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## G = Dic5⋊M4(2)  order 320 = 26·5

### The semidirect product of Dic5 and M4(2) acting via M4(2)/C4=C22

Series: Derived Chief Lower central Upper central

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

Generators and relations for Dic5⋊M4(2)
G = < a,b,c,d | a10=c8=d2=1, b2=a5, bab-1=a-1, cac-1=a7, ad=da, cbc-1=dbd=a5b, dcd=c5 >

Subgroups: 442 in 122 conjugacy classes, 46 normal (42 characteristic)
C1, C2 [×3], C2 [×2], C4 [×7], C22, C22 [×6], C5, C8 [×5], C2×C4 [×2], C2×C4 [×7], D4 [×2], C23, C23, D5, C10 [×3], C10, C42, C22⋊C4, C22⋊C4, C4⋊C4, C2×C8 [×4], M4(2) [×4], C22×C4 [×2], C2×D4, Dic5 [×2], Dic5 [×2], Dic5, C20 [×2], D10 [×3], C2×C10, C2×C10 [×3], C4×C8, C22⋊C8 [×2], C4⋊C8, C4×D4, C2×M4(2) [×2], C5⋊C8 [×2], C5⋊C8 [×3], C4×D5 [×2], C2×Dic5 [×3], C2×Dic5 [×2], C5⋊D4 [×2], C2×C20 [×2], C22×D5, C22×C10, C86D4, C4×Dic5, C10.D4, D10⋊C4, C5×C22⋊C4, C4.F5 [×2], C2×C5⋊C8 [×4], C22.F5 [×2], C2×C4×D5, C22×Dic5, C2×C5⋊D4, C4×C5⋊C8, D10⋊C8, Dic5⋊C8, C23.2F5, Dic54D4, C2×C4.F5, C2×C22.F5, Dic5⋊M4(2)
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C2×C4 [×6], D4 [×2], C23, M4(2) [×2], C22×C4, C2×D4, C4○D4, F5, C4×D4, C2×M4(2), C8○D4, C2×F5 [×3], C86D4, C22×F5, D5⋊M4(2), D4.F5, D4×F5, Dic5⋊M4(2)

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

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

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

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

38 conjugacy classes

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

38 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 4 4 4 4 8 8 type + + + + + + + + + + + + - + image C1 C2 C2 C2 C2 C2 C2 C2 C4 C4 C4 C4 D4 M4(2) C4○D4 C8○D4 F5 C2×F5 C2×F5 D5⋊M4(2) D4.F5 D4×F5 kernel Dic5⋊M4(2) C4×C5⋊C8 D10⋊C8 Dic5⋊C8 C23.2F5 Dic5⋊4D4 C2×C4.F5 C2×C22.F5 C10.D4 D10⋊C4 C5×C22⋊C4 C2×C5⋊D4 C5⋊C8 Dic5 Dic5 C10 C22⋊C4 C2×C4 C23 C2 C2 C2 # reps 1 1 1 1 1 1 1 1 2 2 2 2 2 4 2 4 1 2 1 4 1 1

Matrix representation of Dic5⋊M4(2) in GL6(𝔽41)

 40 0 0 0 0 0 0 40 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 40 40 40 40
,
 1 2 0 0 0 0 40 40 0 0 0 0 0 0 22 0 3 3 0 0 19 22 22 19 0 0 3 3 0 22 0 0 0 38 19 38
,
 40 39 0 0 0 0 0 1 0 0 0 0 0 0 25 9 1 8 0 0 7 40 24 8 0 0 25 17 24 16 0 0 33 17 1 34
,
 1 2 0 0 0 0 0 40 0 0 0 0 0 0 22 0 3 3 0 0 38 19 38 0 0 0 0 38 19 38 0 0 3 3 0 22

G:=sub<GL(6,GF(41))| [40,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,0,40,0,0,1,0,0,40,0,0,0,1,0,40,0,0,0,0,1,40],[1,40,0,0,0,0,2,40,0,0,0,0,0,0,22,19,3,0,0,0,0,22,3,38,0,0,3,22,0,19,0,0,3,19,22,38],[40,0,0,0,0,0,39,1,0,0,0,0,0,0,25,7,25,33,0,0,9,40,17,17,0,0,1,24,24,1,0,0,8,8,16,34],[1,0,0,0,0,0,2,40,0,0,0,0,0,0,22,38,0,3,0,0,0,19,38,3,0,0,3,38,19,0,0,0,3,0,38,22] >;

Dic5⋊M4(2) in GAP, Magma, Sage, TeX

{\rm Dic}_5\rtimes M_4(2)
% in TeX

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

G:=SmallGroup(320,1033);
// by ID

G=gap.SmallGroup(320,1033);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,56,344,1094,219,136,6278,1595]);
// 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^7,a*d=d*a,c*b*c^-1=d*b*d=a^5*b,d*c*d=c^5>;
// generators/relations

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