Copied to
clipboard

## G = C42.141D6order 192 = 26·3

### 141st non-split extension by C42 of D6 acting via D6/C3=C22

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C6 — C42.141D6
 Chief series C1 — C3 — C6 — C2×C6 — C22×S3 — S3×C2×C4 — C2×S3×Q8 — C42.141D6
 Lower central C3 — C2×C6 — C42.141D6
 Upper central C1 — C22 — C4.4D4

Generators and relations for C42.141D6
G = < a,b,c,d | a4=b4=1, c6=d2=a2, ab=ba, cac-1=dad-1=a-1, cbc-1=a2b-1, dbd-1=b-1, dcd-1=c5 >

Subgroups: 656 in 270 conjugacy classes, 103 normal (29 characteristic)
C1, C2, C2 [×2], C2 [×4], C3, C4 [×2], C4 [×12], C22, C22 [×10], S3 [×2], C6, C6 [×2], C6 [×2], C2×C4, C2×C4 [×4], C2×C4 [×19], D4 [×6], Q8 [×10], C23 [×2], C23, Dic3 [×2], Dic3 [×6], C12 [×2], C12 [×4], D6 [×2], D6 [×2], C2×C6, C2×C6 [×6], C42, C42, C22⋊C4 [×4], C22⋊C4 [×6], C4⋊C4 [×10], C22×C4 [×5], C2×D4, C2×D4 [×2], C2×Q8, C2×Q8 [×8], C4○D4 [×4], Dic6 [×8], C4×S3 [×4], C4×S3 [×4], C2×Dic3, C2×Dic3 [×6], C2×Dic3 [×4], C3⋊D4 [×4], C2×C12, C2×C12 [×4], C3×D4 [×2], C3×Q8 [×2], C22×S3, C22×C6 [×2], C42⋊C2, C22⋊Q8 [×4], C22.D4 [×4], C4.4D4, C4.4D4, C4⋊Q8 [×2], C22×Q8, C2×C4○D4, C4×Dic3, Dic3⋊C4 [×6], C4⋊Dic3 [×4], D6⋊C4 [×2], C6.D4 [×4], C4×C12, C3×C22⋊C4 [×4], C2×Dic6 [×2], C2×Dic6 [×2], S3×C2×C4, S3×C2×C4 [×2], D42S3 [×4], S3×Q8 [×4], C22×Dic3 [×2], C2×C3⋊D4 [×2], C6×D4, C6×Q8, C23.38C23, C122Q8, C422S3, Dic3.D4 [×4], C23.9D6 [×4], C23.12D6, Dic3⋊Q8, C3×C4.4D4, C2×D42S3, C2×S3×Q8, C42.141D6
Quotients: C1, C2 [×15], C22 [×35], S3, D4 [×4], C23 [×15], D6 [×7], C2×D4 [×6], C24, C22×S3 [×7], C22×D4, 2- 1+4 [×2], S3×D4 [×2], S3×C23, C23.38C23, C2×S3×D4, Q8○D12 [×2], C42.141D6

Smallest permutation representation of C42.141D6
On 96 points
Generators in S96
```(1 89 7 95)(2 96 8 90)(3 91 9 85)(4 86 10 92)(5 93 11 87)(6 88 12 94)(13 59 19 53)(14 54 20 60)(15 49 21 55)(16 56 22 50)(17 51 23 57)(18 58 24 52)(25 72 31 66)(26 67 32 61)(27 62 33 68)(28 69 34 63)(29 64 35 70)(30 71 36 65)(37 77 43 83)(38 84 44 78)(39 79 45 73)(40 74 46 80)(41 81 47 75)(42 76 48 82)
(1 73 36 19)(2 14 25 80)(3 75 26 21)(4 16 27 82)(5 77 28 23)(6 18 29 84)(7 79 30 13)(8 20 31 74)(9 81 32 15)(10 22 33 76)(11 83 34 17)(12 24 35 78)(37 63 51 87)(38 94 52 70)(39 65 53 89)(40 96 54 72)(41 67 55 91)(42 86 56 62)(43 69 57 93)(44 88 58 64)(45 71 59 95)(46 90 60 66)(47 61 49 85)(48 92 50 68)
(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)
(1 6 7 12)(2 11 8 5)(3 4 9 10)(13 24 19 18)(14 17 20 23)(15 22 21 16)(25 34 31 28)(26 27 32 33)(29 30 35 36)(37 40 43 46)(38 45 44 39)(41 48 47 42)(49 56 55 50)(51 54 57 60)(52 59 58 53)(61 62 67 68)(63 72 69 66)(64 65 70 71)(73 84 79 78)(74 77 80 83)(75 82 81 76)(85 86 91 92)(87 96 93 90)(88 89 94 95)```

`G:=sub<Sym(96)| (1,89,7,95)(2,96,8,90)(3,91,9,85)(4,86,10,92)(5,93,11,87)(6,88,12,94)(13,59,19,53)(14,54,20,60)(15,49,21,55)(16,56,22,50)(17,51,23,57)(18,58,24,52)(25,72,31,66)(26,67,32,61)(27,62,33,68)(28,69,34,63)(29,64,35,70)(30,71,36,65)(37,77,43,83)(38,84,44,78)(39,79,45,73)(40,74,46,80)(41,81,47,75)(42,76,48,82), (1,73,36,19)(2,14,25,80)(3,75,26,21)(4,16,27,82)(5,77,28,23)(6,18,29,84)(7,79,30,13)(8,20,31,74)(9,81,32,15)(10,22,33,76)(11,83,34,17)(12,24,35,78)(37,63,51,87)(38,94,52,70)(39,65,53,89)(40,96,54,72)(41,67,55,91)(42,86,56,62)(43,69,57,93)(44,88,58,64)(45,71,59,95)(46,90,60,66)(47,61,49,85)(48,92,50,68), (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), (1,6,7,12)(2,11,8,5)(3,4,9,10)(13,24,19,18)(14,17,20,23)(15,22,21,16)(25,34,31,28)(26,27,32,33)(29,30,35,36)(37,40,43,46)(38,45,44,39)(41,48,47,42)(49,56,55,50)(51,54,57,60)(52,59,58,53)(61,62,67,68)(63,72,69,66)(64,65,70,71)(73,84,79,78)(74,77,80,83)(75,82,81,76)(85,86,91,92)(87,96,93,90)(88,89,94,95)>;`

`G:=Group( (1,89,7,95)(2,96,8,90)(3,91,9,85)(4,86,10,92)(5,93,11,87)(6,88,12,94)(13,59,19,53)(14,54,20,60)(15,49,21,55)(16,56,22,50)(17,51,23,57)(18,58,24,52)(25,72,31,66)(26,67,32,61)(27,62,33,68)(28,69,34,63)(29,64,35,70)(30,71,36,65)(37,77,43,83)(38,84,44,78)(39,79,45,73)(40,74,46,80)(41,81,47,75)(42,76,48,82), (1,73,36,19)(2,14,25,80)(3,75,26,21)(4,16,27,82)(5,77,28,23)(6,18,29,84)(7,79,30,13)(8,20,31,74)(9,81,32,15)(10,22,33,76)(11,83,34,17)(12,24,35,78)(37,63,51,87)(38,94,52,70)(39,65,53,89)(40,96,54,72)(41,67,55,91)(42,86,56,62)(43,69,57,93)(44,88,58,64)(45,71,59,95)(46,90,60,66)(47,61,49,85)(48,92,50,68), (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), (1,6,7,12)(2,11,8,5)(3,4,9,10)(13,24,19,18)(14,17,20,23)(15,22,21,16)(25,34,31,28)(26,27,32,33)(29,30,35,36)(37,40,43,46)(38,45,44,39)(41,48,47,42)(49,56,55,50)(51,54,57,60)(52,59,58,53)(61,62,67,68)(63,72,69,66)(64,65,70,71)(73,84,79,78)(74,77,80,83)(75,82,81,76)(85,86,91,92)(87,96,93,90)(88,89,94,95) );`

`G=PermutationGroup([(1,89,7,95),(2,96,8,90),(3,91,9,85),(4,86,10,92),(5,93,11,87),(6,88,12,94),(13,59,19,53),(14,54,20,60),(15,49,21,55),(16,56,22,50),(17,51,23,57),(18,58,24,52),(25,72,31,66),(26,67,32,61),(27,62,33,68),(28,69,34,63),(29,64,35,70),(30,71,36,65),(37,77,43,83),(38,84,44,78),(39,79,45,73),(40,74,46,80),(41,81,47,75),(42,76,48,82)], [(1,73,36,19),(2,14,25,80),(3,75,26,21),(4,16,27,82),(5,77,28,23),(6,18,29,84),(7,79,30,13),(8,20,31,74),(9,81,32,15),(10,22,33,76),(11,83,34,17),(12,24,35,78),(37,63,51,87),(38,94,52,70),(39,65,53,89),(40,96,54,72),(41,67,55,91),(42,86,56,62),(43,69,57,93),(44,88,58,64),(45,71,59,95),(46,90,60,66),(47,61,49,85),(48,92,50,68)], [(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)], [(1,6,7,12),(2,11,8,5),(3,4,9,10),(13,24,19,18),(14,17,20,23),(15,22,21,16),(25,34,31,28),(26,27,32,33),(29,30,35,36),(37,40,43,46),(38,45,44,39),(41,48,47,42),(49,56,55,50),(51,54,57,60),(52,59,58,53),(61,62,67,68),(63,72,69,66),(64,65,70,71),(73,84,79,78),(74,77,80,83),(75,82,81,76),(85,86,91,92),(87,96,93,90),(88,89,94,95)])`

36 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 3 4A 4B 4C 4D 4E 4F 4G 4H 4I ··· 4N 6A 6B 6C 6D 6E 12A ··· 12F 12G 12H order 1 2 2 2 2 2 2 2 3 4 4 4 4 4 4 4 4 4 ··· 4 6 6 6 6 6 12 ··· 12 12 12 size 1 1 1 1 4 4 6 6 2 2 2 4 4 4 4 6 6 12 ··· 12 2 2 2 8 8 4 ··· 4 8 8

36 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 4 4 4 type + + + + + + + + + + + + + + + + - + - image C1 C2 C2 C2 C2 C2 C2 C2 C2 C2 S3 D4 D6 D6 D6 D6 2- 1+4 S3×D4 Q8○D12 kernel C42.141D6 C12⋊2Q8 C42⋊2S3 Dic3.D4 C23.9D6 C23.12D6 Dic3⋊Q8 C3×C4.4D4 C2×D4⋊2S3 C2×S3×Q8 C4.4D4 C4×S3 C42 C22⋊C4 C2×D4 C2×Q8 C6 C4 C2 # reps 1 1 1 4 4 1 1 1 1 1 1 4 1 4 1 1 2 2 4

Matrix representation of C42.141D6 in GL8(𝔽13)

 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 12 0 2 2 0 0 0 0 0 12 9 2 0 0 0 0 4 9 1 0 0 0 0 0 8 4 0 1
,
 0 0 12 0 0 0 0 0 0 0 0 12 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 3 6 0 0 0 0 0 0 7 10 0 0 0 0 0 0 0 0 10 6 0 0 0 0 0 0 7 3
,
 0 12 0 0 0 0 0 0 1 12 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 12 1 0 0 0 0 0 0 0 0 12 12 5 3 0 0 0 0 1 0 5 5 0 0 0 0 2 1 0 1 0 0 0 0 1 2 12 1
,
 1 12 0 0 0 0 0 0 0 12 0 0 0 0 0 0 0 0 12 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 12 12 5 3 0 0 0 0 0 1 3 5 0 0 0 0 1 2 12 1 0 0 0 0 2 1 0 1

`G:=sub<GL(8,GF(13))| [1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,12,0,4,8,0,0,0,0,0,12,9,4,0,0,0,0,2,9,1,0,0,0,0,0,2,2,0,1],[0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,12,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,0,0,3,7,0,0,0,0,0,0,6,10,0,0,0,0,0,0,0,0,10,7,0,0,0,0,0,0,6,3],[0,1,0,0,0,0,0,0,12,12,0,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,1,1,0,0,0,0,0,0,0,0,12,1,2,1,0,0,0,0,12,0,1,2,0,0,0,0,5,5,0,12,0,0,0,0,3,5,1,1],[1,0,0,0,0,0,0,0,12,12,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,1,1,0,0,0,0,0,0,0,0,12,0,1,2,0,0,0,0,12,1,2,1,0,0,0,0,5,3,12,0,0,0,0,0,3,5,1,1] >;`

C42.141D6 in GAP, Magma, Sage, TeX

`C_4^2._{141}D_6`
`% in TeX`

`G:=Group("C4^2.141D6");`
`// GroupNames label`

`G:=SmallGroup(192,1234);`
`// by ID`

`G=gap.SmallGroup(192,1234);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,120,219,100,675,297,192,6278]);`
`// Polycyclic`

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

׿
×
𝔽