Copied to
clipboard

G = (C2×C10).40D8order 320 = 26·5

17th non-split extension by C2×C10 of D8 acting via D8/D4=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: (C2×C10).40D8, C10.49(C2×D8), C4⋊C4.227D10, C207D4.9C2, (C2×C20).283D4, D206C425C2, C4.86(C4○D20), C20.55D44C2, C10.D825C2, C22.9(D4⋊D5), (C22×C4).94D10, C54(C22.D8), C20.174(C4○D4), (C2×C20).320C23, (C2×D20).94C22, (C22×C10).185D4, C23.77(C5⋊D4), C2.6(C20.C23), C10.84(C8.C22), C4⋊Dic5.130C22, (C22×C20).135C22, C2.8(C23.23D10), C10.58(C22.D4), (C2×C4⋊C4)⋊3D5, (C10×C4⋊C4)⋊3C2, C2.5(C2×D4⋊D5), (C2×C10).440(C2×D4), (C2×C4).31(C5⋊D4), (C5×C4⋊C4).258C22, (C2×C52C8).81C22, (C2×C4).420(C22×D5), C22.130(C2×C5⋊D4), SmallGroup(320,594)

Series: Derived Chief Lower central Upper central

C1C2×C20 — (C2×C10).40D8
C1C5C10C20C2×C20C2×D20C207D4 — (C2×C10).40D8
C5C10C2×C20 — (C2×C10).40D8
C1C22C22×C4C2×C4⋊C4

Generators and relations for (C2×C10).40D8
 G = < a,b,c,d | a10=b2=c8=1, d2=a5, ab=ba, cac-1=a-1, ad=da, cbc-1=a5b, bd=db, dcd-1=c-1 >

Subgroups: 462 in 114 conjugacy classes, 43 normal (25 characteristic)
C1, C2 [×3], C2 [×3], C4 [×2], C4 [×4], C22, C22 [×2], C22 [×5], C5, C8 [×2], C2×C4 [×2], C2×C4 [×7], D4 [×4], C23, C23, D5, C10 [×3], C10 [×2], C22⋊C4, C4⋊C4 [×2], C4⋊C4 [×2], C2×C8 [×2], C22×C4, C22×C4, C2×D4 [×2], Dic5, C20 [×2], C20 [×3], D10 [×3], C2×C10, C2×C10 [×2], C2×C10 [×2], C22⋊C8, D4⋊C4 [×2], C2.D8 [×2], C2×C4⋊C4, C4⋊D4, C52C8 [×2], D20 [×2], C2×Dic5, C5⋊D4 [×2], C2×C20 [×2], C2×C20 [×6], C22×D5, C22×C10, C22.D8, C2×C52C8 [×2], C4⋊Dic5, D10⋊C4, C5×C4⋊C4 [×2], C5×C4⋊C4, C2×D20, C2×C5⋊D4, C22×C20, C22×C20, C10.D8 [×2], D206C4 [×2], C20.55D4, C207D4, C10×C4⋊C4, (C2×C10).40D8
Quotients: C1, C2 [×7], C22 [×7], D4 [×2], C23, D5, D8 [×2], C2×D4, C4○D4 [×2], D10 [×3], C22.D4, C2×D8, C8.C22, C5⋊D4 [×2], C22×D5, C22.D8, D4⋊D5 [×2], C4○D20 [×2], C2×C5⋊D4, C23.23D10, C2×D4⋊D5, C20.C23, (C2×C10).40D8

Smallest permutation representation of (C2×C10).40D8
On 160 points
Generators in S160
(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)
(1 110)(2 101)(3 102)(4 103)(5 104)(6 105)(7 106)(8 107)(9 108)(10 109)(11 71)(12 72)(13 73)(14 74)(15 75)(16 76)(17 77)(18 78)(19 79)(20 80)(21 97)(22 98)(23 99)(24 100)(25 91)(26 92)(27 93)(28 94)(29 95)(30 96)(31 125)(32 126)(33 127)(34 128)(35 129)(36 130)(37 121)(38 122)(39 123)(40 124)(41 117)(42 118)(43 119)(44 120)(45 111)(46 112)(47 113)(48 114)(49 115)(50 116)(51 145)(52 146)(53 147)(54 148)(55 149)(56 150)(57 141)(58 142)(59 143)(60 144)(61 137)(62 138)(63 139)(64 140)(65 131)(66 132)(67 133)(68 134)(69 135)(70 136)(81 157)(82 158)(83 159)(84 160)(85 151)(86 152)(87 153)(88 154)(89 155)(90 156)
(1 71 45 70 30 85 31 56)(2 80 46 69 21 84 32 55)(3 79 47 68 22 83 33 54)(4 78 48 67 23 82 34 53)(5 77 49 66 24 81 35 52)(6 76 50 65 25 90 36 51)(7 75 41 64 26 89 37 60)(8 74 42 63 27 88 38 59)(9 73 43 62 28 87 39 58)(10 72 44 61 29 86 40 57)(11 116 136 91 151 130 150 105)(12 115 137 100 152 129 141 104)(13 114 138 99 153 128 142 103)(14 113 139 98 154 127 143 102)(15 112 140 97 155 126 144 101)(16 111 131 96 156 125 145 110)(17 120 132 95 157 124 146 109)(18 119 133 94 158 123 147 108)(19 118 134 93 159 122 148 107)(20 117 135 92 160 121 149 106)
(1 131 6 136)(2 132 7 137)(3 133 8 138)(4 134 9 139)(5 135 10 140)(11 45 16 50)(12 46 17 41)(13 47 18 42)(14 48 19 43)(15 49 20 44)(21 146 26 141)(22 147 27 142)(23 148 28 143)(24 149 29 144)(25 150 30 145)(31 156 36 151)(32 157 37 152)(33 158 38 153)(34 159 39 154)(35 160 40 155)(51 91 56 96)(52 92 57 97)(53 93 58 98)(54 94 59 99)(55 95 60 100)(61 101 66 106)(62 102 67 107)(63 103 68 108)(64 104 69 109)(65 105 70 110)(71 111 76 116)(72 112 77 117)(73 113 78 118)(74 114 79 119)(75 115 80 120)(81 121 86 126)(82 122 87 127)(83 123 88 128)(84 124 89 129)(85 125 90 130)

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

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

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

59 conjugacy classes

class 1 2A2B2C2D2E2F4A4B4C···4G4H5A5B8A8B8C8D10A···10N20A···20X
order1222222444···4455888810···1020···20
size11112240224···44022202020202···24···4

59 irreducible representations

dim1111112222222222444
type++++++++++++-+
imageC1C2C2C2C2C2D4D4D5C4○D4D8D10D10C5⋊D4C5⋊D4C4○D20C8.C22D4⋊D5C20.C23
kernel(C2×C10).40D8C10.D8D206C4C20.55D4C207D4C10×C4⋊C4C2×C20C22×C10C2×C4⋊C4C20C2×C10C4⋊C4C22×C4C2×C4C23C4C10C22C2
# reps12211111244424416144

Matrix representation of (C2×C10).40D8 in GL6(𝔽41)

4000000
0400000
0040700
0034700
000010
000001
,
2510000
32160000
001000
000100
000010
000001
,
2090000
24210000
0073400
0013400
00001212
00002912
,
900000
090000
0040000
0004000
00002929
00002912

G:=sub<GL(6,GF(41))| [40,0,0,0,0,0,0,40,0,0,0,0,0,0,40,34,0,0,0,0,7,7,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[25,32,0,0,0,0,1,16,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[20,24,0,0,0,0,9,21,0,0,0,0,0,0,7,1,0,0,0,0,34,34,0,0,0,0,0,0,12,29,0,0,0,0,12,12],[9,0,0,0,0,0,0,9,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,29,29,0,0,0,0,29,12] >;

(C2×C10).40D8 in GAP, Magma, Sage, TeX

(C_2\times C_{10})._{40}D_8
% in TeX

G:=Group("(C2xC10).40D8");
// GroupNames label

G:=SmallGroup(320,594);
// by ID

G=gap.SmallGroup(320,594);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,253,254,100,1123,297,136,12550]);
// Polycyclic

G:=Group<a,b,c,d|a^10=b^2=c^8=1,d^2=a^5,a*b=b*a,c*a*c^-1=a^-1,a*d=d*a,c*b*c^-1=a^5*b,b*d=d*b,d*c*d^-1=c^-1>;
// generators/relations

׿
×
𝔽