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## G = C6.692+ 1+4order 192 = 26·3

### 69th 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 — C6.692+ 1+4
 Chief series C1 — C3 — C6 — C2×C6 — C22×S3 — S3×C2×C4 — D6.D4 — C6.692+ 1+4
 Lower central C3 — C2×C6 — C6.692+ 1+4
 Upper central C1 — C22 — C22.D4

Generators and relations for C6.692+ 1+4
G = < a,b,c,d,e | a6=b4=e2=1, c2=a3, d2=a3b2, ab=ba, ac=ca, dad-1=a-1, ae=ea, cbc-1=b-1, bd=db, ebe=a3b, dcd-1=ece=a3c, ede=a3b2d >

Subgroups: 608 in 220 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×D4, C2×Q8, Dic6, C4×S3, D12, C2×Dic3, C2×Dic3, C3⋊D4, C2×C12, C2×C12, C3×D4, C22×S3, C22×C6, C4⋊D4, C22⋊Q8, C22.D4, C22.D4, C4.4D4, C42.C2, C4×Dic3, Dic3⋊C4, C4⋊Dic3, D6⋊C4, C6.D4, C3×C22⋊C4, C3×C4⋊C4, C2×Dic6, S3×C2×C4, C2×D12, C22×Dic3, C2×C3⋊D4, C22×C12, C6×D4, C22.56C24, Dic3.D4, C23.9D6, Dic3⋊D4, C23.11D6, C23.21D6, C4.Dic6, D6.D4, D6⋊Q8, C4.D12, C12.48D4, C127D4, D63D4, C23.14D6, C3×C22.D4, C6.692+ 1+4
Quotients: C1, C2, C22, S3, C23, D6, C24, C22×S3, 2+ 1+4, 2- 1+4, S3×C23, C22.56C24, D46D6, D4○D12, Q8○D12, C6.692+ 1+4

Smallest permutation representation of C6.692+ 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 7 76)(2 83 8 77)(3 84 9 78)(4 79 10 73)(5 80 11 74)(6 81 12 75)(13 94 19 88)(14 95 20 89)(15 96 21 90)(16 91 22 85)(17 92 23 86)(18 93 24 87)(25 55 31 49)(26 56 32 50)(27 57 33 51)(28 58 34 52)(29 59 35 53)(30 60 36 54)(37 67 43 61)(38 68 44 62)(39 69 45 63)(40 70 46 64)(41 71 47 65)(42 72 48 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 43 34 40)(26 48 35 39)(27 47 36 38)(28 46 31 37)(29 45 32 42)(30 44 33 41)(49 67 58 64)(50 72 59 63)(51 71 60 62)(52 70 55 61)(53 69 56 66)(54 68 57 65)(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 13)(2 14)(3 15)(4 16)(5 17)(6 18)(7 19)(8 20)(9 21)(10 22)(11 23)(12 24)(25 37)(26 38)(27 39)(28 40)(29 41)(30 42)(31 43)(32 44)(33 45)(34 46)(35 47)(36 48)(49 64)(50 65)(51 66)(52 61)(53 62)(54 63)(55 70)(56 71)(57 72)(58 67)(59 68)(60 69)(73 88)(74 89)(75 90)(76 85)(77 86)(78 87)(79 94)(80 95)(81 96)(82 91)(83 92)(84 93)```

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

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

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

33 conjugacy classes

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

33 irreducible representations

 dim 1 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 C2 S3 D6 D6 D6 D6 2+ 1+4 2- 1+4 D4⋊6D6 D4○D12 Q8○D12 kernel C6.692+ 1+4 Dic3.D4 C23.9D6 Dic3⋊D4 C23.11D6 C23.21D6 C4.Dic6 D6.D4 D6⋊Q8 C4.D12 C12.48D4 C12⋊7D4 D6⋊3D4 C23.14D6 C3×C22.D4 C22.D4 C22⋊C4 C4⋊C4 C22×C4 C2×D4 C6 C6 C2 C2 C2 # reps 1 1 1 1 2 1 1 1 1 1 1 1 1 1 1 1 3 2 1 1 2 1 2 2 2

Matrix representation of C6.692+ 1+4 in GL8(𝔽13)

 0 12 0 0 0 0 0 0 1 12 0 0 0 0 0 0 12 0 12 12 0 0 0 0 0 1 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
,
 0 0 12 1 0 0 0 0 12 12 11 12 0 0 0 0 5 5 1 0 0 0 0 0 4 5 1 0 0 0 0 0 0 0 0 0 0 4 10 0 0 0 0 0 9 0 0 3 0 0 0 0 3 0 0 4 0 0 0 0 0 10 9 0
,
 12 0 0 0 0 0 0 0 0 12 0 0 0 0 0 0 5 5 1 0 0 0 0 0 5 5 0 1 0 0 0 0 0 0 0 0 3 0 0 4 0 0 0 0 0 10 9 0 0 0 0 0 0 9 3 0 0 0 0 0 4 0 0 10
,
 4 11 0 0 0 0 0 0 2 9 0 0 0 0 0 0 9 0 11 9 0 0 0 0 11 2 11 2 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
,
 2 9 0 0 0 0 0 0 4 11 0 0 0 0 0 0 9 0 11 9 0 0 0 0 0 4 4 2 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))| [0,1,12,0,0,0,0,0,12,12,0,1,0,0,0,0,0,0,12,1,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,12],[0,12,5,4,0,0,0,0,0,12,5,5,0,0,0,0,12,11,1,1,0,0,0,0,1,12,0,0,0,0,0,0,0,0,0,0,0,9,3,0,0,0,0,0,4,0,0,10,0,0,0,0,10,0,0,9,0,0,0,0,0,3,4,0],[12,0,5,5,0,0,0,0,0,12,5,5,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,3,0,0,4,0,0,0,0,0,10,9,0,0,0,0,0,0,9,3,0,0,0,0,0,4,0,0,10],[4,2,9,11,0,0,0,0,11,9,0,2,0,0,0,0,0,0,11,11,0,0,0,0,0,0,9,2,0,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,12,0],[2,4,9,0,0,0,0,0,9,11,0,4,0,0,0,0,0,0,11,4,0,0,0,0,0,0,9,2,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0] >;`

C6.692+ 1+4 in GAP, Magma, Sage, TeX

`C_6._{69}2_+^{1+4}`
`% in TeX`

`G:=Group("C6.69ES+(2,2)");`
`// GroupNames label`

`G:=SmallGroup(192,1226);`
`// by ID`

`G=gap.SmallGroup(192,1226);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,232,758,219,184,1571,570,6278]);`
`// Polycyclic`

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

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