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## G = Dic5⋊2M4(2)  order 320 = 26·5

### 1st semidirect product of Dic5 and M4(2) acting via M4(2)/C8=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C10 — Dic5⋊2M4(2)
 Chief series C1 — C5 — C10 — C20 — C2×C20 — C2×C4×D5 — C4×C5⋊D4 — Dic5⋊2M4(2)
 Lower central C5 — C2×C10 — Dic5⋊2M4(2)
 Upper central C1 — C2×C4 — C22⋊C8

Generators and relations for Dic52M4(2)
G = < a,b,c,d | a10=c8=d2=1, b2=a5, bab-1=cac-1=a-1, ad=da, cbc-1=dbd=a5b, dcd=c5 >

Subgroups: 398 in 122 conjugacy classes, 51 normal (47 characteristic)
C1, C2 [×3], C2 [×2], C4 [×2], C4 [×5], C22, C22 [×6], C5, C8 [×5], C2×C4 [×2], C2×C4 [×7], D4 [×2], C23, C23, D5, C10 [×3], C10, C42, C22⋊C4 [×2], C4⋊C4, C2×C8 [×2], C2×C8 [×2], M4(2) [×4], C22×C4, C22×C4, C2×D4, Dic5 [×2], Dic5 [×2], C20 [×2], C20, D10 [×3], C2×C10, C2×C10 [×3], C4×C8, C22⋊C8, C22⋊C8, C4⋊C8, C4×D4, C2×M4(2) [×2], C52C8 [×2], C52C8, C40 [×2], C4×D5 [×2], C2×Dic5 [×3], C5⋊D4 [×2], C2×C20 [×2], C2×C20 [×2], C22×D5, C22×C10, C86D4, C8⋊D5 [×2], C2×C52C8 [×2], C4.Dic5 [×2], C4×Dic5, C10.D4, D10⋊C4, C23.D5, C2×C40 [×2], C2×C4×D5, C2×C5⋊D4, C22×C20, C8×Dic5, C20.8Q8, D101C8, C5×C22⋊C8, C2×C8⋊D5, C2×C4.Dic5, C4×C5⋊D4, Dic52M4(2)
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C2×C4 [×6], D4 [×2], C23, D5, M4(2) [×2], C22×C4, C2×D4, C4○D4, D10 [×3], C4×D4, C2×M4(2), C8○D4, C4×D5 [×2], C22×D5, C86D4, C2×C4×D5, D4×D5, D42D5, Dic54D4, D20.3C4, D5×M4(2), Dic52M4(2)

Smallest permutation representation of Dic52M4(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 94 6 99)(2 93 7 98)(3 92 8 97)(4 91 9 96)(5 100 10 95)(11 81 16 86)(12 90 17 85)(13 89 18 84)(14 88 19 83)(15 87 20 82)(21 109 26 104)(22 108 27 103)(23 107 28 102)(24 106 29 101)(25 105 30 110)(31 126 36 121)(32 125 37 130)(33 124 38 129)(34 123 39 128)(35 122 40 127)(41 116 46 111)(42 115 47 120)(43 114 48 119)(44 113 49 118)(45 112 50 117)(51 132 56 137)(52 131 57 136)(53 140 58 135)(54 139 59 134)(55 138 60 133)(61 150 66 145)(62 149 67 144)(63 148 68 143)(64 147 69 142)(65 146 70 141)(71 152 76 157)(72 151 77 156)(73 160 78 155)(74 159 79 154)(75 158 80 153)
(1 79 48 63 27 88 39 54)(2 78 49 62 28 87 40 53)(3 77 50 61 29 86 31 52)(4 76 41 70 30 85 32 51)(5 75 42 69 21 84 33 60)(6 74 43 68 22 83 34 59)(7 73 44 67 23 82 35 58)(8 72 45 66 24 81 36 57)(9 71 46 65 25 90 37 56)(10 80 47 64 26 89 38 55)(11 121 131 97 156 112 150 106)(12 130 132 96 157 111 141 105)(13 129 133 95 158 120 142 104)(14 128 134 94 159 119 143 103)(15 127 135 93 160 118 144 102)(16 126 136 92 151 117 145 101)(17 125 137 91 152 116 146 110)(18 124 138 100 153 115 147 109)(19 123 139 99 154 114 148 108)(20 122 140 98 155 113 149 107)
(1 103)(2 104)(3 105)(4 106)(5 107)(6 108)(7 109)(8 110)(9 101)(10 102)(11 85)(12 86)(13 87)(14 88)(15 89)(16 90)(17 81)(18 82)(19 83)(20 84)(21 98)(22 99)(23 100)(24 91)(25 92)(26 93)(27 94)(28 95)(29 96)(30 97)(31 111)(32 112)(33 113)(34 114)(35 115)(36 116)(37 117)(38 118)(39 119)(40 120)(41 121)(42 122)(43 123)(44 124)(45 125)(46 126)(47 127)(48 128)(49 129)(50 130)(51 131)(52 132)(53 133)(54 134)(55 135)(56 136)(57 137)(58 138)(59 139)(60 140)(61 141)(62 142)(63 143)(64 144)(65 145)(66 146)(67 147)(68 148)(69 149)(70 150)(71 151)(72 152)(73 153)(74 154)(75 155)(76 156)(77 157)(78 158)(79 159)(80 160)

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

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

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

68 conjugacy classes

 class 1 2A 2B 2C 2D 2E 4A 4B 4C 4D 4E 4F 4G 4H 4I 4J 5A 5B 8A 8B 8C 8D 8E 8F 8G 8H 8I 8J 8K 8L 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 4 4 5 5 8 8 8 8 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 4 20 1 1 1 1 4 10 10 10 10 20 2 2 2 2 2 2 4 4 10 10 10 10 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 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 4 4 4 type + + + + + + + + + + + + + - image C1 C2 C2 C2 C2 C2 C2 C2 C4 C4 C4 C4 D4 D5 M4(2) C4○D4 D10 D10 C8○D4 C4×D5 C4×D5 D20.3C4 D4×D5 D4⋊2D5 D5×M4(2) kernel Dic5⋊2M4(2) C8×Dic5 C20.8Q8 D10⋊1C8 C5×C22⋊C8 C2×C8⋊D5 C2×C4.Dic5 C4×C5⋊D4 C10.D4 D10⋊C4 C23.D5 C2×C5⋊D4 C5⋊2C8 C22⋊C8 Dic5 C20 C2×C8 C22×C4 C10 C2×C4 C23 C2 C4 C4 C2 # reps 1 1 1 1 1 1 1 1 2 2 2 2 2 2 4 2 4 2 4 4 4 16 2 2 4

Matrix representation of Dic52M4(2) in GL6(𝔽41)

 40 0 0 0 0 0 0 40 0 0 0 0 0 0 0 1 0 0 0 0 40 34 0 0 0 0 0 0 40 0 0 0 0 0 0 40
,
 9 5 0 0 0 0 0 32 0 0 0 0 0 0 40 0 0 0 0 0 7 1 0 0 0 0 0 0 40 8 0 0 0 0 10 1
,
 1 37 0 0 0 0 21 40 0 0 0 0 0 0 40 0 0 0 0 0 7 1 0 0 0 0 0 0 14 0 0 0 0 0 24 27
,
 32 36 0 0 0 0 16 9 0 0 0 0 0 0 40 0 0 0 0 0 0 40 0 0 0 0 0 0 40 8 0 0 0 0 0 1

G:=sub<GL(6,GF(41))| [40,0,0,0,0,0,0,40,0,0,0,0,0,0,0,40,0,0,0,0,1,34,0,0,0,0,0,0,40,0,0,0,0,0,0,40],[9,0,0,0,0,0,5,32,0,0,0,0,0,0,40,7,0,0,0,0,0,1,0,0,0,0,0,0,40,10,0,0,0,0,8,1],[1,21,0,0,0,0,37,40,0,0,0,0,0,0,40,7,0,0,0,0,0,1,0,0,0,0,0,0,14,24,0,0,0,0,0,27],[32,16,0,0,0,0,36,9,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,8,1] >;

Dic52M4(2) in GAP, Magma, Sage, TeX

{\rm Dic}_5\rtimes_2M_4(2)
% in TeX

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

G:=SmallGroup(320,356);
// by ID

G=gap.SmallGroup(320,356);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,701,422,219,58,136,12550]);
// Polycyclic

G:=Group<a,b,c,d|a^10=c^8=d^2=1,b^2=a^5,b*a*b^-1=c*a*c^-1=a^-1,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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