Copied to
clipboard

## G = C42.154D10order 320 = 26·5

### 154th non-split extension by C42 of D10 acting via D10/C5=C22

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C10 — C42.154D10
 Chief series C1 — C5 — C10 — C2×C10 — C22×D5 — C2×C4×D5 — D10⋊Q8 — C42.154D10
 Lower central C5 — C2×C10 — C42.154D10
 Upper central C1 — C22 — C42.C2

Generators and relations for C42.154D10
G = < a,b,c,d | a4=b4=1, c10=d2=a2, ab=ba, cac-1=a-1b2, dad-1=a-1, cbc-1=b-1, dbd-1=a2b, dcd-1=c9 >

Subgroups: 590 in 192 conjugacy classes, 93 normal (91 characteristic)
C1, C2 [×3], C2, C4 [×15], C22, C22 [×3], C5, C2×C4 [×7], C2×C4 [×9], Q8 [×4], C23, D5, C10 [×3], C42, C42 [×5], C22⋊C4 [×6], C4⋊C4 [×6], C4⋊C4 [×14], C22×C4, C2×Q8 [×2], Dic5 [×2], Dic5 [×6], C20 [×7], D10 [×3], C2×C10, C42⋊C2, C4×Q8 [×2], C22⋊Q8 [×2], C42.C2, C42.C2 [×4], C422C2 [×4], C4⋊Q8, Dic10 [×4], C4×D5 [×2], C2×Dic5 [×7], C2×C20 [×7], C22×D5, C22.35C24, C4×Dic5 [×5], C10.D4 [×10], C4⋊Dic5 [×4], D10⋊C4 [×6], C4×C20, C5×C4⋊C4 [×6], C2×Dic10 [×2], C2×C4×D5, C4×Dic10, C422D5, Dic53Q8, C20⋊Q8, Dic5.Q8 [×3], C4.Dic10, C4⋊C47D5, D10⋊Q8 [×2], C4⋊C4⋊D5 [×3], C5×C42.C2, C42.154D10
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, C23×D5, Q8.10D10, D5×C4○D4, D4.10D10, C42.154D10

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

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

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

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

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 1 1 1 2 2 2 2 4 4 4 4 type + + + + + + + + + + + + + + - - image C1 C2 C2 C2 C2 C2 C2 C2 C2 C2 C2 D5 C4○D4 D10 D10 2- 1+4 Q8.10D10 D5×C4○D4 D4.10D10 kernel C42.154D10 C4×Dic10 C42⋊2D5 Dic5⋊3Q8 C20⋊Q8 Dic5.Q8 C4.Dic10 C4⋊C4⋊7D5 D10⋊Q8 C4⋊C4⋊D5 C5×C42.C2 C42.C2 Dic5 C42 C4⋊C4 C10 C2 C2 C2 # reps 1 1 1 1 1 3 1 1 2 3 1 2 4 2 12 2 4 4 4

Matrix representation of C42.154D10 in GL8(𝔽41)

 9 0 23 0 0 0 0 0 0 9 0 23 0 0 0 0 9 0 32 0 0 0 0 0 0 9 0 32 0 0 0 0 0 0 0 0 3 7 25 1 0 0 0 0 3 19 23 22 0 0 0 0 7 2 1 34 0 0 0 0 40 1 26 18
,
 1 0 39 0 0 0 0 0 0 1 0 39 0 0 0 0 1 0 40 0 0 0 0 0 0 1 0 40 0 0 0 0 0 0 0 0 17 40 22 21 0 0 0 0 1 24 12 7 0 0 0 0 0 0 18 5 0 0 0 0 0 0 1 23
,
 1 29 28 13 0 0 0 0 12 11 28 24 0 0 0 0 15 15 40 12 0 0 0 0 26 23 29 30 0 0 0 0 0 0 0 0 8 9 6 24 0 0 0 0 16 32 33 11 0 0 0 0 35 6 33 1 0 0 0 0 27 33 21 9
,
 1 29 28 13 0 0 0 0 35 40 17 13 0 0 0 0 27 14 29 1 0 0 0 0 12 14 11 12 0 0 0 0 0 0 0 0 27 32 21 5 0 0 0 0 38 9 24 30 0 0 0 0 34 35 11 2 0 0 0 0 1 8 23 35

`G:=sub<GL(8,GF(41))| [9,0,9,0,0,0,0,0,0,9,0,9,0,0,0,0,23,0,32,0,0,0,0,0,0,23,0,32,0,0,0,0,0,0,0,0,3,3,7,40,0,0,0,0,7,19,2,1,0,0,0,0,25,23,1,26,0,0,0,0,1,22,34,18],[1,0,1,0,0,0,0,0,0,1,0,1,0,0,0,0,39,0,40,0,0,0,0,0,0,39,0,40,0,0,0,0,0,0,0,0,17,1,0,0,0,0,0,0,40,24,0,0,0,0,0,0,22,12,18,1,0,0,0,0,21,7,5,23],[1,12,15,26,0,0,0,0,29,11,15,23,0,0,0,0,28,28,40,29,0,0,0,0,13,24,12,30,0,0,0,0,0,0,0,0,8,16,35,27,0,0,0,0,9,32,6,33,0,0,0,0,6,33,33,21,0,0,0,0,24,11,1,9],[1,35,27,12,0,0,0,0,29,40,14,14,0,0,0,0,28,17,29,11,0,0,0,0,13,13,1,12,0,0,0,0,0,0,0,0,27,38,34,1,0,0,0,0,32,9,35,8,0,0,0,0,21,24,11,23,0,0,0,0,5,30,2,35] >;`

C42.154D10 in GAP, Magma, Sage, TeX

`C_4^2._{154}D_{10}`
`% in TeX`

`G:=Group("C4^2.154D10");`
`// GroupNames label`

`G:=SmallGroup(320,1368);`
`// by ID`

`G=gap.SmallGroup(320,1368);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,120,758,555,100,1571,570,136,12550]);`
`// Polycyclic`

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

׿
×
𝔽