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## G = Dic5.13M4(2)  order 320 = 26·5

### 6th non-split extension by Dic5 of M4(2) acting via M4(2)/C2×C4=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C10 — Dic5.13M4(2)
 Chief series C1 — C5 — C10 — Dic5 — C2×Dic5 — C2×C5⋊C8 — C23.2F5 — Dic5.13M4(2)
 Lower central C5 — C2×C10 — Dic5.13M4(2)
 Upper central C1 — C22 — C22×C4

Generators and relations for Dic5.13M4(2)
G = < a,b,c,d | a10=c8=d2=1, b2=a5, bab-1=a-1, cac-1=a3, ad=da, cbc-1=a5b, bd=db, dcd=a5c5 >

Subgroups: 330 in 110 conjugacy classes, 48 normal (30 characteristic)
C1, C2, C2, C4, C22, C22, C22, C5, C8, C2×C4, C2×C4, C23, C10, C10, C42, C2×C8, C22×C4, C22×C4, Dic5, Dic5, C20, C2×C10, C2×C10, C2×C10, C8⋊C4, C22⋊C8, C4⋊C8, C2×C42, C5⋊C8, C2×Dic5, C2×Dic5, C2×C20, C2×C20, C22×C10, C42.6C4, C4×Dic5, C2×C5⋊C8, C22×Dic5, C22×C20, C10.C42, Dic5⋊C8, C23.2F5, C2×C4×Dic5, Dic5.13M4(2)
Quotients: C1, C2, C4, C22, C2×C4, C23, M4(2), C22×C4, C4○D4, F5, C42⋊C2, C2×M4(2), C2×F5, C42.6C4, C22.F5, C22×F5, D5⋊M4(2), D10.C23, C2×C22.F5, Dic5.13M4(2)

Smallest permutation representation of Dic5.13M4(2)
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 89 6 84)(2 88 7 83)(3 87 8 82)(4 86 9 81)(5 85 10 90)(11 106 16 101)(12 105 17 110)(13 104 18 109)(14 103 19 108)(15 102 20 107)(21 72 26 77)(22 71 27 76)(23 80 28 75)(24 79 29 74)(25 78 30 73)(31 70 36 65)(32 69 37 64)(33 68 38 63)(34 67 39 62)(35 66 40 61)(41 53 46 58)(42 52 47 57)(43 51 48 56)(44 60 49 55)(45 59 50 54)(91 159 96 154)(92 158 97 153)(93 157 98 152)(94 156 99 151)(95 155 100 160)(111 146 116 141)(112 145 117 150)(113 144 118 149)(114 143 119 148)(115 142 120 147)(121 137 126 132)(122 136 127 131)(123 135 128 140)(124 134 129 139)(125 133 130 138)
(1 121 50 92 22 113 31 109)(2 128 49 95 23 120 40 102)(3 125 48 98 24 117 39 105)(4 122 47 91 25 114 38 108)(5 129 46 94 26 111 37 101)(6 126 45 97 27 118 36 104)(7 123 44 100 28 115 35 107)(8 130 43 93 29 112 34 110)(9 127 42 96 30 119 33 103)(10 124 41 99 21 116 32 106)(11 90 139 53 156 72 146 69)(12 87 138 56 157 79 145 62)(13 84 137 59 158 76 144 65)(14 81 136 52 159 73 143 68)(15 88 135 55 160 80 142 61)(16 85 134 58 151 77 141 64)(17 82 133 51 152 74 150 67)(18 89 132 54 153 71 149 70)(19 86 131 57 154 78 148 63)(20 83 140 60 155 75 147 66)
(11 151)(12 152)(13 153)(14 154)(15 155)(16 156)(17 157)(18 158)(19 159)(20 160)(91 103)(92 104)(93 105)(94 106)(95 107)(96 108)(97 109)(98 110)(99 101)(100 102)(111 124)(112 125)(113 126)(114 127)(115 128)(116 129)(117 130)(118 121)(119 122)(120 123)(131 143)(132 144)(133 145)(134 146)(135 147)(136 148)(137 149)(138 150)(139 141)(140 142)

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

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

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

44 conjugacy classes

 class 1 2A 2B 2C 2D 2E 4A 4B 4C 4D 4E 4F 4G 4H 4I ··· 4N 5 8A ··· 8H 10A ··· 10G 20A ··· 20H order 1 2 2 2 2 2 4 4 4 4 4 4 4 4 4 ··· 4 5 8 ··· 8 10 ··· 10 20 ··· 20 size 1 1 1 1 2 2 2 2 2 2 5 5 5 5 10 ··· 10 4 20 ··· 20 4 ··· 4 4 ··· 4

44 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 2 4 4 4 4 4 4 type + + + + + + + + - image C1 C2 C2 C2 C2 C4 C4 C4 M4(2) C4○D4 M4(2) F5 C2×F5 C2×F5 C22.F5 D5⋊M4(2) D10.C23 kernel Dic5.13M4(2) C10.C42 Dic5⋊C8 C23.2F5 C2×C4×Dic5 C4×Dic5 C22×Dic5 C22×C20 Dic5 Dic5 C2×C10 C22×C4 C2×C4 C23 C22 C2 C2 # reps 1 2 2 2 1 4 2 2 4 4 4 1 2 1 4 4 4

Matrix representation of Dic5.13M4(2) in GL6(𝔽41)

 40 0 0 0 0 0 0 40 0 0 0 0 0 0 1 1 0 0 0 0 5 6 0 0 0 0 2 1 34 1 0 0 2 1 33 1
,
 32 0 0 0 0 0 0 9 0 0 0 0 0 0 27 13 0 0 0 0 29 14 0 0 0 0 5 15 9 0 0 0 30 34 31 32
,
 0 1 0 0 0 0 1 0 0 0 0 0 0 0 22 6 7 19 0 0 19 18 11 21 0 0 40 10 24 14 0 0 2 24 40 18
,
 1 0 0 0 0 0 0 40 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

G:=sub<GL(6,GF(41))| [40,0,0,0,0,0,0,40,0,0,0,0,0,0,1,5,2,2,0,0,1,6,1,1,0,0,0,0,34,33,0,0,0,0,1,1],[32,0,0,0,0,0,0,9,0,0,0,0,0,0,27,29,5,30,0,0,13,14,15,34,0,0,0,0,9,31,0,0,0,0,0,32],[0,1,0,0,0,0,1,0,0,0,0,0,0,0,22,19,40,2,0,0,6,18,10,24,0,0,7,11,24,40,0,0,19,21,14,18],[1,0,0,0,0,0,0,40,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1] >;

Dic5.13M4(2) in GAP, Magma, Sage, TeX

{\rm Dic}_5._{13}M_4(2)
% in TeX

G:=Group("Dic5.13M4(2)");
// GroupNames label

G:=SmallGroup(320,1095);
// by ID

G=gap.SmallGroup(320,1095);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,56,253,758,100,136,6278,1595]);
// Polycyclic

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

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