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## G = Dic5.12M4(2)  order 320 = 26·5

### 5th 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 — C10 — Dic5.12M4(2)
 Chief series C1 — C5 — C10 — Dic5 — C2×Dic5 — C2×C5⋊C8 — C4×C5⋊C8 — Dic5.12M4(2)
 Lower central C5 — C10 — Dic5.12M4(2)
 Upper central C1 — C2×C4 — C22×C4

Generators and relations for Dic5.12M4(2)
G = < a,b,c,d | a10=c8=d2=1, b2=a5, bab-1=a-1, cac-1=a3, ad=da, bc=cb, bd=db, dcd=a5c5 >

Subgroups: 330 in 118 conjugacy classes, 64 normal (30 characteristic)
C1, C2, C2, C4, C4, C22, C22, C22, C5, C8, C2×C4, C2×C4, C2×C4, C23, C10, C10, C42, C2×C8, C22×C4, C22×C4, Dic5, Dic5, C20, C2×C10, C2×C10, C2×C10, C4×C8, C22⋊C8, C4⋊C8, C2×C42, C5⋊C8, C2×Dic5, C2×Dic5, C2×C20, C2×C20, C22×C10, C42.12C4, C4×Dic5, C2×C5⋊C8, C22×Dic5, C22×C20, C4×C5⋊C8, C20⋊C8, C23.2F5, C2×C4×Dic5, Dic5.12M4(2)
Quotients: C1, C2, C4, C22, C8, C2×C4, C23, C2×C8, M4(2), C22×C4, C4○D4, F5, C42⋊C2, C22×C8, C2×M4(2), C5⋊C8, C2×F5, C42.12C4, C2×C5⋊C8, C22×F5, D5⋊M4(2), D10.C23, C22×C5⋊C8, Dic5.12M4(2)

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

56 conjugacy classes

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

56 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 2 2 4 4 4 4 4 4 type + + + + + + - + + image C1 C2 C2 C2 C2 C4 C4 C4 C8 M4(2) C4○D4 F5 C5⋊C8 C2×F5 C2×F5 D5⋊M4(2) D10.C23 kernel Dic5.12M4(2) C4×C5⋊C8 C20⋊C8 C23.2F5 C2×C4×Dic5 C4×Dic5 C22×Dic5 C22×C20 C2×C20 Dic5 Dic5 C22×C4 C2×C4 C2×C4 C23 C2 C2 # reps 1 2 2 2 1 4 2 2 16 4 4 1 4 2 1 4 4

Matrix representation of Dic5.12M4(2) in GL6(𝔽41)

 40 0 0 0 0 0 0 40 0 0 0 0 0 0 34 1 0 0 0 0 40 0 0 0 0 0 0 0 7 34 0 0 0 0 7 40
,
 9 0 0 0 0 0 0 9 0 0 0 0 0 0 7 40 0 0 0 0 7 34 0 0 0 0 0 0 7 40 0 0 0 0 7 34
,
 24 40 0 0 0 0 1 17 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 19 9 0 0 0 0 19 22 0 0
,
 1 0 0 0 0 0 7 40 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 40 0 0 0 0 0 0 40

G:=sub<GL(6,GF(41))| [40,0,0,0,0,0,0,40,0,0,0,0,0,0,34,40,0,0,0,0,1,0,0,0,0,0,0,0,7,7,0,0,0,0,34,40],[9,0,0,0,0,0,0,9,0,0,0,0,0,0,7,7,0,0,0,0,40,34,0,0,0,0,0,0,7,7,0,0,0,0,40,34],[24,1,0,0,0,0,40,17,0,0,0,0,0,0,0,0,19,19,0,0,0,0,9,22,0,0,1,0,0,0,0,0,0,1,0,0],[1,7,0,0,0,0,0,40,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,40,0,0,0,0,0,0,40] >;

Dic5.12M4(2) in GAP, Magma, Sage, TeX

{\rm Dic}_5._{12}M_4(2)
% in TeX

G:=Group("Dic5.12M4(2)");
// GroupNames label

G:=SmallGroup(320,1086);
// by ID

G=gap.SmallGroup(320,1086);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,56,120,422,136,6278,1595]);
// 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,b*c=c*b,b*d=d*b,d*c*d=a^5*c^5>;
// generators/relations

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