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G = C2×C20.48D4order 320 = 26·5

Direct product of C2 and C20.48D4

direct product, metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C2×C20.48D4, C234Dic10, C24.67D10, (C22×C10)⋊7Q8, C20.424(C2×D4), (C2×C20).477D4, C104(C22⋊Q8), (C23×C4).10D5, (C23×C20).12C2, C4⋊Dic563C22, C223(C2×Dic10), C10.19(C22×Q8), (C2×C20).703C23, (C2×C10).282C24, (C22×C4).446D10, C10.130(C22×D4), (C2×Dic10)⋊58C22, (C22×Dic10)⋊12C2, C22.79(C4○D20), C10.D443C22, C2.20(C22×Dic10), C22.301(C23×D5), C23.231(C22×D5), (C22×C10).411C23, (C23×C10).104C22, (C22×C20).528C22, (C2×Dic5).148C23, C23.D5.129C22, (C22×Dic5).160C22, C55(C2×C22⋊Q8), (C2×C10)⋊6(C2×Q8), (C2×C4⋊Dic5)⋊28C2, C2.69(C2×C4○D20), C10.59(C2×C4○D4), C4.120(C2×C5⋊D4), C2.5(C22×C5⋊D4), (C2×C10).571(C2×D4), (C2×C10.D4)⋊17C2, (C2×C4).262(C5⋊D4), (C2×C4).656(C22×D5), C22.100(C2×C5⋊D4), (C2×C23.D5).23C2, (C2×C10).110(C4○D4), SmallGroup(320,1456)

Series: Derived Chief Lower central Upper central

C1C2×C10 — C2×C20.48D4
C1C5C10C2×C10C2×Dic5C22×Dic5C22×Dic10 — C2×C20.48D4
C5C2×C10 — C2×C20.48D4
C1C23C23×C4

Generators and relations for C2×C20.48D4
 G = < a,b,c,d | a2=b20=c4=1, d2=b10, ab=ba, ac=ca, ad=da, cbc-1=dbd-1=b-1, dcd-1=b10c-1 >

Subgroups: 878 in 322 conjugacy classes, 143 normal (23 characteristic)
C1, C2, C2, C2, C4, C4, C22, C22, C22, C5, C2×C4, C2×C4, Q8, C23, C23, C23, C10, C10, C10, C22⋊C4, C4⋊C4, C22×C4, C22×C4, C22×C4, C2×Q8, C24, Dic5, C20, C20, C2×C10, C2×C10, C2×C10, C2×C22⋊C4, C2×C4⋊C4, C22⋊Q8, C23×C4, C22×Q8, Dic10, C2×Dic5, C2×Dic5, C2×C20, C2×C20, C22×C10, C22×C10, C22×C10, C2×C22⋊Q8, C10.D4, C4⋊Dic5, C23.D5, C2×Dic10, C2×Dic10, C22×Dic5, C22×C20, C22×C20, C22×C20, C23×C10, C2×C10.D4, C20.48D4, C2×C4⋊Dic5, C2×C23.D5, C22×Dic10, C23×C20, C2×C20.48D4
Quotients: C1, C2, C22, D4, Q8, C23, D5, C2×D4, C2×Q8, C4○D4, C24, D10, C22⋊Q8, C22×D4, C22×Q8, C2×C4○D4, Dic10, C5⋊D4, C22×D5, C2×C22⋊Q8, C2×Dic10, C4○D20, C2×C5⋊D4, C23×D5, C20.48D4, C22×Dic10, C2×C4○D20, C22×C5⋊D4, C2×C20.48D4

Smallest permutation representation of C2×C20.48D4
On 160 points
Generators in S160
(1 135)(2 136)(3 137)(4 138)(5 139)(6 140)(7 121)(8 122)(9 123)(10 124)(11 125)(12 126)(13 127)(14 128)(15 129)(16 130)(17 131)(18 132)(19 133)(20 134)(21 48)(22 49)(23 50)(24 51)(25 52)(26 53)(27 54)(28 55)(29 56)(30 57)(31 58)(32 59)(33 60)(34 41)(35 42)(36 43)(37 44)(38 45)(39 46)(40 47)(61 107)(62 108)(63 109)(64 110)(65 111)(66 112)(67 113)(68 114)(69 115)(70 116)(71 117)(72 118)(73 119)(74 120)(75 101)(76 102)(77 103)(78 104)(79 105)(80 106)(81 153)(82 154)(83 155)(84 156)(85 157)(86 158)(87 159)(88 160)(89 141)(90 142)(91 143)(92 144)(93 145)(94 146)(95 147)(96 148)(97 149)(98 150)(99 151)(100 152)
(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)
(1 117 48 159)(2 116 49 158)(3 115 50 157)(4 114 51 156)(5 113 52 155)(6 112 53 154)(7 111 54 153)(8 110 55 152)(9 109 56 151)(10 108 57 150)(11 107 58 149)(12 106 59 148)(13 105 60 147)(14 104 41 146)(15 103 42 145)(16 102 43 144)(17 101 44 143)(18 120 45 142)(19 119 46 141)(20 118 47 160)(21 87 135 71)(22 86 136 70)(23 85 137 69)(24 84 138 68)(25 83 139 67)(26 82 140 66)(27 81 121 65)(28 100 122 64)(29 99 123 63)(30 98 124 62)(31 97 125 61)(32 96 126 80)(33 95 127 79)(34 94 128 78)(35 93 129 77)(36 92 130 76)(37 91 131 75)(38 90 132 74)(39 89 133 73)(40 88 134 72)
(1 149 11 159)(2 148 12 158)(3 147 13 157)(4 146 14 156)(5 145 15 155)(6 144 16 154)(7 143 17 153)(8 142 18 152)(9 141 19 151)(10 160 20 150)(21 61 31 71)(22 80 32 70)(23 79 33 69)(24 78 34 68)(25 77 35 67)(26 76 36 66)(27 75 37 65)(28 74 38 64)(29 73 39 63)(30 72 40 62)(41 114 51 104)(42 113 52 103)(43 112 53 102)(44 111 54 101)(45 110 55 120)(46 109 56 119)(47 108 57 118)(48 107 58 117)(49 106 59 116)(50 105 60 115)(81 121 91 131)(82 140 92 130)(83 139 93 129)(84 138 94 128)(85 137 95 127)(86 136 96 126)(87 135 97 125)(88 134 98 124)(89 133 99 123)(90 132 100 122)

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

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

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

92 conjugacy classes

class 1 2A···2G2H2I2J2K4A···4H4I···4P5A5B10A···10AD20A···20AF
order12···222224···44···45510···1020···20
size11···122222···220···20222···22···2

92 irreducible representations

dim1111111222222222
type++++++++-+++-
imageC1C2C2C2C2C2C2D4Q8D5C4○D4D10D10C5⋊D4Dic10C4○D20
kernelC2×C20.48D4C2×C10.D4C20.48D4C2×C4⋊Dic5C2×C23.D5C22×Dic10C23×C20C2×C20C22×C10C23×C4C2×C10C22×C4C24C2×C4C23C22
# reps12812114424122161616

Matrix representation of C2×C20.48D4 in GL5(𝔽41)

400000
01000
00100
000400
000040
,
400000
04000
003100
00001
000400
,
400000
00100
040000
000032
000320
,
10000
00100
01000
000032
000320

G:=sub<GL(5,GF(41))| [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],[40,0,0,0,0,0,4,0,0,0,0,0,31,0,0,0,0,0,0,40,0,0,0,1,0],[40,0,0,0,0,0,0,40,0,0,0,1,0,0,0,0,0,0,0,32,0,0,0,32,0],[1,0,0,0,0,0,0,1,0,0,0,1,0,0,0,0,0,0,0,32,0,0,0,32,0] >;

C2×C20.48D4 in GAP, Magma, Sage, TeX

C_2\times C_{20}._{48}D_4
% in TeX

G:=Group("C2xC20.48D4");
// GroupNames label

G:=SmallGroup(320,1456);
// by ID

G=gap.SmallGroup(320,1456);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,224,758,184,675,12550]);
// Polycyclic

G:=Group<a,b,c,d|a^2=b^20=c^4=1,d^2=b^10,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c^-1=d*b*d^-1=b^-1,d*c*d^-1=b^10*c^-1>;
// generators/relations

׿
×
𝔽