Copied to
clipboard

## G = C42.176D10order 320 = 26·5

### 176th non-split extension by C42 of D10 acting via D10/C5=C22

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C10 — C42.176D10
 Chief series C1 — C5 — C10 — C2×C10 — C22×D5 — C2×C4×D5 — D10⋊3Q8 — C42.176D10
 Lower central C5 — C2×C10 — C42.176D10
 Upper central C1 — C22 — C4⋊Q8

Generators and relations for C42.176D10
G = < a,b,c,d | a4=b4=1, c10=a2, d2=a2b2, ab=ba, cac-1=dad-1=a-1, cbc-1=b-1, dbd-1=a2b-1, dcd-1=b2c9 >

Subgroups: 558 in 192 conjugacy classes, 95 normal (19 characteristic)
C1, C2, C2 [×2], C2, C4 [×2], C4 [×13], C22, C22 [×3], C5, C2×C4, C2×C4 [×6], C2×C4 [×9], Q8 [×4], C23, D5, C10, C10 [×2], C42, C42 [×5], C22⋊C4 [×6], C4⋊C4 [×4], C4⋊C4 [×16], C22×C4, C2×Q8 [×2], Dic5 [×7], C20 [×2], C20 [×6], D10 [×3], C2×C10, C42⋊C2, C4×Q8 [×2], C22⋊Q8 [×2], C42.C2 [×5], C422C2 [×4], C4⋊Q8, C4×D5 [×2], C2×Dic5, C2×Dic5 [×6], C2×C20, C2×C20 [×6], C5×Q8 [×4], C22×D5, C22.35C24, C4×Dic5, C4×Dic5 [×4], C10.D4 [×10], C4⋊Dic5 [×6], D10⋊C4 [×6], C4×C20, C5×C4⋊C4 [×4], C2×C4×D5, Q8×C10 [×2], C20.6Q8, C42⋊D5, Dic5.Q8 [×4], C4⋊C4⋊D5 [×4], Q8×Dic5 [×2], D103Q8 [×2], C5×C4⋊Q8, C42.176D10
Quotients: C1, C2 [×15], C22 [×35], C23 [×15], D5, C4○D4 [×2], C24, D10 [×7], C2×C4○D4, 2- 1+4 [×2], C22×D5 [×7], C22.35C24, D42D5 [×2], C23×D5, C2×D42D5, Q8.10D10 [×2], C42.176D10

Smallest permutation representation of C42.176D10
On 160 points
Generators in S160
```(1 143 11 153)(2 154 12 144)(3 145 13 155)(4 156 14 146)(5 147 15 157)(6 158 16 148)(7 149 17 159)(8 160 18 150)(9 151 19 141)(10 142 20 152)(21 117 31 107)(22 108 32 118)(23 119 33 109)(24 110 34 120)(25 101 35 111)(26 112 36 102)(27 103 37 113)(28 114 38 104)(29 105 39 115)(30 116 40 106)(41 71 51 61)(42 62 52 72)(43 73 53 63)(44 64 54 74)(45 75 55 65)(46 66 56 76)(47 77 57 67)(48 68 58 78)(49 79 59 69)(50 70 60 80)(81 128 91 138)(82 139 92 129)(83 130 93 140)(84 121 94 131)(85 132 95 122)(86 123 96 133)(87 134 97 124)(88 125 98 135)(89 136 99 126)(90 127 100 137)
(1 86 115 50)(2 51 116 87)(3 88 117 52)(4 53 118 89)(5 90 119 54)(6 55 120 91)(7 92 101 56)(8 57 102 93)(9 94 103 58)(10 59 104 95)(11 96 105 60)(12 41 106 97)(13 98 107 42)(14 43 108 99)(15 100 109 44)(16 45 110 81)(17 82 111 46)(18 47 112 83)(19 84 113 48)(20 49 114 85)(21 62 155 135)(22 136 156 63)(23 64 157 137)(24 138 158 65)(25 66 159 139)(26 140 160 67)(27 68 141 121)(28 122 142 69)(29 70 143 123)(30 124 144 71)(31 72 145 125)(32 126 146 73)(33 74 147 127)(34 128 148 75)(35 76 149 129)(36 130 150 77)(37 78 151 131)(38 132 152 79)(39 80 153 133)(40 134 154 61)
(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 10 105 114)(2 113 106 9)(3 8 107 112)(4 111 108 7)(5 6 109 110)(11 20 115 104)(12 103 116 19)(13 18 117 102)(14 101 118 17)(15 16 119 120)(21 26 145 150)(22 149 146 25)(23 24 147 148)(27 40 151 144)(28 143 152 39)(29 38 153 142)(30 141 154 37)(31 36 155 160)(32 159 156 35)(33 34 157 158)(41 84 87 58)(42 57 88 83)(43 82 89 56)(44 55 90 81)(45 100 91 54)(46 53 92 99)(47 98 93 52)(48 51 94 97)(49 96 95 50)(59 86 85 60)(61 121 124 78)(62 77 125 140)(63 139 126 76)(64 75 127 138)(65 137 128 74)(66 73 129 136)(67 135 130 72)(68 71 131 134)(69 133 132 70)(79 123 122 80)```

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

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

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

50 conjugacy classes

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

50 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 2 2 2 4 4 4 type + + + + + + + + + + + + - - image C1 C2 C2 C2 C2 C2 C2 C2 D5 C4○D4 D10 D10 D10 2- 1+4 D4⋊2D5 Q8.10D10 kernel C42.176D10 C20.6Q8 C42⋊D5 Dic5.Q8 C4⋊C4⋊D5 Q8×Dic5 D10⋊3Q8 C5×C4⋊Q8 C4⋊Q8 C20 C42 C4⋊C4 C2×Q8 C10 C4 C2 # reps 1 1 1 4 4 2 2 1 2 4 2 8 4 2 4 8

Matrix representation of C42.176D10 in GL8(𝔽41)

 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 39 0 35 0 0 0 0 0 0 39 0 35 0 0 0 0 35 0 2 0 0 0 0 0 0 35 0 2
,
 17 40 14 11 0 0 0 0 1 24 14 14 0 0 0 0 13 22 17 1 0 0 0 0 28 13 40 24 0 0 0 0 0 0 0 0 17 35 0 0 0 0 0 0 7 24 0 0 0 0 0 0 0 0 17 35 0 0 0 0 0 0 7 24
,
 10 23 0 11 0 0 0 0 18 13 30 0 0 0 0 0 34 35 28 18 0 0 0 0 39 34 23 31 0 0 0 0 0 0 0 0 0 0 34 35 0 0 0 0 0 0 7 0 0 0 0 0 7 6 0 0 0 0 0 0 34 0 0 0
,
 31 18 0 30 0 0 0 0 6 10 30 36 0 0 0 0 10 19 28 32 0 0 0 0 19 0 23 13 0 0 0 0 0 0 0 0 0 0 7 1 0 0 0 0 0 0 34 34 0 0 0 0 34 40 0 0 0 0 0 0 7 7 0 0

`G:=sub<GL(8,GF(41))| [1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,39,0,35,0,0,0,0,0,0,39,0,35,0,0,0,0,35,0,2,0,0,0,0,0,0,35,0,2],[17,1,13,28,0,0,0,0,40,24,22,13,0,0,0,0,14,14,17,40,0,0,0,0,11,14,1,24,0,0,0,0,0,0,0,0,17,7,0,0,0,0,0,0,35,24,0,0,0,0,0,0,0,0,17,7,0,0,0,0,0,0,35,24],[10,18,34,39,0,0,0,0,23,13,35,34,0,0,0,0,0,30,28,23,0,0,0,0,11,0,18,31,0,0,0,0,0,0,0,0,0,0,7,34,0,0,0,0,0,0,6,0,0,0,0,0,34,7,0,0,0,0,0,0,35,0,0,0],[31,6,10,19,0,0,0,0,18,10,19,0,0,0,0,0,0,30,28,23,0,0,0,0,30,36,32,13,0,0,0,0,0,0,0,0,0,0,34,7,0,0,0,0,0,0,40,7,0,0,0,0,7,34,0,0,0,0,0,0,1,34,0,0] >;`

C42.176D10 in GAP, Magma, Sage, TeX

`C_4^2._{176}D_{10}`
`% in TeX`

`G:=Group("C4^2.176D10");`
`// GroupNames label`

`G:=SmallGroup(320,1403);`
`// by ID`

`G=gap.SmallGroup(320,1403);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,120,758,219,100,675,297,136,12550]);`
`// Polycyclic`

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

׿
×
𝔽