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## G = C6.1452+ 1+4order 192 = 26·3

### 54th 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 — C6.1452+ 1+4
 Chief series C1 — C3 — C6 — C2×C6 — C22×S3 — S3×C23 — C2×S3×D4 — C6.1452+ 1+4
 Lower central C3 — C2×C6 — C6.1452+ 1+4
 Upper central C1 — C22 — C2×C4○D4

Generators and relations for C6.1452+ 1+4
G = < a,b,c,d,e | a6=b4=c2=1, d2=b2, e2=a3, ab=ba, ac=ca, ad=da, eae-1=a-1, cbc=b-1, bd=db, ebe-1=a3b, cd=dc, ce=ec, ede-1=a3b2d >

Subgroups: 952 in 334 conjugacy classes, 113 normal (43 characteristic)
C1, C2 [×3], C2 [×9], C3, C4 [×2], C4 [×8], C22, C22 [×4], C22 [×25], S3 [×4], C6 [×3], C6 [×5], C2×C4 [×2], C2×C4 [×2], C2×C4 [×15], D4 [×4], D4 [×14], Q8 [×2], C23, C23 [×2], C23 [×13], Dic3 [×5], C12 [×2], C12 [×3], D6 [×2], D6 [×16], C2×C6, C2×C6 [×4], C2×C6 [×7], C42, C22⋊C4 [×12], C4⋊C4 [×4], C22×C4, C22×C4 [×2], C22×C4 [×3], C2×D4, C2×D4 [×2], C2×D4 [×10], C2×Q8, C4○D4 [×4], C24 [×2], C4×S3 [×2], D12 [×2], C2×Dic3 [×3], C2×Dic3 [×2], C2×Dic3 [×2], C3⋊D4 [×8], C2×C12 [×2], C2×C12 [×2], C2×C12 [×6], C3×D4 [×4], C3×D4 [×4], C3×Q8 [×2], C22×S3, C22×S3 [×2], C22×S3 [×10], C22×C6, C22×C6 [×2], C2×C22⋊C4 [×2], C4×D4 [×2], C22≀C2 [×2], C4⋊D4 [×3], C22⋊Q8, C22.D4 [×2], C4.4D4, C22×D4, C2×C4○D4, C4×Dic3, Dic3⋊C4, Dic3⋊C4 [×2], C4⋊Dic3, D6⋊C4, D6⋊C4 [×8], C6.D4, C6.D4 [×2], S3×C2×C4, C2×D12, S3×D4 [×4], C22×Dic3 [×2], C2×C3⋊D4, C2×C3⋊D4 [×4], C22×C12, C22×C12 [×2], C6×D4, C6×D4 [×2], C6×Q8, C3×C4○D4 [×4], S3×C23 [×2], D45D4, C2×D6⋊C4 [×2], C4×C3⋊D4, C23.28D6 [×2], C127D4, D4×Dic3, C232D6 [×2], C23.14D6 [×2], D63Q8, C12.23D4, C2×S3×D4, C6×C4○D4, C6.1452+ 1+4
Quotients: C1, C2 [×15], C22 [×35], S3, D4 [×4], C23 [×15], D6 [×7], C2×D4 [×6], C4○D4 [×2], C24, C3⋊D4 [×4], C22×S3 [×7], C22×D4, C2×C4○D4, 2+ 1+4, C2×C3⋊D4 [×6], S3×C23, D45D4, S3×C4○D4, D4○D12, C22×C3⋊D4, C6.1452+ 1+4

Smallest permutation representation of C6.1452+ 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 16 46)(2 7 17 47)(3 8 18 48)(4 9 13 43)(5 10 14 44)(6 11 15 45)(19 41 29 31)(20 42 30 32)(21 37 25 33)(22 38 26 34)(23 39 27 35)(24 40 28 36)
(1 13)(2 14)(3 15)(4 16)(5 17)(6 18)(7 10)(8 11)(9 12)(19 26)(20 27)(21 28)(22 29)(23 30)(24 25)(31 34)(32 35)(33 36)(37 40)(38 41)(39 42)(43 46)(44 47)(45 48)
(1 22 16 26)(2 23 17 27)(3 24 18 28)(4 19 13 29)(5 20 14 30)(6 21 15 25)(7 39 47 35)(8 40 48 36)(9 41 43 31)(10 42 44 32)(11 37 45 33)(12 38 46 34)
(1 22 4 19)(2 21 5 24)(3 20 6 23)(7 40 10 37)(8 39 11 42)(9 38 12 41)(13 29 16 26)(14 28 17 25)(15 27 18 30)(31 43 34 46)(32 48 35 45)(33 47 36 44)```

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

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

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

45 conjugacy classes

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

45 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 4 4 4 type + + + + + + + + + + + + + + + + + + + image C1 C2 C2 C2 C2 C2 C2 C2 C2 C2 C2 C2 S3 D4 D6 D6 D6 C4○D4 C3⋊D4 2+ 1+4 S3×C4○D4 D4○D12 kernel C6.1452+ 1+4 C2×D6⋊C4 C4×C3⋊D4 C23.28D6 C12⋊7D4 D4×Dic3 C23⋊2D6 C23.14D6 D6⋊3Q8 C12.23D4 C2×S3×D4 C6×C4○D4 C2×C4○D4 C3×D4 C22×C4 C2×D4 C2×Q8 D6 D4 C6 C2 C2 # reps 1 2 1 2 1 1 2 2 1 1 1 1 1 4 3 3 1 4 8 1 2 2

Matrix representation of C6.1452+ 1+4 in GL4(𝔽13) generated by

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

C6.1452+ 1+4 in GAP, Magma, Sage, TeX

`C_6._{145}2_+^{1+4}`
`% in TeX`

`G:=Group("C6.145ES+(2,2)");`
`// GroupNames label`

`G:=SmallGroup(192,1388);`
`// by ID`

`G=gap.SmallGroup(192,1388);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,387,675,136,6278]);`
`// Polycyclic`

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

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