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G = (C2×C10)⋊8Q16order 320 = 26·5

2nd semidirect product of C2×C10 and Q16 acting via Q16/Q8=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: (C2×C10)⋊8Q16, (C5×Q8).31D4, (C2×C20).304D4, C20.211(C2×D4), C10.47(C2×Q16), C55(C22⋊Q16), C10.75C22≀C2, (C22×Q8).4D5, Q8⋊Dic539C2, (C2×Q8).169D10, Q8.13(C5⋊D4), C223(C5⋊Q16), (C2×C20).478C23, (C22×C4).157D10, (C22×C10).201D4, C2.9(C242D5), C23.89(C5⋊D4), C20.48D4.14C2, C20.55D4.10C2, C4⋊Dic5.188C22, (Q8×C10).204C22, C2.23(C20.C23), C10.103(C8.C22), (C22×C20).204C22, (C2×Dic10).142C22, (Q8×C2×C10).4C2, C4.61(C2×C5⋊D4), (C2×C5⋊Q16)⋊23C2, C2.17(C2×C5⋊Q16), (C2×C10).561(C2×D4), (C2×C4).87(C5⋊D4), (C2×C4).563(C22×D5), C22.221(C2×C5⋊D4), (C2×C52C8).176C22, SmallGroup(320,855)

Series: Derived Chief Lower central Upper central

C1C2×C20 — (C2×C10)⋊8Q16
C1C5C10C20C2×C20C2×Dic10C20.48D4 — (C2×C10)⋊8Q16
C5C10C2×C20 — (C2×C10)⋊8Q16
C1C22C22×C4C22×Q8

Generators and relations for (C2×C10)⋊8Q16
 G = < a,b,c,d | a2=b10=c8=1, d2=c4, ab=ba, cac-1=ab5, ad=da, cbc-1=b-1, bd=db, dcd-1=c-1 >

Subgroups: 414 in 148 conjugacy classes, 51 normal (25 characteristic)
C1, C2 [×3], C2 [×2], C4 [×2], C4 [×7], C22, C22 [×2], C22 [×2], C5, C8 [×2], C2×C4 [×2], C2×C4 [×10], Q8 [×4], Q8 [×8], C23, C10 [×3], C10 [×2], C22⋊C4, C4⋊C4 [×2], C2×C8 [×2], Q16 [×4], C22×C4, C22×C4, C2×Q8 [×2], C2×Q8 [×6], Dic5 [×2], C20 [×2], C20 [×5], C2×C10, C2×C10 [×2], C2×C10 [×2], C22⋊C8, Q8⋊C4 [×2], C22⋊Q8, C2×Q16 [×2], C22×Q8, C52C8 [×2], Dic10 [×2], C2×Dic5 [×2], C2×C20 [×2], C2×C20 [×8], C5×Q8 [×4], C5×Q8 [×6], C22×C10, C22⋊Q16, C2×C52C8 [×2], C10.D4, C4⋊Dic5, C5⋊Q16 [×4], C23.D5, C2×Dic10, C22×C20, C22×C20, Q8×C10 [×2], Q8×C10 [×5], C20.55D4, Q8⋊Dic5 [×2], C20.48D4, C2×C5⋊Q16 [×2], Q8×C2×C10, (C2×C10)⋊8Q16
Quotients: C1, C2 [×7], C22 [×7], D4 [×6], C23, D5, Q16 [×2], C2×D4 [×3], D10 [×3], C22≀C2, C2×Q16, C8.C22, C5⋊D4 [×6], C22×D5, C22⋊Q16, C5⋊Q16 [×2], C2×C5⋊D4 [×3], C20.C23, C2×C5⋊Q16, C242D5, (C2×C10)⋊8Q16

Smallest permutation representation of (C2×C10)⋊8Q16
On 160 points
Generators in S160
(1 30)(2 21)(3 22)(4 23)(5 24)(6 25)(7 26)(8 27)(9 28)(10 29)(11 160)(12 151)(13 152)(14 153)(15 154)(16 155)(17 156)(18 157)(19 158)(20 159)(31 45)(32 46)(33 47)(34 48)(35 49)(36 50)(37 41)(38 42)(39 43)(40 44)(51 65)(52 66)(53 67)(54 68)(55 69)(56 70)(57 61)(58 62)(59 63)(60 64)(71 86)(72 87)(73 88)(74 89)(75 90)(76 81)(77 82)(78 83)(79 84)(80 85)(91 107)(92 108)(93 109)(94 110)(95 101)(96 102)(97 103)(98 104)(99 105)(100 106)(111 127)(112 128)(113 129)(114 130)(115 121)(116 122)(117 123)(118 124)(119 125)(120 126)(131 147)(132 148)(133 149)(134 150)(135 141)(136 142)(137 143)(138 144)(139 145)(140 146)
(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)
(1 154 36 145 30 20 50 134)(2 153 37 144 21 19 41 133)(3 152 38 143 22 18 42 132)(4 151 39 142 23 17 43 131)(5 160 40 141 24 16 44 140)(6 159 31 150 25 15 45 139)(7 158 32 149 26 14 46 138)(8 157 33 148 27 13 47 137)(9 156 34 147 28 12 48 136)(10 155 35 146 29 11 49 135)(51 119 86 99 65 130 71 110)(52 118 87 98 66 129 72 109)(53 117 88 97 67 128 73 108)(54 116 89 96 68 127 74 107)(55 115 90 95 69 126 75 106)(56 114 81 94 70 125 76 105)(57 113 82 93 61 124 77 104)(58 112 83 92 62 123 78 103)(59 111 84 91 63 122 79 102)(60 120 85 100 64 121 80 101)
(1 70 30 56)(2 61 21 57)(3 62 22 58)(4 63 23 59)(5 64 24 60)(6 65 25 51)(7 66 26 52)(8 67 27 53)(9 68 28 54)(10 69 29 55)(11 106 155 95)(12 107 156 96)(13 108 157 97)(14 109 158 98)(15 110 159 99)(16 101 160 100)(17 102 151 91)(18 103 152 92)(19 104 153 93)(20 105 154 94)(31 86 45 71)(32 87 46 72)(33 88 47 73)(34 89 48 74)(35 90 49 75)(36 81 50 76)(37 82 41 77)(38 83 42 78)(39 84 43 79)(40 85 44 80)(111 131 122 142)(112 132 123 143)(113 133 124 144)(114 134 125 145)(115 135 126 146)(116 136 127 147)(117 137 128 148)(118 138 129 149)(119 139 130 150)(120 140 121 141)

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

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

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

59 conjugacy classes

class 1 2A2B2C2D2E4A4B4C···4G4H4I5A5B8A8B8C8D10A···10N20A···20X
order122222444···44455888810···1020···20
size111122224···4404022202020202···24···4

59 irreducible representations

dim1111112222222222444
type++++++++++-++--
imageC1C2C2C2C2C2D4D4D4D5Q16D10D10C5⋊D4C5⋊D4C5⋊D4C8.C22C5⋊Q16C20.C23
kernel(C2×C10)⋊8Q16C20.55D4Q8⋊Dic5C20.48D4C2×C5⋊Q16Q8×C2×C10C2×C20C5×Q8C22×C10C22×Q8C2×C10C22×C4C2×Q8C2×C4Q8C23C10C22C2
# reps11212114124244164144

Matrix representation of (C2×C10)⋊8Q16 in GL4(𝔽41) generated by

1000
344000
00400
00040
,
31000
8400
0010
0001
,
343900
24700
001212
002912
,
1000
0100
001140
004030
G:=sub<GL(4,GF(41))| [1,34,0,0,0,40,0,0,0,0,40,0,0,0,0,40],[31,8,0,0,0,4,0,0,0,0,1,0,0,0,0,1],[34,24,0,0,39,7,0,0,0,0,12,29,0,0,12,12],[1,0,0,0,0,1,0,0,0,0,11,40,0,0,40,30] >;

(C2×C10)⋊8Q16 in GAP, Magma, Sage, TeX

(C_2\times C_{10})\rtimes_8Q_{16}
% in TeX

G:=Group("(C2xC10):8Q16");
// GroupNames label

G:=SmallGroup(320,855);
// by ID

G=gap.SmallGroup(320,855);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,224,253,254,184,1123,297,136,12550]);
// Polycyclic

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

׿
×
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