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G = C10.(C4○D8)  order 320 = 26·5

17th non-split extension by C10 of C4○D8 acting via C4○D8/C4○D4=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C4⋊C4.65D10, C22⋊Q8.4D5, (C2×C20).265D4, (C2×Q8).27D10, C10.99(C4○D8), Q8⋊Dic514C2, C10.D839C2, C20.Q838C2, (C22×C10).91D4, C20.189(C4○D4), C4.95(D42D5), (C2×C20).364C23, C20.55D4.8C2, (C22×C4).126D10, C23.26(C5⋊D4), C57(C23.20D4), (Q8×C10).45C22, C10.89(C8.C22), C4⋊Dic5.339C22, C2.18(D4.8D10), C2.10(C20.C23), (C22×C20).168C22, C23.21D10.14C2, C10.82(C22.D4), C2.16(C23.18D10), (C5×C22⋊Q8).3C2, (C2×C10).495(C2×D4), (C2×C4).172(C5⋊D4), (C5×C4⋊C4).112C22, (C2×C4).464(C22×D5), C22.170(C2×C5⋊D4), (C2×C52C8).114C22, SmallGroup(320,672)

Series: Derived Chief Lower central Upper central

C1C2×C20 — C10.(C4○D8)
C1C5C10C20C2×C20C4⋊Dic5C23.21D10 — C10.(C4○D8)
C5C10C2×C20 — C10.(C4○D8)
C1C22C22×C4C22⋊Q8

Generators and relations for C10.(C4○D8)
 G = < a,b,c,d | a10=b8=d2=1, c2=a5, bab-1=cac-1=a-1, ad=da, cbc-1=b-1, dbd=a5b, dcd=b4c >

Subgroups: 286 in 96 conjugacy classes, 39 normal (all characteristic)
C1, C2 [×3], C2, C4 [×2], C4 [×5], C22, C22 [×3], C5, C8 [×2], C2×C4 [×2], C2×C4 [×6], Q8 [×2], C23, C10 [×3], C10, C42, C22⋊C4 [×2], C4⋊C4, C4⋊C4 [×3], C2×C8 [×2], C22×C4, C2×Q8, Dic5 [×2], C20 [×2], C20 [×3], C2×C10, C2×C10 [×3], C22⋊C8, Q8⋊C4 [×2], C4.Q8, C2.D8, C42⋊C2, C22⋊Q8, C52C8 [×2], C2×Dic5 [×2], C2×C20 [×2], C2×C20 [×4], C5×Q8 [×2], C22×C10, C23.20D4, C2×C52C8 [×2], C4×Dic5, C4⋊Dic5 [×2], C23.D5, C5×C22⋊C4, C5×C4⋊C4, C5×C4⋊C4, C22×C20, Q8×C10, C10.D8, C20.Q8, C20.55D4, Q8⋊Dic5 [×2], C23.21D10, C5×C22⋊Q8, C10.(C4○D8)
Quotients: C1, C2 [×7], C22 [×7], D4 [×2], C23, D5, C2×D4, C4○D4 [×2], D10 [×3], C22.D4, C4○D8, C8.C22, C5⋊D4 [×2], C22×D5, C23.20D4, D42D5 [×2], C2×C5⋊D4, C23.18D10, C20.C23, D4.8D10, C10.(C4○D8)

Smallest permutation representation of C10.(C4○D8)
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)
(1 136 40 156 25 150 45 17)(2 135 31 155 26 149 46 16)(3 134 32 154 27 148 47 15)(4 133 33 153 28 147 48 14)(5 132 34 152 29 146 49 13)(6 131 35 151 30 145 50 12)(7 140 36 160 21 144 41 11)(8 139 37 159 22 143 42 20)(9 138 38 158 23 142 43 19)(10 137 39 157 24 141 44 18)(51 100 85 124 66 104 71 120)(52 99 86 123 67 103 72 119)(53 98 87 122 68 102 73 118)(54 97 88 121 69 101 74 117)(55 96 89 130 70 110 75 116)(56 95 90 129 61 109 76 115)(57 94 81 128 62 108 77 114)(58 93 82 127 63 107 78 113)(59 92 83 126 64 106 79 112)(60 91 84 125 65 105 80 111)
(1 91 6 96)(2 100 7 95)(3 99 8 94)(4 98 9 93)(5 97 10 92)(11 90 16 85)(12 89 17 84)(13 88 18 83)(14 87 19 82)(15 86 20 81)(21 109 26 104)(22 108 27 103)(23 107 28 102)(24 106 29 101)(25 105 30 110)(31 120 36 115)(32 119 37 114)(33 118 38 113)(34 117 39 112)(35 116 40 111)(41 129 46 124)(42 128 47 123)(43 127 48 122)(44 126 49 121)(45 125 50 130)(51 140 56 135)(52 139 57 134)(53 138 58 133)(54 137 59 132)(55 136 60 131)(61 149 66 144)(62 148 67 143)(63 147 68 142)(64 146 69 141)(65 145 70 150)(71 160 76 155)(72 159 77 154)(73 158 78 153)(74 157 79 152)(75 156 80 151)
(11 16)(12 17)(13 18)(14 19)(15 20)(51 61)(52 62)(53 63)(54 64)(55 65)(56 66)(57 67)(58 68)(59 69)(60 70)(71 90)(72 81)(73 82)(74 83)(75 84)(76 85)(77 86)(78 87)(79 88)(80 89)(91 105)(92 106)(93 107)(94 108)(95 109)(96 110)(97 101)(98 102)(99 103)(100 104)(111 125)(112 126)(113 127)(114 128)(115 129)(116 130)(117 121)(118 122)(119 123)(120 124)(131 136)(132 137)(133 138)(134 139)(135 140)(141 146)(142 147)(143 148)(144 149)(145 150)(151 156)(152 157)(153 158)(154 159)(155 160)

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

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

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

47 conjugacy classes

class 1 2A2B2C2D4A4B4C4D4E4F4G4H4I4J5A5B8A8B8C8D10A···10F10G10H10I10J20A···20H20I···20P
order12222444444444455888810···101010101020···2020···20
size111142222882020202022202020202···244444···48···8

47 irreducible representations

dim111111122222222224444
type+++++++++++++--
imageC1C2C2C2C2C2C2D4D4D5C4○D4D10D10D10C4○D8C5⋊D4C5⋊D4C8.C22D42D5C20.C23D4.8D10
kernelC10.(C4○D8)C10.D8C20.Q8C20.55D4Q8⋊Dic5C23.21D10C5×C22⋊Q8C2×C20C22×C10C22⋊Q8C20C4⋊C4C22×C4C2×Q8C10C2×C4C23C10C4C2C2
# reps111121111242224441444

Matrix representation of C10.(C4○D8) in GL6(𝔽41)

4000000
0400000
0037000
00231000
000010
000001
,
7220000
9340000
0030400
00111100
0000380
00002127
,
3200000
0320000
0030400
00111100
00004022
000001
,
130000
0400000
001000
000100
000010
00001540

G:=sub<GL(6,GF(41))| [40,0,0,0,0,0,0,40,0,0,0,0,0,0,37,23,0,0,0,0,0,10,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[7,9,0,0,0,0,22,34,0,0,0,0,0,0,30,11,0,0,0,0,4,11,0,0,0,0,0,0,38,21,0,0,0,0,0,27],[32,0,0,0,0,0,0,32,0,0,0,0,0,0,30,11,0,0,0,0,4,11,0,0,0,0,0,0,40,0,0,0,0,0,22,1],[1,0,0,0,0,0,3,40,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,15,0,0,0,0,0,40] >;

C10.(C4○D8) in GAP, Magma, Sage, TeX

C_{10}.(C_4\circ D_8)
% in TeX

G:=Group("C10.(C4oD8)");
// GroupNames label

G:=SmallGroup(320,672);
// by ID

G=gap.SmallGroup(320,672);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,112,232,254,219,184,1123,297,136,12550]);
// Polycyclic

G:=Group<a,b,c,d|a^10=b^8=d^2=1,c^2=a^5,b*a*b^-1=c*a*c^-1=a^-1,a*d=d*a,c*b*c^-1=b^-1,d*b*d=a^5*b,d*c*d=b^4*c>;
// generators/relations

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