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## G = C6.2+ 1+4order 192 = 26·3

### 9th 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 — C6.2+ 1+4
 Chief series C1 — C3 — C6 — C2×C6 — C22×S3 — S3×C2×C4 — C2×C4○D12 — C6.2+ 1+4
 Lower central C3 — C2×C6 — C6.2+ 1+4
 Upper central C1 — C22 — C2×C4⋊C4

Generators and relations for C6.2+ 1+4
G = < a,b,c,d,e | a6=b4=e2=1, c2=a3, d2=b2, bab-1=dad-1=eae=a-1, ac=ca, cbc-1=a3b-1, dbd-1=a3b, be=eb, dcd-1=ece=a3c, ede=a3b2d >

Subgroups: 824 in 294 conjugacy classes, 111 normal (17 characteristic)
C1, C2 [×3], C2 [×6], C3, C4 [×4], C4 [×8], C22, C22 [×2], C22 [×14], S3 [×4], C6 [×3], C6 [×2], C2×C4 [×10], C2×C4 [×14], D4 [×16], Q8 [×4], C23, C23 [×4], Dic3 [×4], C12 [×4], C12 [×4], D6 [×12], C2×C6, C2×C6 [×2], C2×C6 [×2], C22⋊C4 [×8], C4⋊C4 [×4], C4⋊C4 [×4], C22×C4, C22×C4 [×2], C22×C4 [×4], C2×D4 [×10], C2×Q8 [×2], C4○D4 [×8], Dic6 [×4], C4×S3 [×8], D12 [×8], C2×Dic3 [×4], C3⋊D4 [×8], C2×C12 [×10], C2×C12 [×2], C22×S3 [×4], C22×C6, C2×C4⋊C4, C4⋊D4 [×8], C22⋊Q8 [×4], C2×C4○D4 [×2], C4⋊Dic3 [×4], D6⋊C4 [×8], C3×C4⋊C4 [×4], C2×Dic6 [×2], S3×C2×C4 [×4], C2×D12 [×6], C4○D12 [×8], C2×C3⋊D4 [×4], C22×C12, C22×C12 [×2], C22.31C24, C12⋊D4 [×4], C4.D12 [×4], C127D4 [×4], C6×C4⋊C4, C2×C4○D12 [×2], C6.2+ 1+4
Quotients: C1, C2 [×15], C22 [×35], S3, D4 [×4], C23 [×15], D6 [×7], C2×D4 [×6], C24, D12 [×4], C22×S3 [×7], C22×D4, 2+ 1+4, 2- 1+4, C2×D12 [×6], S3×C23, C22.31C24, C22×D12, D46D6, Q8.15D6, C6.2+ 1+4

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

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

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

42 conjugacy classes

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

42 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 2 4 4 4 4 type + + + + + + + + + + + + - image C1 C2 C2 C2 C2 C2 S3 D4 D6 D6 D12 2+ 1+4 2- 1+4 D4⋊6D6 Q8.15D6 kernel C6.2+ 1+4 C12⋊D4 C4.D12 C12⋊7D4 C6×C4⋊C4 C2×C4○D12 C2×C4⋊C4 C2×C12 C4⋊C4 C22×C4 C2×C4 C6 C6 C2 C2 # reps 1 4 4 4 1 2 1 4 4 3 8 1 1 2 2

Matrix representation of C6.2+ 1+4 in GL6(𝔽13)

 0 1 0 0 0 0 12 1 0 0 0 0 0 0 0 1 0 0 0 0 12 1 0 0 0 0 0 0 0 1 0 0 0 0 12 1
,
 0 12 0 0 0 0 12 0 0 0 0 0 0 0 11 11 0 0 0 0 9 2 0 0 0 0 0 0 11 11 0 0 0 0 9 2
,
 10 6 0 0 0 0 7 3 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 12 0 0 0 0 0 0 12 0 0
,
 7 3 0 0 0 0 10 6 0 0 0 0 0 0 9 4 10 3 0 0 0 4 0 3 0 0 10 3 4 9 0 0 0 3 0 9
,
 0 12 0 0 0 0 12 0 0 0 0 0 0 0 6 6 5 5 0 0 12 7 10 8 0 0 5 5 7 7 0 0 10 8 1 6

`G:=sub<GL(6,GF(13))| [0,12,0,0,0,0,1,1,0,0,0,0,0,0,0,12,0,0,0,0,1,1,0,0,0,0,0,0,0,12,0,0,0,0,1,1],[0,12,0,0,0,0,12,0,0,0,0,0,0,0,11,9,0,0,0,0,11,2,0,0,0,0,0,0,11,9,0,0,0,0,11,2],[10,7,0,0,0,0,6,3,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,1,0,0,0,0,0,0,1,0,0],[7,10,0,0,0,0,3,6,0,0,0,0,0,0,9,0,10,0,0,0,4,4,3,3,0,0,10,0,4,0,0,0,3,3,9,9],[0,12,0,0,0,0,12,0,0,0,0,0,0,0,6,12,5,10,0,0,6,7,5,8,0,0,5,10,7,1,0,0,5,8,7,6] >;`

C6.2+ 1+4 in GAP, Magma, Sage, TeX

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

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

`G:=SmallGroup(192,1069);`
`// by ID`

`G=gap.SmallGroup(192,1069);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,232,758,675,570,80,6278]);`
`// Polycyclic`

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