(C422C2), C22.93(D42D5), (C22×C10).323C23, C2.12(C23.D10), (C22×Dic5).39C22, C2.21(Dic5.14D4), (C2×C4⋊Dic5)⋊8C2, (C2×C10).33(C2×Q8), (C2×C10).317(C2×D4), (C2×C4).28(C5⋊D4), (C2×C22⋊C4).11D5, (C2×C10).77(C4○D4), (C2×C10.D4)⋊20C2, (C10×C22⋊C4).13C2, C22.123(C2×C5⋊D4), (C2×C23.D5).10C2, SmallGroup(320,575)

Series: Derived Chief Lower central Upper central

C1C22×C10 — C24.6D10
C1C5C10C2×C10C22×C10C22×Dic5C2×C10.D4 — C24.6D10
C5C22×C10 — C24.6D10
C1C23C2×C22⋊C4

Generators and relations for C24.6D10
 G = < a,b,c,d,e,f | a2=b2=c2=d2=1, e10=f2=dc=cd, ab=ba, ac=ca, eae-1=ad=da, faf-1=abd, bc=cb, bd=db, be=eb, bf=fb, ce=ec, cf=fc, de=ed, df=fd, fef-1=ce9 >

Subgroups: 614 in 186 conjugacy classes, 63 normal (51 characteristic)
C1, C2 [×7], C2 [×2], C4 [×9], C22 [×7], C22 [×10], C5, C2×C4 [×2], C2×C4 [×19], C23, C23 [×2], C23 [×6], C10 [×7], C10 [×2], C22⋊C4 [×6], C4⋊C4 [×6], C22×C4 [×2], C22×C4 [×4], C24, Dic5 [×6], C20 [×3], C2×C10 [×7], C2×C10 [×10], C2.C42, C2×C22⋊C4, C2×C22⋊C4 [×2], C2×C4⋊C4 [×3], C2×Dic5 [×4], C2×Dic5 [×10], C2×C20 [×2], C2×C20 [×5], C22×C10, C22×C10 [×2], C22×C10 [×6], C23.Q8, C10.D4 [×4], C4⋊Dic5 [×2], C23.D5 [×4], C5×C22⋊C4 [×2], C22×Dic5 [×4], C22×C20 [×2], C23×C10, C10.10C42, C2×C10.D4 [×2], C2×C4⋊Dic5, C2×C23.D5 [×2], C10×C22⋊C4, C24.6D10
Quotients: C1, C2 [×7], C22 [×7], D4 [×6], Q8 [×2], C23, D5, C2×D4 [×3], C2×Q8, C4○D4 [×3], D10 [×3], C4⋊D4 [×3], C22⋊Q8 [×3], C422C2, Dic10 [×2], C5⋊D4 [×2], C22×D5, C23.Q8, C2×Dic10, C4○D20, D4×D5 [×2], D42D5 [×2], C2×C5⋊D4, Dic5.14D4 [×2], C23.D10, D10⋊D4, C20.48D4, C202D4, Dic5⋊D4, C24.6D10

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

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

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

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

62 conjugacy classes

class 1 2A···2G2H2I4A4B4C4D4E···4L5A5B10A···10N10O···10V20A···20P
order12···22244444···45510···1010···1020···20
size11···144444420···20222···24···44···4

62 irreducible representations

dim111111222222222244
type++++++++-+++-+-
imageC1C2C2C2C2C2D4D4Q8D5C4○D4D10D10C5⋊D4Dic10C4○D20D4×D5D42D5
kernelC24.6D10C10.10C42C2×C10.D4C2×C4⋊Dic5C2×C23.D5C10×C22⋊C4C2×Dic5C2×C20C22×C10C2×C22⋊C4C2×C10C22×C4C24C2×C4C23C22C22C22
# reps112121422264288844

Matrix representation of C24.6D10 in GL6(𝔽41)

100000
010000
001000
00244000
000010
00001240
,
100000
010000
0040000
0004000
0000400
0000040
,
4000000
0400000
001000
000100
0000400
0000040
,
100000
010000
001000
000100
0000400
0000040
,
2000000
16390000
0025000
00242300
00001239
00001029
,
26360000
37150000
00402400
000100
0000400
0000291

G:=sub<GL(6,GF(41))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,24,0,0,0,0,0,40,0,0,0,0,0,0,1,12,0,0,0,0,0,40],[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],[40,0,0,0,0,0,0,40,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,40,0,0,0,0,0,0,40],[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],[20,16,0,0,0,0,0,39,0,0,0,0,0,0,25,24,0,0,0,0,0,23,0,0,0,0,0,0,12,10,0,0,0,0,39,29],[26,37,0,0,0,0,36,15,0,0,0,0,0,0,40,0,0,0,0,0,24,1,0,0,0,0,0,0,40,29,0,0,0,0,0,1] >;

C24.6D10 in GAP, Magma, Sage, TeX

C_2^4._6D_{10}
% in TeX

G:=Group("C2^4.6D10");
// GroupNames label

G:=SmallGroup(320,575);
// by ID

G=gap.SmallGroup(320,575);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,112,253,344,254,387,12550]);
// Polycyclic

G:=Group<a,b,c,d,e,f|a^2=b^2=c^2=d^2=1,e^10=f^2=d*c=c*d,a*b=b*a,a*c=c*a,e*a*e^-1=a*d=d*a,f*a*f^-1=a*b*d,b*c=c*b,b*d=d*b,b*e=e*b,b*f=f*b,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

׿
×
𝔽