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## G = C6.712- 1+4order 192 = 26·3

### 26th non-split extension by C6 of 2- 1+4 acting via 2- 1+4/C4○D4=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C6 — C6.712- 1+4
 Chief series C1 — C3 — C6 — C2×C6 — C2×Dic3 — C22×Dic3 — D4×Dic3 — C6.712- 1+4
 Lower central C3 — C2×C6 — C6.712- 1+4
 Upper central C1 — C22 — C4⋊D4

Generators and relations for C6.712- 1+4
G = < a,b,c,d,e | a6=b4=c2=1, d2=e2=a3b2, bab-1=dad-1=a-1, ac=ca, ae=ea, cbc=a3b-1, dbd-1=a3b, be=eb, dcd-1=a3c, ce=ec, ede-1=a3b2d >

Subgroups: 496 in 216 conjugacy classes, 95 normal (43 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, C6, C6, C2×C4, C2×C4, C2×C4, D4, Q8, C23, C23, Dic3, C12, C12, C2×C6, C2×C6, C42, C22⋊C4, C22⋊C4, C4⋊C4, C4⋊C4, C22×C4, C22×C4, C2×D4, C2×D4, C2×Q8, Dic6, C2×Dic3, C2×Dic3, C2×Dic3, C2×C12, C2×C12, C2×C12, C3×D4, C22×C6, C22×C6, C42⋊C2, C4×D4, C4×Q8, C4⋊D4, C22⋊Q8, C22.D4, C4.4D4, C422C2, C4⋊Q8, C4×Dic3, C4×Dic3, Dic3⋊C4, C4⋊Dic3, C6.D4, C6.D4, C3×C22⋊C4, C3×C4⋊C4, C2×Dic6, C2×Dic6, C22×Dic3, C22×C12, C6×D4, C6×D4, C22.36C24, Dic3.D4, C23.8D6, Dic6⋊C4, C12⋊Q8, C12.48D4, C23.26D6, D4×Dic3, C23.23D6, C23.12D6, C23.12D6, C3×C4⋊D4, C6.712- 1+4
Quotients: C1, C2, C22, S3, C23, D6, C4○D4, C24, C22×S3, C2×C4○D4, 2+ 1+4, 2- 1+4, D42S3, S3×C23, C22.36C24, C2×D42S3, D46D6, Q8○D12, C6.712- 1+4

Smallest permutation representation of C6.712- 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 95 18 7)(2 94 13 12)(3 93 14 11)(4 92 15 10)(5 91 16 9)(6 96 17 8)(19 80 29 85)(20 79 30 90)(21 84 25 89)(22 83 26 88)(23 82 27 87)(24 81 28 86)(31 76 41 71)(32 75 42 70)(33 74 37 69)(34 73 38 68)(35 78 39 67)(36 77 40 72)(43 59 54 64)(44 58 49 63)(45 57 50 62)(46 56 51 61)(47 55 52 66)(48 60 53 65)
(7 92)(8 93)(9 94)(10 95)(11 96)(12 91)(19 29)(20 30)(21 25)(22 26)(23 27)(24 28)(31 41)(32 42)(33 37)(34 38)(35 39)(36 40)(55 63)(56 64)(57 65)(58 66)(59 61)(60 62)(67 70)(68 71)(69 72)(73 76)(74 77)(75 78)(79 82)(80 83)(81 84)(85 88)(86 89)(87 90)
(1 83 15 85)(2 82 16 90)(3 81 17 89)(4 80 18 88)(5 79 13 87)(6 84 14 86)(7 19 92 26)(8 24 93 25)(9 23 94 30)(10 22 95 29)(11 21 96 28)(12 20 91 27)(31 56 38 64)(32 55 39 63)(33 60 40 62)(34 59 41 61)(35 58 42 66)(36 57 37 65)(43 73 51 71)(44 78 52 70)(45 77 53 69)(46 76 54 68)(47 75 49 67)(48 74 50 72)
(1 51 15 43)(2 52 16 44)(3 53 17 45)(4 54 18 46)(5 49 13 47)(6 50 14 48)(7 56 92 64)(8 57 93 65)(9 58 94 66)(10 59 95 61)(11 60 96 62)(12 55 91 63)(19 31 26 38)(20 32 27 39)(21 33 28 40)(22 34 29 41)(23 35 30 42)(24 36 25 37)(67 79 75 87)(68 80 76 88)(69 81 77 89)(70 82 78 90)(71 83 73 85)(72 84 74 86)```

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

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

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

36 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 3 4A 4B 4C 4D 4E 4F 4G 4H 4I 4J ··· 4O 6A 6B 6C 6D 6E 6F 6G 12A 12B 12C 12D 12E 12F order 1 2 2 2 2 2 2 3 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 2 2 2 4 4 4 6 6 6 6 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 2 2 2 2 2 2 4 4 4 4 4 type + + + + + + + + + + + + + + + + + - - - image C1 C2 C2 C2 C2 C2 C2 C2 C2 C2 C2 S3 D6 D6 D6 D6 C4○D4 2+ 1+4 2- 1+4 D4⋊2S3 D4⋊6D6 Q8○D12 kernel C6.712- 1+4 Dic3.D4 C23.8D6 Dic6⋊C4 C12⋊Q8 C12.48D4 C23.26D6 D4×Dic3 C23.23D6 C23.12D6 C3×C4⋊D4 C4⋊D4 C22⋊C4 C4⋊C4 C22×C4 C2×D4 C12 C6 C6 C4 C2 C2 # reps 1 2 2 1 1 1 1 1 2 3 1 1 2 1 1 3 4 1 1 2 2 2

Matrix representation of C6.712- 1+4 in GL6(𝔽13)

 12 0 0 0 0 0 0 12 0 0 0 0 0 0 4 0 0 0 0 0 0 4 0 0 0 0 0 0 10 0 0 0 0 0 0 10
,
 8 0 0 0 0 0 0 5 0 0 0 0 0 0 0 0 4 2 0 0 0 0 11 9 0 0 9 11 0 0 0 0 2 4 0 0
,
 12 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 12 0 0 0 0 0 0 12
,
 0 5 0 0 0 0 8 0 0 0 0 0 0 0 0 0 2 4 0 0 0 0 9 11 0 0 11 9 0 0 0 0 4 2 0 0
,
 12 0 0 0 0 0 0 12 0 0 0 0 0 0 4 2 0 0 0 0 11 9 0 0 0 0 0 0 4 2 0 0 0 0 11 9

`G:=sub<GL(6,GF(13))| [12,0,0,0,0,0,0,12,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,10,0,0,0,0,0,0,10],[8,0,0,0,0,0,0,5,0,0,0,0,0,0,0,0,9,2,0,0,0,0,11,4,0,0,4,11,0,0,0,0,2,9,0,0],[12,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,12,0,0,0,0,0,0,12],[0,8,0,0,0,0,5,0,0,0,0,0,0,0,0,0,11,4,0,0,0,0,9,2,0,0,2,9,0,0,0,0,4,11,0,0],[12,0,0,0,0,0,0,12,0,0,0,0,0,0,4,11,0,0,0,0,2,9,0,0,0,0,0,0,4,11,0,0,0,0,2,9] >;`

C6.712- 1+4 in GAP, Magma, Sage, TeX

`C_6._{71}2_-^{1+4}`
`% in TeX`

`G:=Group("C6.71ES-(2,2)");`
`// GroupNames label`

`G:=SmallGroup(192,1162);`
`// by ID`

`G=gap.SmallGroup(192,1162);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,224,219,675,570,297,6278]);`
`// Polycyclic`

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

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