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G = C2×C20.10D4order 320 = 26·5

Direct product of C2 and C20.10D4

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

Aliases: C2×C20.10D4, C20.209(C2×D4), (C2×C20).194D4, (Q8×C10).20C4, (C2×Q8).6Dic5, (C22×Q8).3D5, (C22×C20).38C4, (C2×Q8).167D10, C103(C4.10D4), (C22×C4).5Dic5, C20.82(C22⋊C4), (C2×C20).476C23, (C22×C4).155D10, C4.12(C23.D5), C23.33(C2×Dic5), (Q8×C10).202C22, C4.Dic5.46C22, C22.7(C22×Dic5), (C22×C20).202C22, C22.36(C23.D5), (Q8×C2×C10).3C2, C55(C2×C4.10D4), C4.93(C2×C5⋊D4), (C2×C20).294(C2×C4), (C2×C4).26(C2×Dic5), C2.15(C2×C23.D5), (C2×C4).199(C5⋊D4), C10.120(C2×C22⋊C4), (C2×C4).130(C22×D5), (C2×C4.Dic5).28C2, (C2×C10).299(C22×C4), (C22×C10).205(C2×C4), (C2×C10).180(C22⋊C4), SmallGroup(320,853)

Series: Derived Chief Lower central Upper central

C1C2×C10 — C2×C20.10D4
C1C5C10C20C2×C20C4.Dic5C2×C4.Dic5 — C2×C20.10D4
C5C10C2×C10 — C2×C20.10D4
C1C22C22×C4C22×Q8

Generators and relations for C2×C20.10D4
 G = < a,b,c,d | a2=b20=1, c4=b10, d2=b5, ab=ba, ac=ca, ad=da, cbc-1=b-1, dbd-1=b9, dcd-1=b5c3 >

Subgroups: 318 in 146 conjugacy classes, 71 normal (21 characteristic)
C1, C2, C2 [×2], C2 [×2], C4 [×4], C4 [×4], C22 [×3], C22 [×2], C5, C8 [×4], C2×C4 [×2], C2×C4 [×8], C2×C4 [×4], Q8 [×8], C23, C10, C10 [×2], C10 [×2], C2×C8 [×2], M4(2) [×6], C22×C4, C22×C4 [×2], C2×Q8 [×4], C2×Q8 [×4], C20 [×4], C20 [×4], C2×C10 [×3], C2×C10 [×2], C4.10D4 [×4], C2×M4(2) [×2], C22×Q8, C52C8 [×4], C2×C20 [×2], C2×C20 [×8], C2×C20 [×4], C5×Q8 [×8], C22×C10, C2×C4.10D4, C2×C52C8 [×2], C4.Dic5 [×4], C4.Dic5 [×2], C22×C20, C22×C20 [×2], Q8×C10 [×4], Q8×C10 [×4], C20.10D4 [×4], C2×C4.Dic5 [×2], Q8×C2×C10, C2×C20.10D4
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C2×C4 [×6], D4 [×4], C23, D5, C22⋊C4 [×4], C22×C4, C2×D4 [×2], Dic5 [×4], D10 [×3], C4.10D4 [×2], C2×C22⋊C4, C2×Dic5 [×6], C5⋊D4 [×4], C22×D5, C2×C4.10D4, C23.D5 [×4], C22×Dic5, C2×C5⋊D4 [×2], C20.10D4 [×2], C2×C23.D5, C2×C20.10D4

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

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

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

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

62 conjugacy classes

class 1 2A2B2C2D2E4A4B4C4D4E4F4G4H5A5B8A···8H10A···10N20A···20X
order12222244444444558···810···1020···20
size111122222244442220···202···24···4

62 irreducible representations

dim111111222222244
type++++++-+-+-
imageC1C2C2C2C4C4D4D5Dic5D10Dic5D10C5⋊D4C4.10D4C20.10D4
kernelC2×C20.10D4C20.10D4C2×C4.Dic5Q8×C2×C10C22×C20Q8×C10C2×C20C22×Q8C22×C4C22×C4C2×Q8C2×Q8C2×C4C10C2
# reps1421444242441628

Matrix representation of C2×C20.10D4 in GL6(𝔽41)

4000000
0400000
0040000
0004000
0000400
0000040
,
610000
4000000
0003700
004000
00195031
00014100
,
5330000
3360000
00934360
0002036
0089327
00339039
,
3680000
3850000
0016302326
0027212618
00154538
002393540

G:=sub<GL(6,GF(41))| [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],[6,40,0,0,0,0,1,0,0,0,0,0,0,0,0,4,19,0,0,0,37,0,5,14,0,0,0,0,0,10,0,0,0,0,31,0],[5,3,0,0,0,0,33,36,0,0,0,0,0,0,9,0,8,33,0,0,34,2,9,9,0,0,36,0,32,0,0,0,0,36,7,39],[36,38,0,0,0,0,8,5,0,0,0,0,0,0,16,27,15,23,0,0,30,21,4,9,0,0,23,26,5,35,0,0,26,18,38,40] >;

C2×C20.10D4 in GAP, Magma, Sage, TeX

C_2\times C_{20}._{10}D_4
% in TeX

G:=Group("C2xC20.10D4");
// GroupNames label

G:=SmallGroup(320,853);
// by ID

G=gap.SmallGroup(320,853);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,56,422,184,297,136,1684,12550]);
// Polycyclic

G:=Group<a,b,c,d|a^2=b^20=1,c^4=b^10,d^2=b^5,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c^-1=b^-1,d*b*d^-1=b^9,d*c*d^-1=b^5*c^3>;
// generators/relations

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