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## G = C42.104D10order 320 = 26·5

### 104th non-split extension by C42 of D10 acting via D10/C5=C22

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C10 — C42.104D10
 Chief series C1 — C5 — C10 — C2×C10 — C2×Dic5 — C22×Dic5 — C2×C10.D4 — C42.104D10
 Lower central C5 — C2×C10 — C42.104D10
 Upper central C1 — C22 — C4×D4

Generators and relations for C42.104D10
G = < a,b,c,d | a4=b4=c10=1, d2=b2, ab=ba, ac=ca, dad-1=ab2, cbc-1=dbd-1=a2b-1, dcd-1=c-1 >

Subgroups: 838 in 238 conjugacy classes, 97 normal (91 characteristic)
C1, C2 [×3], C2 [×5], C4 [×12], C22, C22 [×2], C22 [×11], C5, C2×C4 [×5], C2×C4 [×14], D4 [×10], C23 [×2], C23 [×2], D5 [×2], C10 [×3], C10 [×3], C42, C42 [×2], C22⋊C4 [×2], C22⋊C4 [×8], C4⋊C4, C4⋊C4 [×9], C22×C4 [×2], C22×C4 [×4], C2×D4, C2×D4 [×5], Dic5 [×2], Dic5 [×5], C20 [×5], D10 [×6], C2×C10, C2×C10 [×2], C2×C10 [×5], C2×C4⋊C4, C42⋊C2, C4×D4, C4×D4 [×3], C4⋊D4 [×4], C22.D4 [×2], C42.C2, C422C2 [×2], C4×D5 [×3], D20, C2×Dic5 [×6], C2×Dic5 [×3], C5⋊D4 [×7], C2×C20 [×5], C2×C20 [×2], C5×D4 [×2], C22×D5 [×2], C22×C10 [×2], C22.47C24, C4×Dic5 [×2], C10.D4 [×8], C4⋊Dic5, D10⋊C4 [×6], C23.D5 [×2], C4×C20, C5×C22⋊C4 [×2], C5×C4⋊C4, C2×C4×D5 [×2], C2×D20, C22×Dic5 [×2], C2×C5⋊D4 [×4], C22×C20 [×2], D4×C10, C42⋊D5, C422D5, C23.D10, Dic54D4 [×2], D10⋊D4, Dic5.Q8, D10.13D4, C2×C10.D4, C4×C5⋊D4, C23.23D10, C207D4, Dic5⋊D4 [×2], D4×C20, C42.104D10
Quotients: C1, C2 [×15], C22 [×35], C23 [×15], D5, C4○D4 [×4], C24, D10 [×7], C2×C4○D4 [×2], 2+ 1+4, C22×D5 [×7], C22.47C24, C4○D20 [×2], C23×D5, C2×C4○D20, D46D10, D5×C4○D4, C42.104D10

Smallest permutation representation of C42.104D10
On 160 points
Generators in S160
```(1 119 99 36)(2 120 100 37)(3 111 91 38)(4 112 92 39)(5 113 93 40)(6 114 94 31)(7 115 95 32)(8 116 96 33)(9 117 97 34)(10 118 98 35)(11 101 48 28)(12 102 49 29)(13 103 50 30)(14 104 41 21)(15 105 42 22)(16 106 43 23)(17 107 44 24)(18 108 45 25)(19 109 46 26)(20 110 47 27)(51 144 124 71)(52 145 125 72)(53 146 126 73)(54 147 127 74)(55 148 128 75)(56 149 129 76)(57 150 130 77)(58 141 121 78)(59 142 122 79)(60 143 123 80)(61 154 134 87)(62 155 135 88)(63 156 136 89)(64 157 137 90)(65 158 138 81)(66 159 139 82)(67 160 140 83)(68 151 131 84)(69 152 132 85)(70 153 133 86)
(1 133 123 26)(2 110 124 61)(3 135 125 28)(4 102 126 63)(5 137 127 30)(6 104 128 65)(7 139 129 22)(8 106 130 67)(9 131 121 24)(10 108 122 69)(11 111 88 72)(12 146 89 39)(13 113 90 74)(14 148 81 31)(15 115 82 76)(16 150 83 33)(17 117 84 78)(18 142 85 35)(19 119 86 80)(20 144 87 37)(21 55 138 94)(23 57 140 96)(25 59 132 98)(27 51 134 100)(29 53 136 92)(32 159 149 42)(34 151 141 44)(36 153 143 46)(38 155 145 48)(40 157 147 50)(41 75 158 114)(43 77 160 116)(45 79 152 118)(47 71 154 120)(49 73 156 112)(52 101 91 62)(54 103 93 64)(56 105 95 66)(58 107 97 68)(60 109 99 70)
(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 25 123 132)(2 24 124 131)(3 23 125 140)(4 22 126 139)(5 21 127 138)(6 30 128 137)(7 29 129 136)(8 28 130 135)(9 27 121 134)(10 26 122 133)(11 116 88 77)(12 115 89 76)(13 114 90 75)(14 113 81 74)(15 112 82 73)(16 111 83 72)(17 120 84 71)(18 119 85 80)(19 118 86 79)(20 117 87 78)(31 157 148 50)(32 156 149 49)(33 155 150 48)(34 154 141 47)(35 153 142 46)(36 152 143 45)(37 151 144 44)(38 160 145 43)(39 159 146 42)(40 158 147 41)(51 68 100 107)(52 67 91 106)(53 66 92 105)(54 65 93 104)(55 64 94 103)(56 63 95 102)(57 62 96 101)(58 61 97 110)(59 70 98 109)(60 69 99 108)```

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

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

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

65 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 2H 4A ··· 4F 4G 4H 4I 4J 4K 4L 4M 4N 4O 4P 5A 5B 10A ··· 10F 10G ··· 10N 20A ··· 20H 20I ··· 20X order 1 2 2 2 2 2 2 2 2 4 ··· 4 4 4 4 4 4 4 4 4 4 4 5 5 10 ··· 10 10 ··· 10 20 ··· 20 20 ··· 20 size 1 1 1 1 2 2 4 20 20 2 ··· 2 4 4 10 10 10 10 20 20 20 20 2 2 2 ··· 2 4 ··· 4 2 ··· 2 4 ··· 4

65 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 4 4 4 type + + + + + + + + + + + + + + + + + + + + + image C1 C2 C2 C2 C2 C2 C2 C2 C2 C2 C2 C2 C2 C2 D5 C4○D4 C4○D4 D10 D10 D10 D10 D10 C4○D20 2+ 1+4 D4⋊6D10 D5×C4○D4 kernel C42.104D10 C42⋊D5 C42⋊2D5 C23.D10 Dic5⋊4D4 D10⋊D4 Dic5.Q8 D10.13D4 C2×C10.D4 C4×C5⋊D4 C23.23D10 C20⋊7D4 Dic5⋊D4 D4×C20 C4×D4 Dic5 C2×C10 C42 C22⋊C4 C4⋊C4 C22×C4 C2×D4 C22 C10 C2 C2 # reps 1 1 1 1 2 1 1 1 1 1 1 1 2 1 2 4 4 2 4 2 4 2 16 1 4 4

Matrix representation of C42.104D10 in GL4(𝔽41) generated by

 30 9 0 0 32 11 0 0 0 0 9 0 0 0 0 9
,
 9 0 0 0 0 9 0 0 0 0 40 20 0 0 0 1
,
 1 34 0 0 7 34 0 0 0 0 1 0 0 0 37 40
,
 19 9 0 0 19 22 0 0 0 0 40 0 0 0 4 1
`G:=sub<GL(4,GF(41))| [30,32,0,0,9,11,0,0,0,0,9,0,0,0,0,9],[9,0,0,0,0,9,0,0,0,0,40,0,0,0,20,1],[1,7,0,0,34,34,0,0,0,0,1,37,0,0,0,40],[19,19,0,0,9,22,0,0,0,0,40,4,0,0,0,1] >;`

C42.104D10 in GAP, Magma, Sage, TeX

`C_4^2._{104}D_{10}`
`% in TeX`

`G:=Group("C4^2.104D10");`
`// GroupNames label`

`G:=SmallGroup(320,1212);`
`// by ID`

`G=gap.SmallGroup(320,1212);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,224,758,100,794,136,12550]);`
`// Polycyclic`

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

׿
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