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## G = C6.252- 1+4order 192 = 26·3

### 25th non-split extension by C6 of 2- 1+4 acting via 2- 1+4/C2×Q8=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C6 — C6.252- 1+4
 Chief series C1 — C3 — C6 — C2×C6 — C22×S3 — S3×C2×C4 — C23.9D6 — C6.252- 1+4
 Lower central C3 — C2×C6 — C6.252- 1+4
 Upper central C1 — C22 — C22⋊Q8

Generators and relations for C6.252- 1+4
G = < a,b,c,d,e | a6=b4=1, c2=e2=a3, d2=a3b2, ab=ba, ac=ca, dad-1=a-1, ae=ea, cbc-1=a3b-1, bd=db, ebe-1=a3b, dcd-1=ece-1=a3c, ede-1=b2d >

Subgroups: 464 in 196 conjugacy classes, 91 normal (all characteristic)
C1, C2, C2, C3, C4, C22, C22, S3, C6, C6, C2×C4, C2×C4, D4, Q8, C23, C23, Dic3, C12, D6, C2×C6, C2×C6, C42, C22⋊C4, C22⋊C4, C4⋊C4, C4⋊C4, C22×C4, C22×C4, C2×D4, C2×Q8, C2×Q8, Dic6, C4×S3, C2×Dic3, C3⋊D4, C2×C12, C2×C12, C3×Q8, C22×S3, C22×C6, C22⋊Q8, C22⋊Q8, C22.D4, C4.4D4, C42.C2, C422C2, C4⋊Q8, C4×Dic3, Dic3⋊C4, C4⋊Dic3, D6⋊C4, C6.D4, C3×C22⋊C4, C3×C4⋊C4, C2×Dic6, S3×C2×C4, C2×C3⋊D4, C22×C12, C6×Q8, C22.57C24, C23.8D6, C23.9D6, C23.11D6, C12⋊Q8, Dic3.Q8, C4.Dic6, D6⋊Q8, C4⋊C4⋊S3, C12.48D4, C23.28D6, Dic3⋊Q8, D63Q8, C3×C22⋊Q8, C6.252- 1+4
Quotients: C1, C2, C22, S3, C23, D6, C24, C22×S3, 2+ 1+4, 2- 1+4, S3×C23, C22.57C24, D46D6, Q8.15D6, Q8○D12, C6.252- 1+4

Smallest permutation representation of C6.252- 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 79 10 76)(2 80 11 77)(3 81 12 78)(4 82 7 73)(5 83 8 74)(6 84 9 75)(13 91 22 88)(14 92 23 89)(15 93 24 90)(16 94 19 85)(17 95 20 86)(18 96 21 87)(25 55 34 52)(26 56 35 53)(27 57 36 54)(28 58 31 49)(29 59 32 50)(30 60 33 51)(37 67 46 64)(38 68 47 65)(39 69 48 66)(40 70 43 61)(41 71 44 62)(42 72 45 63)
(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 19 7 13)(2 24 8 18)(3 23 9 17)(4 22 10 16)(5 21 11 15)(6 20 12 14)(25 46 31 40)(26 45 32 39)(27 44 33 38)(28 43 34 37)(29 48 35 42)(30 47 36 41)(49 70 55 64)(50 69 56 63)(51 68 57 62)(52 67 58 61)(53 72 59 66)(54 71 60 65)(73 91 79 85)(74 96 80 90)(75 95 81 89)(76 94 82 88)(77 93 83 87)(78 92 84 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 40 28 37)(26 41 29 38)(27 42 30 39)(31 46 34 43)(32 47 35 44)(33 48 36 45)(49 61 52 64)(50 62 53 65)(51 63 54 66)(55 67 58 70)(56 68 59 71)(57 69 60 72)(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), (1,79,10,76)(2,80,11,77)(3,81,12,78)(4,82,7,73)(5,83,8,74)(6,84,9,75)(13,91,22,88)(14,92,23,89)(15,93,24,90)(16,94,19,85)(17,95,20,86)(18,96,21,87)(25,55,34,52)(26,56,35,53)(27,57,36,54)(28,58,31,49)(29,59,32,50)(30,60,33,51)(37,67,46,64)(38,68,47,65)(39,69,48,66)(40,70,43,61)(41,71,44,62)(42,72,45,63), (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), (1,19,7,13)(2,24,8,18)(3,23,9,17)(4,22,10,16)(5,21,11,15)(6,20,12,14)(25,46,31,40)(26,45,32,39)(27,44,33,38)(28,43,34,37)(29,48,35,42)(30,47,36,41)(49,70,55,64)(50,69,56,63)(51,68,57,62)(52,67,58,61)(53,72,59,66)(54,71,60,65)(73,91,79,85)(74,96,80,90)(75,95,81,89)(76,94,82,88)(77,93,83,87)(78,92,84,86), (1,16,4,13)(2,17,5,14)(3,18,6,15)(7,22,10,19)(8,23,11,20)(9,24,12,21)(25,40,28,37)(26,41,29,38)(27,42,30,39)(31,46,34,43)(32,47,35,44)(33,48,36,45)(49,61,52,64)(50,62,53,65)(51,63,54,66)(55,67,58,70)(56,68,59,71)(57,69,60,72)(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), (1,79,10,76)(2,80,11,77)(3,81,12,78)(4,82,7,73)(5,83,8,74)(6,84,9,75)(13,91,22,88)(14,92,23,89)(15,93,24,90)(16,94,19,85)(17,95,20,86)(18,96,21,87)(25,55,34,52)(26,56,35,53)(27,57,36,54)(28,58,31,49)(29,59,32,50)(30,60,33,51)(37,67,46,64)(38,68,47,65)(39,69,48,66)(40,70,43,61)(41,71,44,62)(42,72,45,63), (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), (1,19,7,13)(2,24,8,18)(3,23,9,17)(4,22,10,16)(5,21,11,15)(6,20,12,14)(25,46,31,40)(26,45,32,39)(27,44,33,38)(28,43,34,37)(29,48,35,42)(30,47,36,41)(49,70,55,64)(50,69,56,63)(51,68,57,62)(52,67,58,61)(53,72,59,66)(54,71,60,65)(73,91,79,85)(74,96,80,90)(75,95,81,89)(76,94,82,88)(77,93,83,87)(78,92,84,86), (1,16,4,13)(2,17,5,14)(3,18,6,15)(7,22,10,19)(8,23,11,20)(9,24,12,21)(25,40,28,37)(26,41,29,38)(27,42,30,39)(31,46,34,43)(32,47,35,44)(33,48,36,45)(49,61,52,64)(50,62,53,65)(51,63,54,66)(55,67,58,70)(56,68,59,71)(57,69,60,72)(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)], [(1,79,10,76),(2,80,11,77),(3,81,12,78),(4,82,7,73),(5,83,8,74),(6,84,9,75),(13,91,22,88),(14,92,23,89),(15,93,24,90),(16,94,19,85),(17,95,20,86),(18,96,21,87),(25,55,34,52),(26,56,35,53),(27,57,36,54),(28,58,31,49),(29,59,32,50),(30,60,33,51),(37,67,46,64),(38,68,47,65),(39,69,48,66),(40,70,43,61),(41,71,44,62),(42,72,45,63)], [(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)], [(1,19,7,13),(2,24,8,18),(3,23,9,17),(4,22,10,16),(5,21,11,15),(6,20,12,14),(25,46,31,40),(26,45,32,39),(27,44,33,38),(28,43,34,37),(29,48,35,42),(30,47,36,41),(49,70,55,64),(50,69,56,63),(51,68,57,62),(52,67,58,61),(53,72,59,66),(54,71,60,65),(73,91,79,85),(74,96,80,90),(75,95,81,89),(76,94,82,88),(77,93,83,87),(78,92,84,86)], [(1,16,4,13),(2,17,5,14),(3,18,6,15),(7,22,10,19),(8,23,11,20),(9,24,12,21),(25,40,28,37),(26,41,29,38),(27,42,30,39),(31,46,34,43),(32,47,35,44),(33,48,36,45),(49,61,52,64),(50,62,53,65),(51,63,54,66),(55,67,58,70),(56,68,59,71),(57,69,60,72),(73,85,76,88),(74,86,77,89),(75,87,78,90),(79,91,82,94),(80,92,83,95),(81,93,84,96)]])`

33 conjugacy classes

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

33 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 4 4 4 4 4 type + + + + + + + + + + + + + + + + + + + + - - image C1 C2 C2 C2 C2 C2 C2 C2 C2 C2 C2 C2 C2 C2 S3 D6 D6 D6 D6 2+ 1+4 2- 1+4 D4⋊6D6 Q8.15D6 Q8○D12 kernel C6.252- 1+4 C23.8D6 C23.9D6 C23.11D6 C12⋊Q8 Dic3.Q8 C4.Dic6 D6⋊Q8 C4⋊C4⋊S3 C12.48D4 C23.28D6 Dic3⋊Q8 D6⋊3Q8 C3×C22⋊Q8 C22⋊Q8 C22⋊C4 C4⋊C4 C22×C4 C2×Q8 C6 C6 C2 C2 C2 # reps 1 2 1 1 1 1 1 1 2 1 1 1 1 1 1 2 3 1 1 1 2 2 2 2

Matrix representation of C6.252- 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 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
,
 3 1 8 0 0 0 0 0 12 2 0 8 0 0 0 0 7 1 10 12 0 0 0 0 12 6 1 11 0 0 0 0 0 0 0 0 0 12 10 0 0 0 0 0 12 0 0 10 0 0 0 0 5 0 0 1 0 0 0 0 0 5 1 0
,
 11 3 10 7 0 0 0 0 10 8 6 3 0 0 0 0 3 0 2 10 0 0 0 0 0 3 3 5 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 8 12 0 0 0 0 0 8 0 0 12
,
 1 0 11 0 0 0 0 0 12 12 2 2 0 0 0 0 0 0 12 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 12 0 0 0 0 0 0 0 5 0 0 1 0 0 0 0 0 8 12 0
,
 1 0 11 0 0 0 0 0 0 1 0 11 0 0 0 0 1 0 12 0 0 0 0 0 0 1 0 12 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 0 0 1 0 0 0 0 0 0 1 0

`G:=sub<GL(8,GF(13))| [1,12,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,12,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[3,12,7,12,0,0,0,0,1,2,1,6,0,0,0,0,8,0,10,1,0,0,0,0,0,8,12,11,0,0,0,0,0,0,0,0,0,12,5,0,0,0,0,0,12,0,0,5,0,0,0,0,10,0,0,1,0,0,0,0,0,10,1,0],[11,10,3,0,0,0,0,0,3,8,0,3,0,0,0,0,10,6,2,3,0,0,0,0,7,3,10,5,0,0,0,0,0,0,0,0,1,0,0,8,0,0,0,0,0,1,8,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,12],[1,12,0,0,0,0,0,0,0,12,0,0,0,0,0,0,11,2,12,1,0,0,0,0,0,2,0,1,0,0,0,0,0,0,0,0,0,12,5,0,0,0,0,0,1,0,0,8,0,0,0,0,0,0,0,12,0,0,0,0,0,0,1,0],[1,0,1,0,0,0,0,0,0,1,0,1,0,0,0,0,11,0,12,0,0,0,0,0,0,11,0,12,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0] >;`

C6.252- 1+4 in GAP, Magma, Sage, TeX

`C_6._{25}2_-^{1+4}`
`% in TeX`

`G:=Group("C6.25ES-(2,2)");`
`// GroupNames label`

`G:=SmallGroup(192,1205);`
`// by ID`

`G=gap.SmallGroup(192,1205);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,232,758,219,1571,570,136,6278]);`
`// Polycyclic`

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

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