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G = (C2×Dic5)⋊C8order 320 = 26·5

1st semidirect product of C2×Dic5 and C8 acting via C8/C2=C4

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: (C2×Dic5)⋊1C8, C22⋊C8.2D5, C22.4(C8×D5), (C2×C4).108D20, (C2×C20).439D4, (C22×C4).2D10, C23.41(C4×D5), C2.6(D101C8), C10.19(C22⋊C8), C10.26(C23⋊C4), C22.4(C8⋊D5), (C2×C10).11M4(2), (C22×Dic5).2C4, C20.55D4.11C2, C10.7(C4.10D4), C2.1(C4.12D20), (C22×C20).323C22, C2.2(C23.1D10), C53(C22.M4(2)), C22.33(D10⋊C4), (C2×C10).17(C2×C8), (C5×C22⋊C8).2C2, (C2×C4).210(C5⋊D4), (C22×C10).95(C2×C4), (C2×C10.D4).26C2, (C2×C10).105(C22⋊C4), SmallGroup(320,27)

Series: Derived Chief Lower central Upper central

C1C2×C10 — (C2×Dic5)⋊C8
C1C5C10C2×C10C2×C20C22×C20C2×C10.D4 — (C2×Dic5)⋊C8
C5C10C2×C10 — (C2×Dic5)⋊C8
C1C22C22×C4C22⋊C8

Generators and relations for (C2×Dic5)⋊C8
 G = < a,b,c,d | a2=b10=d8=1, c2=b5, ab=ba, ac=ca, dad-1=ab5, cbc-1=b-1, bd=db, dcd-1=ab5c >

Subgroups: 278 in 78 conjugacy classes, 31 normal (29 characteristic)
C1, C2 [×3], C2 [×2], C4 [×5], C22 [×3], C22 [×2], C5, C8 [×2], C2×C4 [×2], C2×C4 [×7], C23, C10 [×3], C10 [×2], C4⋊C4 [×2], C2×C8 [×2], C22×C4, C22×C4 [×2], Dic5 [×3], C20 [×2], C2×C10 [×3], C2×C10 [×2], C22⋊C8, C22⋊C8, C2×C4⋊C4, C52C8, C40, C2×Dic5 [×2], C2×Dic5 [×4], C2×C20 [×2], C2×C20, C22×C10, C22.M4(2), C2×C52C8, C10.D4 [×2], C2×C40, C22×Dic5 [×2], C22×C20, C20.55D4, C5×C22⋊C8, C2×C10.D4, (C2×Dic5)⋊C8
Quotients: C1, C2 [×3], C4 [×2], C22, C8 [×2], C2×C4, D4 [×2], D5, C22⋊C4, C2×C8, M4(2), D10, C22⋊C8, C23⋊C4, C4.10D4, C4×D5, D20, C5⋊D4, C22.M4(2), C8×D5, C8⋊D5, D10⋊C4, C23.1D10, D101C8, C4.12D20, (C2×Dic5)⋊C8

Smallest permutation representation of (C2×Dic5)⋊C8
On 160 points
Generators in S160
(41 46)(42 47)(43 48)(44 49)(45 50)(51 56)(52 57)(53 58)(54 59)(55 60)(61 66)(62 67)(63 68)(64 69)(65 70)(71 76)(72 77)(73 78)(74 79)(75 80)(121 126)(122 127)(123 128)(124 129)(125 130)(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)
(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 84 6 89)(2 83 7 88)(3 82 8 87)(4 81 9 86)(5 90 10 85)(11 94 16 99)(12 93 17 98)(13 92 18 97)(14 91 19 96)(15 100 20 95)(21 104 26 109)(22 103 27 108)(23 102 28 107)(24 101 29 106)(25 110 30 105)(31 114 36 119)(32 113 37 118)(33 112 38 117)(34 111 39 116)(35 120 40 115)(41 124 46 129)(42 123 47 128)(43 122 48 127)(44 121 49 126)(45 130 50 125)(51 134 56 139)(52 133 57 138)(53 132 58 137)(54 131 59 136)(55 140 60 135)(61 144 66 149)(62 143 67 148)(63 142 68 147)(64 141 69 146)(65 150 70 145)(71 154 76 159)(72 153 77 158)(73 152 78 157)(74 151 79 156)(75 160 80 155)
(1 71 31 51 11 61 21 41)(2 72 32 52 12 62 22 42)(3 73 33 53 13 63 23 43)(4 74 34 54 14 64 24 44)(5 75 35 55 15 65 25 45)(6 76 36 56 16 66 26 46)(7 77 37 57 17 67 27 47)(8 78 38 58 18 68 28 48)(9 79 39 59 19 69 29 49)(10 80 40 60 20 70 30 50)(81 156 116 131 91 146 106 121)(82 157 117 132 92 147 107 122)(83 158 118 133 93 148 108 123)(84 159 119 134 94 149 109 124)(85 160 120 135 95 150 110 125)(86 151 111 136 96 141 101 126)(87 152 112 137 97 142 102 127)(88 153 113 138 98 143 103 128)(89 154 114 139 99 144 104 129)(90 155 115 140 100 145 105 130)

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

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

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

62 conjugacy classes

class 1 2A2B2C2D2E4A4B4C4D4E4F4G4H5A5B8A8B8C8D8E8F8G8H10A···10F10G10H10I10J20A···20H20I20J20K20L40A···40P
order12222244444444558888888810···101010101020···202020202040···40
size111122222220202020224444202020202···244442···244444···4

62 irreducible representations

dim1111112222222224444
type+++++++++--
imageC1C2C2C2C4C8D4D5M4(2)D10D20C5⋊D4C4×D5C8×D5C8⋊D5C23⋊C4C4.10D4C23.1D10C4.12D20
kernel(C2×Dic5)⋊C8C20.55D4C5×C22⋊C8C2×C10.D4C22×Dic5C2×Dic5C2×C20C22⋊C8C2×C10C22×C4C2×C4C2×C4C23C22C22C10C10C2C2
# reps1111482222444881144

Matrix representation of (C2×Dic5)⋊C8 in GL8(𝔽41)

400000000
040000000
004000000
000400000
00001000
00000100
0000160400
0000131040
,
334000000
343000000
00100000
00010000
000040000
000004000
000000400
000000040
,
1835000000
623000000
00100000
000400000
0000271100
0000381400
00000341126
00002623030
,
040000000
400000000
000140000
001400000
0000320370
0000011739
000035190
000060340

G:=sub<GL(8,GF(41))| [40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,1,0,16,13,0,0,0,0,0,1,0,1,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40],[3,34,0,0,0,0,0,0,34,3,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40],[18,6,0,0,0,0,0,0,35,23,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,27,38,0,26,0,0,0,0,11,14,34,2,0,0,0,0,0,0,11,30,0,0,0,0,0,0,26,30],[0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,0,0,14,0,0,0,0,0,0,14,0,0,0,0,0,0,0,0,0,32,0,35,6,0,0,0,0,0,1,1,0,0,0,0,0,37,17,9,3,0,0,0,0,0,39,0,40] >;

(C2×Dic5)⋊C8 in GAP, Magma, Sage, TeX

(C_2\times {\rm Dic}_5)\rtimes C_8
% in TeX

G:=Group("(C2xDic5):C8");
// GroupNames label

G:=SmallGroup(320,27);
// by ID

G=gap.SmallGroup(320,27);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,224,141,36,758,100,570,12550]);
// Polycyclic

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

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