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## G = M4(2).16D10order 320 = 26·5

### 16th non-split extension by M4(2) of D10 acting via D10/C5=C22

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C20 — M4(2).16D10
 Chief series C1 — C5 — C10 — C20 — C2×C20 — C2×Dic10 — D4.9D10 — M4(2).16D10
 Lower central C5 — C10 — C2×C20 — M4(2).16D10
 Upper central C1 — C2 — C2×C4 — C8.C22

Generators and relations for M4(2).16D10
G = < a,b,c,d | a8=b2=c10=1, d2=a6, bab=a5, cac-1=a3, dad-1=a3b, cbc-1=dbd-1=a4b, dcd-1=a2c-1 >

Subgroups: 318 in 100 conjugacy classes, 37 normal (all characteristic)
C1, C2, C2, C4, C4, C22, C22, C5, C8, C2×C4, C2×C4, D4, D4, Q8, Q8, C10, C10, C2×C8, M4(2), M4(2), SD16, Q16, C2×Q8, C2×Q8, C4○D4, Dic5, C20, C20, C2×C10, C2×C10, C4.10D4, C8.C4, C8○D4, C2×Q16, C8.C22, C8.C22, C52C8, C52C8, C40, Dic10, C2×Dic5, C2×C20, C2×C20, C5×D4, C5×D4, C5×Q8, C5×Q8, D4.5D4, C2×C52C8, C2×C52C8, C4.Dic5, C4.Dic5, D4.D5, C5⋊Q16, C5×M4(2), C5×SD16, C5×Q16, C2×Dic10, Q8×C10, C5×C4○D4, C20.53D4, C4.12D20, C20.10D4, C2×C5⋊Q16, D4.Dic5, D4.9D10, C5×C8.C22, M4(2).16D10
Quotients: C1, C2, C22, D4, C23, D5, C2×D4, C4○D4, D10, C4⋊D4, C5⋊D4, C22×D5, D4.5D4, D4×D5, D42D5, C2×C5⋊D4, Dic5⋊D4, M4(2).16D10

Smallest permutation representation of M4(2).16D10
On 160 points
Generators in S160
```(1 123 44 128 31 98 49 93)(2 129 50 124 32 94 45 99)(3 125 46 130 33 100 41 95)(4 121 42 126 34 96 47 91)(5 127 48 122 35 92 43 97)(6 73 143 78 16 89 148 84)(7 79 149 74 17 85 144 90)(8 75 145 80 18 81 150 86)(9 71 141 76 19 87 146 82)(10 77 147 72 20 83 142 88)(11 65 104 70 39 134 109 139)(12 61 110 66 40 140 105 135)(13 67 106 62 36 136 101 131)(14 63 102 68 37 132 107 137)(15 69 108 64 38 138 103 133)(21 58 116 53 28 152 111 157)(22 54 112 59 29 158 117 153)(23 60 118 55 30 154 113 159)(24 56 114 51 26 160 119 155)(25 52 120 57 27 156 115 151)
(1 114)(2 120)(3 116)(4 112)(5 118)(6 102)(7 108)(8 104)(9 110)(10 106)(11 150)(12 146)(13 142)(14 148)(15 144)(16 107)(17 103)(18 109)(19 105)(20 101)(21 41)(22 47)(23 43)(24 49)(25 45)(26 44)(27 50)(28 46)(29 42)(30 48)(31 119)(32 115)(33 111)(34 117)(35 113)(36 147)(37 143)(38 149)(39 145)(40 141)(51 98)(52 124)(53 100)(54 126)(55 92)(56 128)(57 94)(58 130)(59 96)(60 122)(61 76)(62 83)(63 78)(64 85)(65 80)(66 87)(67 72)(68 89)(69 74)(70 81)(71 135)(73 137)(75 139)(77 131)(79 133)(82 140)(84 132)(86 134)(88 136)(90 138)(91 158)(93 160)(95 152)(97 154)(99 156)(121 153)(123 155)(125 157)(127 159)(129 151)
(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 68 49 63 31 137 44 132)(2 62 45 67 32 131 50 136)(3 66 41 61 33 135 46 140)(4 70 47 65 34 139 42 134)(5 64 43 69 35 133 48 138)(6 128 148 123 16 93 143 98)(7 122 144 127 17 97 149 92)(8 126 150 121 18 91 145 96)(9 130 146 125 19 95 141 100)(10 124 142 129 20 99 147 94)(11 59 109 54 39 153 104 158)(12 53 105 58 40 157 110 152)(13 57 101 52 36 151 106 156)(14 51 107 56 37 155 102 160)(15 55 103 60 38 159 108 154)(21 82 111 87 28 76 116 71)(22 86 117 81 29 80 112 75)(23 90 113 85 30 74 118 79)(24 84 119 89 26 78 114 73)(25 88 115 83 27 72 120 77)```

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

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

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

38 conjugacy classes

 class 1 2A 2B 2C 4A 4B 4C 4D 4E 5A 5B 8A 8B 8C 8D 8E 8F 8G 10A 10B 10C 10D 10E 10F 20A 20B 20C 20D 20E ··· 20J 40A 40B 40C 40D order 1 2 2 2 4 4 4 4 4 5 5 8 8 8 8 8 8 8 10 10 10 10 10 10 20 20 20 20 20 ··· 20 40 40 40 40 size 1 1 2 4 2 2 4 8 40 2 2 8 10 10 20 20 20 40 2 2 4 4 8 8 4 4 4 4 8 ··· 8 8 8 8 8

38 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 4 4 4 8 type + + + + + + + + + + + + + + + - + - - image C1 C2 C2 C2 C2 C2 C2 C2 D4 D4 D4 D5 C4○D4 D10 D10 D10 C5⋊D4 C5⋊D4 D4.5D4 D4×D5 D4⋊2D5 M4(2).16D10 kernel M4(2).16D10 C20.53D4 C4.12D20 C20.10D4 C2×C5⋊Q16 D4.Dic5 D4.9D10 C5×C8.C22 C5⋊2C8 C5×D4 C5×Q8 C8.C22 C2×C10 M4(2) C2×Q8 C4○D4 D4 Q8 C5 C4 C22 C1 # reps 1 1 1 1 1 1 1 1 2 1 1 2 2 2 2 2 4 4 2 2 2 2

Matrix representation of M4(2).16D10 in GL6(𝔽41)

 1 0 0 0 0 0 29 40 0 0 0 0 0 0 23 15 17 17 0 0 4 0 19 0 0 0 17 12 18 18 0 0 11 0 17 0
,
 40 0 0 0 0 0 0 40 0 0 0 0 0 0 0 20 40 39 0 0 0 32 5 5 0 0 1 25 0 0 0 0 40 0 9 9
,
 23 0 0 0 0 0 12 25 0 0 0 0 0 0 40 16 0 0 0 0 0 1 0 0 0 0 0 20 40 39 0 0 0 0 0 1
,
 2 14 0 0 0 0 32 39 0 0 0 0 0 0 0 36 0 29 0 0 25 25 30 19 0 0 0 37 0 10 0 0 10 8 8 16

`G:=sub<GL(6,GF(41))| [1,29,0,0,0,0,0,40,0,0,0,0,0,0,23,4,17,11,0,0,15,0,12,0,0,0,17,19,18,17,0,0,17,0,18,0],[40,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,1,40,0,0,20,32,25,0,0,0,40,5,0,9,0,0,39,5,0,9],[23,12,0,0,0,0,0,25,0,0,0,0,0,0,40,0,0,0,0,0,16,1,20,0,0,0,0,0,40,0,0,0,0,0,39,1],[2,32,0,0,0,0,14,39,0,0,0,0,0,0,0,25,0,10,0,0,36,25,37,8,0,0,0,30,0,8,0,0,29,19,10,16] >;`

M4(2).16D10 in GAP, Magma, Sage, TeX

`M_4(2)._{16}D_{10}`
`% in TeX`

`G:=Group("M4(2).16D10");`
`// GroupNames label`

`G:=SmallGroup(320,831);`
`// by ID`

`G=gap.SmallGroup(320,831);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,224,253,254,219,184,1123,297,136,1684,851,438,102,12550]);`
`// Polycyclic`

`G:=Group<a,b,c,d|a^8=b^2=c^10=1,d^2=a^6,b*a*b=a^5,c*a*c^-1=a^3,d*a*d^-1=a^3*b,c*b*c^-1=d*b*d^-1=a^4*b,d*c*d^-1=a^2*c^-1>;`
`// generators/relations`

׿
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