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## G = C42.148D10order 320 = 26·5

### 148th non-split extension by C42 of D10 acting via D10/C5=C22

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C10 — C42.148D10
 Chief series C1 — C5 — C10 — C2×C10 — C22×D5 — C2×C4×D5 — D5×C4⋊C4 — C42.148D10
 Lower central C5 — C2×C10 — C42.148D10
 Upper central C1 — C22 — C42.C2

Generators and relations for C42.148D10
G = < a,b,c,d | a4=b4=1, c10=d2=b2, ab=ba, cac-1=dad-1=a-1, cbc-1=a2b-1, dbd-1=b-1, dcd-1=c9 >

Subgroups: 686 in 206 conjugacy classes, 103 normal (43 characteristic)
C1, C2 [×3], C2 [×2], C4 [×2], C4 [×14], C22, C22 [×4], C5, C2×C4 [×3], C2×C4 [×4], C2×C4 [×13], Q8 [×4], C23, D5 [×2], C10 [×3], C42, C42 [×3], C22⋊C4 [×4], C4⋊C4 [×2], C4⋊C4 [×4], C4⋊C4 [×14], C22×C4 [×3], C2×Q8 [×4], Dic5 [×2], Dic5 [×6], C20 [×2], C20 [×6], D10 [×2], D10 [×2], C2×C10, C2×C4⋊C4, C42⋊C2 [×2], C22⋊Q8 [×4], C42.C2, C42.C2 [×3], C4⋊Q8 [×4], Dic10 [×4], C4×D5 [×4], C4×D5 [×2], C2×Dic5 [×3], C2×Dic5 [×4], C2×C20 [×3], C2×C20 [×4], C22×D5, C23.41C23, C4×Dic5, C4×Dic5 [×2], C10.D4 [×2], C10.D4 [×6], C4⋊Dic5 [×2], C4⋊Dic5 [×4], D10⋊C4 [×2], D10⋊C4 [×2], C4×C20, C5×C4⋊C4 [×2], C5×C4⋊C4 [×4], C2×Dic10 [×4], C2×C4×D5, C2×C4×D5 [×2], C202Q8, C42⋊D5, C20⋊Q8, C20⋊Q8 [×2], Dic5.Q8 [×2], C4.Dic10, D5×C4⋊C4, C4⋊C47D5, D10⋊Q8 [×2], D102Q8 [×2], C5×C42.C2, C42.148D10
Quotients: C1, C2 [×15], C22 [×35], Q8 [×4], C23 [×15], D5, C2×Q8 [×6], C24, D10 [×7], C22×Q8, 2+ 1+4, 2- 1+4, C22×D5 [×7], C23.41C23, Q8×D5 [×2], C23×D5, C2×Q8×D5, D48D10, D4.10D10, C42.148D10

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

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

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

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

50 conjugacy classes

 class 1 2A 2B 2C 2D 2E 4A 4B 4C ··· 4H 4I 4J 4K ··· 4P 5A 5B 10A ··· 10F 20A ··· 20L 20M ··· 20T order 1 2 2 2 2 2 4 4 4 ··· 4 4 4 4 ··· 4 5 5 10 ··· 10 20 ··· 20 20 ··· 20 size 1 1 1 1 10 10 2 2 4 ··· 4 10 10 20 ··· 20 2 2 2 ··· 2 4 ··· 4 8 ··· 8

50 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 4 4 4 4 4 type + + + + + + + + + + + - + + + + - - + - image C1 C2 C2 C2 C2 C2 C2 C2 C2 C2 C2 Q8 D5 D10 D10 2+ 1+4 2- 1+4 Q8×D5 D4⋊8D10 D4.10D10 kernel C42.148D10 C20⋊2Q8 C42⋊D5 C20⋊Q8 Dic5.Q8 C4.Dic10 D5×C4⋊C4 C4⋊C4⋊7D5 D10⋊Q8 D10⋊2Q8 C5×C42.C2 C4×D5 C42.C2 C42 C4⋊C4 C10 C10 C4 C2 C2 # reps 1 1 1 3 2 1 1 1 2 2 1 4 2 2 12 1 1 4 4 4

Matrix representation of C42.148D10 in GL6(𝔽41)

 1 0 0 0 0 0 0 1 0 0 0 0 0 0 19 7 9 37 0 0 0 34 32 9 0 0 26 7 22 0 0 0 26 6 15 7
,
 0 1 0 0 0 0 40 0 0 0 0 0 0 0 2 28 0 0 0 0 13 39 0 0 0 0 0 33 39 28 0 0 33 34 13 2
,
 6 39 0 0 0 0 39 35 0 0 0 0 0 0 34 7 17 13 0 0 1 0 8 17 0 0 21 10 1 1 0 0 34 19 13 6
,
 35 2 0 0 0 0 2 6 0 0 0 0 0 0 34 1 13 17 0 0 1 34 17 8 0 0 21 10 1 1 0 0 34 39 6 13

`G:=sub<GL(6,GF(41))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,19,0,26,26,0,0,7,34,7,6,0,0,9,32,22,15,0,0,37,9,0,7],[0,40,0,0,0,0,1,0,0,0,0,0,0,0,2,13,0,33,0,0,28,39,33,34,0,0,0,0,39,13,0,0,0,0,28,2],[6,39,0,0,0,0,39,35,0,0,0,0,0,0,34,1,21,34,0,0,7,0,10,19,0,0,17,8,1,13,0,0,13,17,1,6],[35,2,0,0,0,0,2,6,0,0,0,0,0,0,34,1,21,34,0,0,1,34,10,39,0,0,13,17,1,6,0,0,17,8,1,13] >;`

C42.148D10 in GAP, Magma, Sage, TeX

`C_4^2._{148}D_{10}`
`% in TeX`

`G:=Group("C4^2.148D10");`
`// GroupNames label`

`G:=SmallGroup(320,1361);`
`// by ID`

`G=gap.SmallGroup(320,1361);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,120,219,268,675,297,192,12550]);`
`// Polycyclic`

`G:=Group<a,b,c,d|a^4=b^4=1,c^10=d^2=b^2,a*b=b*a,c*a*c^-1=d*a*d^-1=a^-1,c*b*c^-1=a^2*b^-1,d*b*d^-1=b^-1,d*c*d^-1=c^9>;`
`// generators/relations`

׿
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