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## G = Dic5.9M4(2)  order 320 = 26·5

### 2nd 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 — Dic5.9M4(2)
 Chief series C1 — C5 — C10 — C20 — C2×C20 — C4×Dic5 — C2×C4×Dic5 — Dic5.9M4(2)
 Lower central C5 — C2×C10 — Dic5.9M4(2)
 Upper central C1 — C2×C4 — C22⋊C8

Generators and relations for Dic5.9M4(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=a5c5 >

Subgroups: 302 in 110 conjugacy classes, 53 normal (33 characteristic)
C1, C2, C2, C4, C4, C22, C22, C22, C5, C8, C2×C4, C2×C4, C23, C10, C10, C42, C2×C8, C2×C8, C22×C4, C22×C4, Dic5, Dic5, C20, C20, C2×C10, C2×C10, C2×C10, C8⋊C4, C22⋊C8, C22⋊C8, C4⋊C8, C2×C42, C52C8, C40, C2×Dic5, C2×Dic5, C2×C20, C2×C20, C22×C10, C42.6C4, C2×C52C8, C4×Dic5, C2×C40, C22×Dic5, C22×C20, C20.8Q8, C408C4, C20.55D4, C5×C22⋊C8, C2×C4×Dic5, Dic5.9M4(2)
Quotients: C1, C2, C4, C22, C2×C4, C23, D5, M4(2), C22×C4, C4○D4, D10, C42⋊C2, C2×M4(2), C4×D5, C22×D5, C42.6C4, C8⋊D5, C2×C4×D5, D42D5, C23.11D10, C2×C8⋊D5, D5×M4(2), Dic5.9M4(2)

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

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

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

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

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 1 2 2 2 2 2 2 2 2 2 4 4 type + + + + + + + + + - image C1 C2 C2 C2 C2 C2 C4 C4 D5 M4(2) C4○D4 M4(2) D10 D10 C4×D5 C4×D5 C8⋊D5 D4⋊2D5 D5×M4(2) kernel Dic5.9M4(2) C20.8Q8 C40⋊8C4 C20.55D4 C5×C22⋊C8 C2×C4×Dic5 C4×Dic5 C22×Dic5 C22⋊C8 Dic5 C20 C2×C10 C2×C8 C22×C4 C2×C4 C23 C22 C4 C2 # reps 1 2 2 1 1 1 4 4 2 4 4 4 4 2 4 4 16 4 4

Matrix representation of Dic5.9M4(2) in GL4(𝔽41) generated by

 34 40 0 0 8 1 0 0 0 0 40 0 0 0 0 40
,
 13 39 0 0 3 28 0 0 0 0 9 0 0 0 18 32
,
 33 39 0 0 16 8 0 0 0 0 4 37 0 0 14 37
,
 1 0 0 0 0 1 0 0 0 0 1 0 0 0 2 40
G:=sub<GL(4,GF(41))| [34,8,0,0,40,1,0,0,0,0,40,0,0,0,0,40],[13,3,0,0,39,28,0,0,0,0,9,18,0,0,0,32],[33,16,0,0,39,8,0,0,0,0,4,14,0,0,37,37],[1,0,0,0,0,1,0,0,0,0,1,2,0,0,0,40] >;

Dic5.9M4(2) in GAP, Magma, Sage, TeX

{\rm Dic}_5._9M_4(2)
% in TeX

G:=Group("Dic5.9M4(2)");
// GroupNames label

G:=SmallGroup(320,346);
// by ID

G=gap.SmallGroup(320,346);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,112,758,219,58,136,12550]);
// Polycyclic

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