Copied to
clipboard

G = Dic5.23D8order 320 = 26·5

10th non-split extension by Dic5 of D8 acting via D8/D4=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: Dic5.23D8, C20.3M4(2), Dic5.18SD16, D4⋊(C5⋊C8), C52(D4⋊C8), (C5×D4)⋊1C8, C10.6C4≀C2, C20.4(C2×C8), C20⋊C82C2, (C2×D4).4F5, (D4×C10).7C4, C4⋊Dic5.9C4, C2.3(D20⋊C4), C2.3(D4⋊F5), (D4×Dic5).16C2, C10.12(C22⋊C8), C4.1(C22.F5), (C2×Dic5).107D4, C10.14(D4⋊C4), C22.41(C22⋊F5), C2.5(C23.2F5), (C4×Dic5).188C22, (C4×C5⋊C8)⋊2C2, C4.1(C2×C5⋊C8), (C2×C4).71(C2×F5), (C2×C20).39(C2×C4), (C2×C10).37(C22⋊C4), SmallGroup(320,262)

Series: Derived Chief Lower central Upper central

C1C20 — Dic5.23D8
C1C5C10C2×C10C2×Dic5C4×Dic5C20⋊C8 — Dic5.23D8
C5C10C20 — Dic5.23D8
C1C22C2×C4C2×D4

Generators and relations for Dic5.23D8
 G = < a,b,c,d | a10=c8=1, b2=a5, d2=b, bab-1=a-1, cac-1=dad-1=a3, bc=cb, bd=db, dcd-1=bc-1 >

Subgroups: 322 in 82 conjugacy classes, 32 normal (28 characteristic)
C1, C2 [×3], C2 [×2], C4 [×2], C4 [×4], C22, C22 [×4], C5, C8 [×3], C2×C4, C2×C4 [×5], D4 [×2], D4, C23, C10 [×3], C10 [×2], C42, C22⋊C4, C4⋊C4, C2×C8 [×2], C22×C4, C2×D4, Dic5 [×2], Dic5 [×2], C20 [×2], C2×C10, C2×C10 [×4], C4×C8, C4⋊C8, C4×D4, C5⋊C8 [×3], C2×Dic5 [×2], C2×Dic5 [×3], C2×C20, C5×D4 [×2], C5×D4, C22×C10, D4⋊C8, C4×Dic5, C4⋊Dic5, C23.D5, C2×C5⋊C8 [×2], C22×Dic5, D4×C10, C4×C5⋊C8, C20⋊C8, D4×Dic5, Dic5.23D8
Quotients: C1, C2 [×3], C4 [×2], C22, C8 [×2], C2×C4, D4 [×2], C22⋊C4, C2×C8, M4(2), D8, SD16, F5, C22⋊C8, D4⋊C4, C4≀C2, C5⋊C8 [×2], C2×F5, D4⋊C8, C2×C5⋊C8, C22.F5, C22⋊F5, D20⋊C4, D4⋊F5, C23.2F5, Dic5.23D8

Smallest permutation representation of Dic5.23D8
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 55 6 60)(2 54 7 59)(3 53 8 58)(4 52 9 57)(5 51 10 56)(11 121 16 126)(12 130 17 125)(13 129 18 124)(14 128 19 123)(15 127 20 122)(21 69 26 64)(22 68 27 63)(23 67 28 62)(24 66 29 61)(25 65 30 70)(31 74 36 79)(32 73 37 78)(33 72 38 77)(34 71 39 76)(35 80 40 75)(41 88 46 83)(42 87 47 82)(43 86 48 81)(44 85 49 90)(45 84 50 89)(91 134 96 139)(92 133 97 138)(93 132 98 137)(94 131 99 136)(95 140 100 135)(101 149 106 144)(102 148 107 143)(103 147 108 142)(104 146 109 141)(105 145 110 150)(111 154 116 159)(112 153 117 158)(113 152 118 157)(114 151 119 156)(115 160 120 155)
(1 13 75 110 23 160 89 100)(2 20 74 103 24 157 88 93)(3 17 73 106 25 154 87 96)(4 14 72 109 26 151 86 99)(5 11 71 102 27 158 85 92)(6 18 80 105 28 155 84 95)(7 15 79 108 29 152 83 98)(8 12 78 101 30 159 82 91)(9 19 77 104 21 156 81 94)(10 16 76 107 22 153 90 97)(31 142 61 118 41 137 59 127)(32 149 70 111 42 134 58 130)(33 146 69 114 43 131 57 123)(34 143 68 117 44 138 56 126)(35 150 67 120 45 135 55 129)(36 147 66 113 46 132 54 122)(37 144 65 116 47 139 53 125)(38 141 64 119 48 136 52 128)(39 148 63 112 49 133 51 121)(40 145 62 115 50 140 60 124)
(1 135 55 95 6 140 60 100)(2 132 54 98 7 137 59 93)(3 139 53 91 8 134 58 96)(4 136 52 94 9 131 57 99)(5 133 51 97 10 138 56 92)(11 90 121 44 16 85 126 49)(12 87 130 47 17 82 125 42)(13 84 129 50 18 89 124 45)(14 81 128 43 19 86 123 48)(15 88 127 46 20 83 122 41)(21 146 69 109 26 141 64 104)(22 143 68 102 27 148 63 107)(23 150 67 105 28 145 62 110)(24 147 66 108 29 142 61 103)(25 144 65 101 30 149 70 106)(31 152 74 118 36 157 79 113)(32 159 73 111 37 154 78 116)(33 156 72 114 38 151 77 119)(34 153 71 117 39 158 76 112)(35 160 80 120 40 155 75 115)

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

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

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

38 conjugacy classes

class 1 2A2B2C2D2E4A4B4C4D4E4F4G4H4I4J 5 8A···8H8I8J8K8L10A10B10C10D10E10F10G20A20B
order122222444444444458···88888101010101010102020
size11114422555510102020410···1020202020444888888

38 irreducible representations

dim1111111222224444488
type++++++++--++-
imageC1C2C2C2C4C4C8D4D8SD16M4(2)C4≀C2F5C2×F5C5⋊C8C22.F5C22⋊F5D20⋊C4D4⋊F5
kernelDic5.23D8C4×C5⋊C8C20⋊C8D4×Dic5C4⋊Dic5D4×C10C5×D4C2×Dic5Dic5Dic5C20C10C2×D4C2×C4D4C4C22C2C2
# reps1111228222241122211

Matrix representation of Dic5.23D8 in GL8(𝔽41)

10000000
01000000
004000000
000400000
000040100
000053500
000027254034
0000321477
,
400000000
040000000
003200000
000320000
0000182100
0000102300
0000602126
00003233220
,
1526000000
1515000000
004370000
0012370000
000000401
000011253934
00003220160
00001440160
,
1526000000
2626000000
004370000
0037370000
000000401
000011253934
0000205160
00003826160

G:=sub<GL(8,GF(41))| [1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40,5,27,32,0,0,0,0,1,35,25,14,0,0,0,0,0,0,40,7,0,0,0,0,0,0,34,7],[40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,32,0,0,0,0,0,0,0,0,32,0,0,0,0,0,0,0,0,18,10,6,32,0,0,0,0,21,23,0,33,0,0,0,0,0,0,21,2,0,0,0,0,0,0,26,20],[15,15,0,0,0,0,0,0,26,15,0,0,0,0,0,0,0,0,4,12,0,0,0,0,0,0,37,37,0,0,0,0,0,0,0,0,0,11,32,14,0,0,0,0,0,25,20,40,0,0,0,0,40,39,16,16,0,0,0,0,1,34,0,0],[15,26,0,0,0,0,0,0,26,26,0,0,0,0,0,0,0,0,4,37,0,0,0,0,0,0,37,37,0,0,0,0,0,0,0,0,0,11,20,38,0,0,0,0,0,25,5,26,0,0,0,0,40,39,16,16,0,0,0,0,1,34,0,0] >;

Dic5.23D8 in GAP, Magma, Sage, TeX

{\rm Dic}_5._{23}D_8
% in TeX

G:=Group("Dic5.23D8");
// GroupNames label

G:=SmallGroup(320,262);
// by ID

G=gap.SmallGroup(320,262);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,28,141,100,1123,570,136,6278,3156]);
// Polycyclic

G:=Group<a,b,c,d|a^10=c^8=1,b^2=a^5,d^2=b,b*a*b^-1=a^-1,c*a*c^-1=d*a*d^-1=a^3,b*c=c*b,b*d=d*b,d*c*d^-1=b*c^-1>;
// generators/relations

׿
×
𝔽