Copied to
clipboard

## G = C6.442+ 1+4order 192 = 26·3

### 44th non-split extension by C6 of 2+ 1+4 acting via 2+ 1+4/C2×D4=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C6 — C6.442+ 1+4
 Chief series C1 — C3 — C6 — C2×C6 — C22×S3 — S3×C2×C4 — C23.9D6 — C6.442+ 1+4
 Lower central C3 — C2×C6 — C6.442+ 1+4
 Upper central C1 — C22 — C4⋊D4

Generators and relations for C6.442+ 1+4
G = < a,b,c,d,e | a6=b4=1, c2=e2=a3, d2=b2, ab=ba, ac=ca, dad-1=a-1, ae=ea, cbc-1=a3b-1, bd=db, ebe-1=a3b, cd=dc, ce=ec, ede-1=a3b2d >

Subgroups: 656 in 240 conjugacy classes, 93 normal (91 characteristic)
C1, C2, C2, C3, C4, C22, C22, S3, C6, C6, C2×C4, C2×C4, D4, C23, C23, Dic3, Dic3, C12, D6, C2×C6, C2×C6, C42, C22⋊C4, C22⋊C4, C4⋊C4, C4⋊C4, C22×C4, C22×C4, C2×D4, C2×D4, C4×S3, D12, C2×Dic3, C2×Dic3, C3⋊D4, C2×C12, C2×C12, C3×D4, C22×S3, C22×C6, C42⋊C2, C4×D4, C4⋊D4, C4⋊D4, C22.D4, C42.C2, C41D4, C4×Dic3, Dic3⋊C4, C4⋊Dic3, D6⋊C4, C6.D4, C3×C22⋊C4, C3×C4⋊C4, S3×C2×C4, C2×D12, C22×Dic3, C2×C3⋊D4, C22×C12, C6×D4, C22.34C24, C23.16D6, Dic34D4, C23.9D6, Dic3⋊D4, Dic3.Q8, D6.D4, C4×C3⋊D4, C23.28D6, C23.23D6, D63D4, C23.14D6, C123D4, C3×C4⋊D4, C6.442+ 1+4
Quotients: C1, C2, C22, S3, C23, D6, C4○D4, C24, C22×S3, C2×C4○D4, 2+ 1+4, S3×C23, C22.34C24, D46D6, S3×C4○D4, C6.442+ 1+4

Smallest permutation representation of C6.442+ 1+4
On 96 points
Generators in S96
```(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 82 10 73)(2 83 11 74)(3 84 12 75)(4 79 7 76)(5 80 8 77)(6 81 9 78)(13 94 22 85)(14 95 23 86)(15 96 24 87)(16 91 19 88)(17 92 20 89)(18 93 21 90)(25 58 34 49)(26 59 35 50)(27 60 36 51)(28 55 31 52)(29 56 32 53)(30 57 33 54)(37 70 46 61)(38 71 47 62)(39 72 48 63)(40 67 43 64)(41 68 44 65)(42 69 45 66)
(1 52 4 49)(2 53 5 50)(3 54 6 51)(7 58 10 55)(8 59 11 56)(9 60 12 57)(13 64 16 61)(14 65 17 62)(15 66 18 63)(19 70 22 67)(20 71 23 68)(21 72 24 69)(25 76 28 73)(26 77 29 74)(27 78 30 75)(31 82 34 79)(32 83 35 80)(33 84 36 81)(37 88 40 85)(38 89 41 86)(39 90 42 87)(43 94 46 91)(44 95 47 92)(45 96 48 93)
(1 22 10 13)(2 21 11 18)(3 20 12 17)(4 19 7 16)(5 24 8 15)(6 23 9 14)(25 46 34 37)(26 45 35 42)(27 44 36 41)(28 43 31 40)(29 48 32 39)(30 47 33 38)(49 70 58 61)(50 69 59 66)(51 68 60 65)(52 67 55 64)(53 72 56 63)(54 71 57 62)(73 94 82 85)(74 93 83 90)(75 92 84 89)(76 91 79 88)(77 96 80 87)(78 95 81 86)
(1 16 4 13)(2 17 5 14)(3 18 6 15)(7 22 10 19)(8 23 11 20)(9 24 12 21)(25 37 28 40)(26 38 29 41)(27 39 30 42)(31 43 34 46)(32 44 35 47)(33 45 36 48)(49 64 52 61)(50 65 53 62)(51 66 54 63)(55 70 58 67)(56 71 59 68)(57 72 60 69)(73 85 76 88)(74 86 77 89)(75 87 78 90)(79 91 82 94)(80 92 83 95)(81 93 84 96)```

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

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

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

36 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 2H 3 4A 4B 4C 4D 4E 4F 4G 4H 4I 4J 4K 4L 4M 6A 6B 6C 6D 6E 6F 6G 12A 12B 12C 12D 12E 12F order 1 2 2 2 2 2 2 2 2 3 4 4 4 4 4 4 4 4 4 4 4 4 4 6 6 6 6 6 6 6 12 12 12 12 12 12 size 1 1 1 1 4 4 4 12 12 2 2 2 4 4 4 6 6 6 6 12 12 12 12 2 2 2 4 4 8 8 4 4 4 4 8 8

36 irreducible representations

 dim 1 1 1 1 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 C2 C2 C2 C2 S3 D6 D6 D6 D6 C4○D4 2+ 1+4 D4⋊6D6 S3×C4○D4 kernel C6.442+ 1+4 C23.16D6 Dic3⋊4D4 C23.9D6 Dic3⋊D4 Dic3.Q8 D6.D4 C4×C3⋊D4 C23.28D6 C23.23D6 D6⋊3D4 C23.14D6 C12⋊3D4 C3×C4⋊D4 C4⋊D4 C22⋊C4 C4⋊C4 C22×C4 C2×D4 Dic3 C6 C2 C2 # reps 1 1 1 1 1 1 1 1 1 1 1 3 1 1 1 2 1 1 3 4 2 4 2

Matrix representation of C6.442+ 1+4 in GL8(𝔽13)

 1 1 0 0 0 0 0 0 12 0 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 12 0 0 0 0 0 0 0 0 0 12 0 0 0 0 0 0 0 0 12 0 0 0 0 0 0 0 0 12 0 0 0 0 0 0 0 0 12
,
 10 7 0 0 0 0 0 0 6 3 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 2 2 12 0 0 0 0 0 2 11 0 12 0 0 0 0 7 0 11 11 0 0 0 0 0 7 11 2
,
 8 0 0 0 0 0 0 0 0 8 0 0 0 0 0 0 0 0 8 0 0 0 0 0 0 0 0 8 0 0 0 0 0 0 0 0 9 1 0 7 0 0 0 0 12 9 7 0 0 0 0 0 10 7 4 1 0 0 0 0 7 3 12 4
,
 0 0 1 0 0 0 0 0 0 0 12 12 0 0 0 0 12 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 4 0 1 0 0 0 0 9 0 1 0
,
 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 0 1 0 0 0 0 0 0 12 0 0 0 0 0 0 0 0 0 0 12 0 0 0 0 0 0 1 0

`G:=sub<GL(8,GF(13))| [1,12,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,12,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,12],[10,6,0,0,0,0,0,0,7,3,0,0,0,0,0,0,0,0,3,7,0,0,0,0,0,0,6,10,0,0,0,0,0,0,0,0,2,2,7,0,0,0,0,0,2,11,0,7,0,0,0,0,12,0,11,11,0,0,0,0,0,12,11,2],[8,0,0,0,0,0,0,0,0,8,0,0,0,0,0,0,0,0,8,0,0,0,0,0,0,0,0,8,0,0,0,0,0,0,0,0,9,12,10,7,0,0,0,0,1,9,7,3,0,0,0,0,0,7,4,12,0,0,0,0,7,0,1,4],[0,0,12,1,0,0,0,0,0,0,0,1,0,0,0,0,1,12,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,0,0,0,1,0,9,0,0,0,0,1,0,4,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0],[0,0,12,0,0,0,0,0,0,0,0,12,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,12,0] >;`

C6.442+ 1+4 in GAP, Magma, Sage, TeX

`C_6._{44}2_+^{1+4}`
`% in TeX`

`G:=Group("C6.44ES+(2,2)");`
`// GroupNames label`

`G:=SmallGroup(192,1174);`
`// by ID`

`G=gap.SmallGroup(192,1174);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,232,758,219,675,297,136,6278]);`
`// Polycyclic`

`G:=Group<a,b,c,d,e|a^6=b^4=1,c^2=e^2=a^3,d^2=b^2,a*b=b*a,a*c=c*a,d*a*d^-1=a^-1,a*e=e*a,c*b*c^-1=a^3*b^-1,b*d=d*b,e*b*e^-1=a^3*b,c*d=d*c,c*e=e*c,e*d*e^-1=a^3*b^2*d>;`
`// generators/relations`

׿
×
𝔽