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## G = C6.1462+ 1+4order 192 = 26·3

### 55th 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 — C6.1462+ 1+4
 Chief series C1 — C3 — C6 — C2×C6 — C22×S3 — S3×C23 — C22×D12 — C6.1462+ 1+4
 Lower central C3 — C2×C6 — C6.1462+ 1+4
 Upper central C1 — C22 — C2×C4○D4

Generators and relations for C6.1462+ 1+4
G = < a,b,c,d,e | a6=b4=e2=1, c2=a3, d2=a3b2, ab=ba, cac-1=dad-1=a-1, ae=ea, cbc-1=a3b-1, dbd-1=a3b, be=eb, cd=dc, ece=a3c, ede=a3b2d >

Subgroups: 1032 in 334 conjugacy classes, 111 normal (23 characteristic)
C1, C2, C2 [×2], C2 [×8], C3, C4 [×4], C4 [×6], C22, C22 [×2], C22 [×28], S3 [×4], C6, C6 [×2], C6 [×4], C2×C4 [×2], C2×C4 [×6], C2×C4 [×8], D4 [×22], Q8 [×2], C23, C23 [×2], C23 [×12], Dic3 [×4], C12 [×4], C12 [×2], D6 [×20], C2×C6, C2×C6 [×2], C2×C6 [×8], C42 [×2], C22⋊C4 [×10], C4⋊C4 [×2], C22×C4, C22×C4 [×2], C2×D4, C2×D4 [×2], C2×D4 [×16], C2×Q8, C4○D4 [×4], C24 [×2], D12 [×8], C2×Dic3 [×4], C3⋊D4 [×8], C2×C12 [×2], C2×C12 [×6], C2×C12 [×4], C3×D4 [×6], C3×Q8 [×2], C22×S3 [×4], C22×S3 [×8], C22×C6, C22×C6 [×2], C42⋊C2, C22≀C2 [×4], C4⋊D4 [×4], C4.4D4 [×2], C41D4 [×2], C22×D4, C2×C4○D4, C4×Dic3 [×2], C4⋊Dic3 [×2], D6⋊C4 [×8], C6.D4 [×2], C2×D12 [×4], C2×D12 [×4], C2×C3⋊D4 [×8], C22×C12, C22×C12 [×2], C6×D4, C6×D4 [×2], C6×Q8, C3×C4○D4 [×4], S3×C23 [×2], C22.29C24, C23.26D6, C127D4 [×4], C232D6 [×4], C123D4 [×2], C12.23D4 [×2], C22×D12, C6×C4○D4, C6.1462+ 1+4
Quotients: C1, C2 [×15], C22 [×35], S3, D4 [×4], C23 [×15], D6 [×7], C2×D4 [×6], C24, C3⋊D4 [×4], C22×S3 [×7], C22×D4, 2+ 1+4 [×2], C2×C3⋊D4 [×6], S3×C23, C22.29C24, D4○D12 [×2], C22×C3⋊D4, C6.1462+ 1+4

Smallest permutation representation of C6.1462+ 1+4
On 48 points
Generators in S48
```(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)
(1 12 17 45)(2 7 18 46)(3 8 13 47)(4 9 14 48)(5 10 15 43)(6 11 16 44)(19 39 30 34)(20 40 25 35)(21 41 26 36)(22 42 27 31)(23 37 28 32)(24 38 29 33)
(1 45 4 48)(2 44 5 47)(3 43 6 46)(7 13 10 16)(8 18 11 15)(9 17 12 14)(19 35 22 32)(20 34 23 31)(21 33 24 36)(25 39 28 42)(26 38 29 41)(27 37 30 40)
(1 29 14 21)(2 28 15 20)(3 27 16 19)(4 26 17 24)(5 25 18 23)(6 30 13 22)(7 35 43 37)(8 34 44 42)(9 33 45 41)(10 32 46 40)(11 31 47 39)(12 36 48 38)
(1 21)(2 22)(3 23)(4 24)(5 19)(6 20)(7 42)(8 37)(9 38)(10 39)(11 40)(12 41)(13 28)(14 29)(15 30)(16 25)(17 26)(18 27)(31 46)(32 47)(33 48)(34 43)(35 44)(36 45)```

`G:=sub<Sym(48)| (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), (1,12,17,45)(2,7,18,46)(3,8,13,47)(4,9,14,48)(5,10,15,43)(6,11,16,44)(19,39,30,34)(20,40,25,35)(21,41,26,36)(22,42,27,31)(23,37,28,32)(24,38,29,33), (1,45,4,48)(2,44,5,47)(3,43,6,46)(7,13,10,16)(8,18,11,15)(9,17,12,14)(19,35,22,32)(20,34,23,31)(21,33,24,36)(25,39,28,42)(26,38,29,41)(27,37,30,40), (1,29,14,21)(2,28,15,20)(3,27,16,19)(4,26,17,24)(5,25,18,23)(6,30,13,22)(7,35,43,37)(8,34,44,42)(9,33,45,41)(10,32,46,40)(11,31,47,39)(12,36,48,38), (1,21)(2,22)(3,23)(4,24)(5,19)(6,20)(7,42)(8,37)(9,38)(10,39)(11,40)(12,41)(13,28)(14,29)(15,30)(16,25)(17,26)(18,27)(31,46)(32,47)(33,48)(34,43)(35,44)(36,45)>;`

`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), (1,12,17,45)(2,7,18,46)(3,8,13,47)(4,9,14,48)(5,10,15,43)(6,11,16,44)(19,39,30,34)(20,40,25,35)(21,41,26,36)(22,42,27,31)(23,37,28,32)(24,38,29,33), (1,45,4,48)(2,44,5,47)(3,43,6,46)(7,13,10,16)(8,18,11,15)(9,17,12,14)(19,35,22,32)(20,34,23,31)(21,33,24,36)(25,39,28,42)(26,38,29,41)(27,37,30,40), (1,29,14,21)(2,28,15,20)(3,27,16,19)(4,26,17,24)(5,25,18,23)(6,30,13,22)(7,35,43,37)(8,34,44,42)(9,33,45,41)(10,32,46,40)(11,31,47,39)(12,36,48,38), (1,21)(2,22)(3,23)(4,24)(5,19)(6,20)(7,42)(8,37)(9,38)(10,39)(11,40)(12,41)(13,28)(14,29)(15,30)(16,25)(17,26)(18,27)(31,46)(32,47)(33,48)(34,43)(35,44)(36,45) );`

`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)], [(1,12,17,45),(2,7,18,46),(3,8,13,47),(4,9,14,48),(5,10,15,43),(6,11,16,44),(19,39,30,34),(20,40,25,35),(21,41,26,36),(22,42,27,31),(23,37,28,32),(24,38,29,33)], [(1,45,4,48),(2,44,5,47),(3,43,6,46),(7,13,10,16),(8,18,11,15),(9,17,12,14),(19,35,22,32),(20,34,23,31),(21,33,24,36),(25,39,28,42),(26,38,29,41),(27,37,30,40)], [(1,29,14,21),(2,28,15,20),(3,27,16,19),(4,26,17,24),(5,25,18,23),(6,30,13,22),(7,35,43,37),(8,34,44,42),(9,33,45,41),(10,32,46,40),(11,31,47,39),(12,36,48,38)], [(1,21),(2,22),(3,23),(4,24),(5,19),(6,20),(7,42),(8,37),(9,38),(10,39),(11,40),(12,41),(13,28),(14,29),(15,30),(16,25),(17,26),(18,27),(31,46),(32,47),(33,48),(34,43),(35,44),(36,45)])`

42 conjugacy classes

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

42 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 2 2 2 2 4 4 type + + + + + + + + + + + + + + + image C1 C2 C2 C2 C2 C2 C2 C2 S3 D4 D6 D6 D6 C3⋊D4 2+ 1+4 D4○D12 kernel C6.1462+ 1+4 C23.26D6 C12⋊7D4 C23⋊2D6 C12⋊3D4 C12.23D4 C22×D12 C6×C4○D4 C2×C4○D4 C2×C12 C22×C4 C2×D4 C2×Q8 C2×C4 C6 C2 # reps 1 1 4 4 2 2 1 1 1 4 3 3 1 8 2 4

Matrix representation of C6.1462+ 1+4 in GL6(𝔽13)

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

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

C6.1462+ 1+4 in GAP, Magma, Sage, TeX

`C_6._{146}2_+^{1+4}`
`% in TeX`

`G:=Group("C6.146ES+(2,2)");`
`// GroupNames label`

`G:=SmallGroup(192,1389);`
`// by ID`

`G=gap.SmallGroup(192,1389);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,758,184,675,570,6278]);`
`// 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,c*a*c^-1=d*a*d^-1=a^-1,a*e=e*a,c*b*c^-1=a^3*b^-1,d*b*d^-1=a^3*b,b*e=e*b,c*d=d*c,e*c*e=a^3*c,e*d*e=a^3*b^2*d>;`
`// generators/relations`

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