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## G = Dic5⋊5M4(2)  order 320 = 26·5

### 2nd semidirect product of Dic5 and M4(2) acting via M4(2)/C2×C4=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C10 — Dic5⋊5M4(2)
 Chief series C1 — C5 — C10 — C20 — C2×C20 — C4×Dic5 — C2×C4×Dic5 — Dic5⋊5M4(2)
 Lower central C5 — C2×C10 — Dic5⋊5M4(2)
 Upper central C1 — C2×C4 — C2×M4(2)

Generators and relations for Dic55M4(2)
G = < a,b,c,d | a10=c8=d2=1, b2=a5, bab-1=a-1, ac=ca, ad=da, cbc-1=a5b, bd=db, dcd=c5 >

Subgroups: 334 in 126 conjugacy classes, 67 normal (25 characteristic)
C1, C2, C2 [×2], C2 [×2], C4 [×2], C4 [×2], C4 [×6], C22, C22 [×2], C22 [×2], C5, C8 [×4], C2×C4 [×2], C2×C4 [×4], C2×C4 [×8], C23, C10, C10 [×2], C10 [×2], C42 [×4], C2×C8 [×2], C2×C8 [×2], M4(2) [×4], C22×C4, C22×C4 [×2], Dic5 [×4], Dic5 [×2], C20 [×2], C20 [×2], C2×C10, C2×C10 [×2], C2×C10 [×2], C4⋊C8 [×4], C2×C42, C2×M4(2), C2×M4(2), C52C8 [×2], C40 [×2], C2×Dic5 [×4], C2×Dic5 [×4], C2×C20 [×2], C2×C20 [×4], C22×C10, C4⋊M4(2), C2×C52C8 [×2], C4.Dic5 [×2], C4×Dic5 [×2], C4×Dic5 [×2], C2×C40 [×2], C5×M4(2) [×2], C22×Dic5 [×2], C22×C20, C20.8Q8 [×4], C2×C4.Dic5, C2×C4×Dic5, C10×M4(2), Dic55M4(2)
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C2×C4 [×6], D4 [×2], Q8 [×2], C23, D5, C4⋊C4 [×4], M4(2) [×4], C22×C4, C2×D4, C2×Q8, D10 [×3], C2×C4⋊C4, C2×M4(2) [×2], Dic10 [×2], C4×D5 [×2], C5⋊D4 [×2], C22×D5, C4⋊M4(2), C10.D4 [×4], C2×Dic10, C2×C4×D5, C2×C5⋊D4, D5×M4(2) [×2], C2×C10.D4, Dic55M4(2)

Smallest permutation representation of Dic55M4(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)
(1 111 6 116)(2 120 7 115)(3 119 8 114)(4 118 9 113)(5 117 10 112)(11 63 16 68)(12 62 17 67)(13 61 18 66)(14 70 19 65)(15 69 20 64)(21 123 26 128)(22 122 27 127)(23 121 28 126)(24 130 29 125)(25 129 30 124)(31 94 36 99)(32 93 37 98)(33 92 38 97)(34 91 39 96)(35 100 40 95)(41 109 46 104)(42 108 47 103)(43 107 48 102)(44 106 49 101)(45 105 50 110)(51 154 56 159)(52 153 57 158)(53 152 58 157)(54 151 59 156)(55 160 60 155)(71 150 76 145)(72 149 77 144)(73 148 78 143)(74 147 79 142)(75 146 80 141)(81 140 86 135)(82 139 87 134)(83 138 88 133)(84 137 89 132)(85 136 90 131)
(1 90 43 54 27 74 34 63)(2 81 44 55 28 75 35 64)(3 82 45 56 29 76 36 65)(4 83 46 57 30 77 37 66)(5 84 47 58 21 78 38 67)(6 85 48 59 22 79 39 68)(7 86 49 60 23 80 40 69)(8 87 50 51 24 71 31 70)(9 88 41 52 25 72 32 61)(10 89 42 53 26 73 33 62)(11 111 136 107 156 127 142 91)(12 112 137 108 157 128 143 92)(13 113 138 109 158 129 144 93)(14 114 139 110 159 130 145 94)(15 115 140 101 160 121 146 95)(16 116 131 102 151 122 147 96)(17 117 132 103 152 123 148 97)(18 118 133 104 153 124 149 98)(19 119 134 105 154 125 150 99)(20 120 135 106 155 126 141 100)
(11 156)(12 157)(13 158)(14 159)(15 160)(16 151)(17 152)(18 153)(19 154)(20 155)(51 70)(52 61)(53 62)(54 63)(55 64)(56 65)(57 66)(58 67)(59 68)(60 69)(71 87)(72 88)(73 89)(74 90)(75 81)(76 82)(77 83)(78 84)(79 85)(80 86)(131 147)(132 148)(133 149)(134 150)(135 141)(136 142)(137 143)(138 144)(139 145)(140 146)

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

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

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

68 conjugacy classes

 class 1 2A 2B 2C 2D 2E 4A 4B 4C 4D 4E 4F 4G ··· 4N 5A 5B 8A 8B 8C 8D 8E 8F 8G 8H 10A ··· 10F 10G 10H 10I 10J 20A ··· 20H 20I 20J 20K 20L 40A ··· 40P order 1 2 2 2 2 2 4 4 4 4 4 4 4 ··· 4 5 5 8 8 8 8 8 8 8 8 10 ··· 10 10 10 10 10 20 ··· 20 20 20 20 20 40 ··· 40 size 1 1 1 1 2 2 1 1 1 1 2 2 10 ··· 10 2 2 4 4 4 4 20 20 20 20 2 ··· 2 4 4 4 4 2 ··· 2 4 4 4 4 4 ··· 4

68 irreducible representations

 dim 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 4 type + + + + + + - + + + - image C1 C2 C2 C2 C2 C4 C4 D4 Q8 D5 M4(2) D10 D10 Dic10 C4×D5 C5⋊D4 C4×D5 D5×M4(2) kernel Dic5⋊5M4(2) C20.8Q8 C2×C4.Dic5 C2×C4×Dic5 C10×M4(2) C4×Dic5 C22×Dic5 C2×C20 C2×C20 C2×M4(2) Dic5 C2×C8 C22×C4 C2×C4 C2×C4 C2×C4 C23 C2 # reps 1 4 1 1 1 4 4 2 2 2 8 4 2 8 4 8 4 8

Matrix representation of Dic55M4(2) in GL4(𝔽41) generated by

 40 0 0 0 0 40 0 0 0 0 7 40 0 0 1 0
,
 32 16 0 0 0 9 0 0 0 0 27 28 0 0 12 14
,
 9 2 0 0 5 32 0 0 0 0 17 1 0 0 40 24
,
 1 21 0 0 0 40 0 0 0 0 1 0 0 0 0 1
G:=sub<GL(4,GF(41))| [40,0,0,0,0,40,0,0,0,0,7,1,0,0,40,0],[32,0,0,0,16,9,0,0,0,0,27,12,0,0,28,14],[9,5,0,0,2,32,0,0,0,0,17,40,0,0,1,24],[1,0,0,0,21,40,0,0,0,0,1,0,0,0,0,1] >;

Dic55M4(2) in GAP, Magma, Sage, TeX

{\rm Dic}_5\rtimes_5M_4(2)
% in TeX

G:=Group("Dic5:5M4(2)");
// GroupNames label

G:=SmallGroup(320,745);
// by ID

G=gap.SmallGroup(320,745);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,112,422,387,58,136,12550]);
// Polycyclic

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

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