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## G = Dic5.14M4(2)  order 320 = 26·5

### 1st non-split extension by Dic5 of M4(2) acting through Inn(Dic5)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C10 — Dic5.14M4(2)
 Chief series C1 — C5 — C10 — C20 — C2×C20 — C4×Dic5 — C2×C4×Dic5 — Dic5.14M4(2)
 Lower central C5 — C10 — Dic5.14M4(2)
 Upper central C1 — C2×C4 — C22⋊C8

Generators and relations for Dic5.14M4(2)
G = < a,b,c,d | a10=c8=d2=1, b2=a5, bab-1=a-1, ac=ca, ad=da, bc=cb, bd=db, dcd=a5c5 >

Subgroups: 302 in 118 conjugacy classes, 61 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, C4×C8, C22⋊C8, C22⋊C8, C4⋊C8, C2×C42, C52C8, C40, C2×Dic5, C2×Dic5, C2×Dic5, C2×C20, C2×C20, C22×C10, C42.12C4, C2×C52C8, C4×Dic5, C2×C40, C22×Dic5, C22×C20, C8×Dic5, C20.8Q8, C20.55D4, C5×C22⋊C8, C2×C4×Dic5, Dic5.14M4(2)
Quotients: C1, C2, C4, C22, C8, C2×C4, C23, D5, C2×C8, M4(2), C22×C4, C4○D4, D10, C42⋊C2, C22×C8, C2×M4(2), C4×D5, C22×D5, C42.12C4, C8×D5, C2×C4×D5, D42D5, C23.11D10, D5×C2×C8, D5×M4(2), Dic5.14M4(2)

Smallest permutation representation of Dic5.14M4(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 63 16 68)(12 62 17 67)(13 61 18 66)(14 70 19 65)(15 69 20 64)(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)(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 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)(11 122 131 91 156 116 147 107)(12 123 132 92 157 117 148 108)(13 124 133 93 158 118 149 109)(14 125 134 94 159 119 150 110)(15 126 135 95 160 120 141 101)(16 127 136 96 151 111 142 102)(17 128 137 97 152 112 143 103)(18 129 138 98 153 113 144 104)(19 130 139 99 154 114 145 105)(20 121 140 100 155 115 146 106)
(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)(11,63,16,68)(12,62,17,67)(13,61,18,66)(14,70,19,65)(15,69,20,64)(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)(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), (1,85,48,54,21,79,39,63)(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)(11,122,131,91,156,116,147,107)(12,123,132,92,157,117,148,108)(13,124,133,93,158,118,149,109)(14,125,134,94,159,119,150,110)(15,126,135,95,160,120,141,101)(16,127,136,96,151,111,142,102)(17,128,137,97,152,112,143,103)(18,129,138,98,153,113,144,104)(19,130,139,99,154,114,145,105)(20,121,140,100,155,115,146,106), (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)(11,63,16,68)(12,62,17,67)(13,61,18,66)(14,70,19,65)(15,69,20,64)(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)(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), (1,85,48,54,21,79,39,63)(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)(11,122,131,91,156,116,147,107)(12,123,132,92,157,117,148,108)(13,124,133,93,158,118,149,109)(14,125,134,94,159,119,150,110)(15,126,135,95,160,120,141,101)(16,127,136,96,151,111,142,102)(17,128,137,97,152,112,143,103)(18,129,138,98,153,113,144,104)(19,130,139,99,154,114,145,105)(20,121,140,100,155,115,146,106), (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),(11,63,16,68),(12,62,17,67),(13,61,18,66),(14,70,19,65),(15,69,20,64),(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),(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)], [(1,85,48,54,21,79,39,63),(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),(11,122,131,91,156,116,147,107),(12,123,132,92,157,117,148,108),(13,124,133,93,158,118,149,109),(14,125,134,94,159,119,150,110),(15,126,135,95,160,120,141,101),(16,127,136,96,151,111,142,102),(17,128,137,97,152,112,143,103),(18,129,138,98,153,113,144,104),(19,130,139,99,154,114,145,105),(20,121,140,100,155,115,146,106)], [(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)]])

80 conjugacy classes

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

80 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 4 4 type + + + + + + + + + - image C1 C2 C2 C2 C2 C2 C4 C4 C8 D5 M4(2) C4○D4 D10 D10 C4×D5 C4×D5 C8×D5 D4⋊2D5 D5×M4(2) kernel Dic5.14M4(2) C8×Dic5 C20.8Q8 C20.55D4 C5×C22⋊C8 C2×C4×Dic5 C4×Dic5 C22×Dic5 C2×Dic5 C22⋊C8 Dic5 C20 C2×C8 C22×C4 C2×C4 C23 C22 C4 C2 # reps 1 2 2 1 1 1 4 4 16 2 4 4 4 2 4 4 16 4 4

Matrix representation of Dic5.14M4(2) in GL4(𝔽41) generated by

 7 40 0 0 1 0 0 0 0 0 1 0 0 0 0 1
,
 1 17 0 0 24 40 0 0 0 0 40 0 0 0 0 40
,
 27 0 0 0 0 27 0 0 0 0 38 9 0 0 0 3
,
 1 0 0 0 0 1 0 0 0 0 1 0 0 0 28 40
G:=sub<GL(4,GF(41))| [7,1,0,0,40,0,0,0,0,0,1,0,0,0,0,1],[1,24,0,0,17,40,0,0,0,0,40,0,0,0,0,40],[27,0,0,0,0,27,0,0,0,0,38,0,0,0,9,3],[1,0,0,0,0,1,0,0,0,0,1,28,0,0,0,40] >;

Dic5.14M4(2) in GAP, Magma, Sage, TeX

{\rm Dic}_5._{14}M_4(2)
% in TeX

G:=Group("Dic5.14M4(2)");
// GroupNames label

G:=SmallGroup(320,345);
// by ID

G=gap.SmallGroup(320,345);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,112,219,58,136,12550]);
// 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,b*c=c*b,b*d=d*b,d*c*d=a^5*c^5>;
// generators/relations

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