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## G = C6.372+ 1+4order 192 = 26·3

### 37th 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 — C6.372+ 1+4
 Chief series C1 — C3 — C6 — C2×C6 — C22×S3 — S3×C23 — C2×S3×D4 — C6.372+ 1+4
 Lower central C3 — C2×C6 — C6.372+ 1+4
 Upper central C1 — C22 — C4⋊D4

Generators and relations for C6.372+ 1+4
G = < a,b,c,d,e | a6=b4=c2=1, d2=a3b2, e2=a3, ab=ba, ac=ca, dad-1=a-1, ae=ea, cbc=b-1, dbd-1=ebe-1=a3b, cd=dc, ce=ec, ede-1=b2d >

Subgroups: 1136 in 346 conjugacy classes, 103 normal (31 characteristic)
C1, C2, C2, C3, C4, C22, C22, C22, S3, C6, C6, C2×C4, C2×C4, C2×C4, D4, C23, C23, C23, Dic3, C12, D6, D6, C2×C6, C2×C6, C2×C6, C22⋊C4, C22⋊C4, C4⋊C4, C4⋊C4, C22×C4, C22×C4, C2×D4, C2×D4, C2×D4, C24, C4×S3, D12, C2×Dic3, C2×Dic3, C3⋊D4, C2×C12, C2×C12, C2×C12, C3×D4, C22×S3, C22×S3, C22×C6, C22×C6, C2×C22⋊C4, C22≀C2, C4⋊D4, C4⋊D4, C22.D4, C22×D4, Dic3⋊C4, C4⋊Dic3, D6⋊C4, C6.D4, C3×C22⋊C4, C3×C4⋊C4, S3×C2×C4, C2×D12, C2×D12, S3×D4, C22×Dic3, C2×C3⋊D4, C22×C12, C6×D4, C6×D4, S3×C23, S3×C23, C233D4, D6⋊D4, C23.9D6, D6.D4, C2×D6⋊C4, C127D4, C232D6, C23.14D6, C3×C4⋊D4, C2×S3×D4, C6.372+ 1+4
Quotients: C1, C2, C22, S3, D4, C23, D6, C2×D4, C24, C22×S3, C22×D4, 2+ 1+4, S3×D4, S3×C23, C233D4, C2×S3×D4, D46D6, D4○D12, C6.372+ 1+4

Smallest permutation representation of C6.372+ 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 46 7 40)(2 47 8 41)(3 48 9 42)(4 43 10 37)(5 44 11 38)(6 45 12 39)(13 34 19 28)(14 35 20 29)(15 36 21 30)(16 31 22 25)(17 32 23 26)(18 33 24 27)
(1 25)(2 26)(3 27)(4 28)(5 29)(6 30)(7 31)(8 32)(9 33)(10 34)(11 35)(12 36)(13 37)(14 38)(15 39)(16 40)(17 41)(18 42)(19 43)(20 44)(21 45)(22 46)(23 47)(24 48)
(1 46 10 37)(2 45 11 42)(3 44 12 41)(4 43 7 40)(5 48 8 39)(6 47 9 38)(13 25 22 34)(14 30 23 33)(15 29 24 32)(16 28 19 31)(17 27 20 36)(18 26 21 35)
(1 16 4 13)(2 17 5 14)(3 18 6 15)(7 22 10 19)(8 23 11 20)(9 24 12 21)(25 40 28 37)(26 41 29 38)(27 42 30 39)(31 46 34 43)(32 47 35 44)(33 48 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), (1,46,7,40)(2,47,8,41)(3,48,9,42)(4,43,10,37)(5,44,11,38)(6,45,12,39)(13,34,19,28)(14,35,20,29)(15,36,21,30)(16,31,22,25)(17,32,23,26)(18,33,24,27), (1,25)(2,26)(3,27)(4,28)(5,29)(6,30)(7,31)(8,32)(9,33)(10,34)(11,35)(12,36)(13,37)(14,38)(15,39)(16,40)(17,41)(18,42)(19,43)(20,44)(21,45)(22,46)(23,47)(24,48), (1,46,10,37)(2,45,11,42)(3,44,12,41)(4,43,7,40)(5,48,8,39)(6,47,9,38)(13,25,22,34)(14,30,23,33)(15,29,24,32)(16,28,19,31)(17,27,20,36)(18,26,21,35), (1,16,4,13)(2,17,5,14)(3,18,6,15)(7,22,10,19)(8,23,11,20)(9,24,12,21)(25,40,28,37)(26,41,29,38)(27,42,30,39)(31,46,34,43)(32,47,35,44)(33,48,36,45)>;`

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

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

36 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 2H 2I 2J 2K 2L 2M 3 4A 4B 4C 4D 4E 4F 4G 4H 6A 6B 6C 6D 6E 6F 6G 12A 12B 12C 12D 12E 12F order 1 2 2 2 2 2 2 2 2 2 2 2 2 2 3 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 2 2 4 4 6 6 6 6 12 12 2 4 4 4 4 12 12 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 2 2 2 2 2 2 4 4 4 4 type + + + + + + + + + + + + + + + + + + + image C1 C2 C2 C2 C2 C2 C2 C2 C2 C2 S3 D4 D6 D6 D6 D6 2+ 1+4 S3×D4 D4⋊6D6 D4○D12 kernel C6.372+ 1+4 D6⋊D4 C23.9D6 D6.D4 C2×D6⋊C4 C12⋊7D4 C23⋊2D6 C23.14D6 C3×C4⋊D4 C2×S3×D4 C4⋊D4 C22×S3 C22⋊C4 C4⋊C4 C22×C4 C2×D4 C6 C22 C2 C2 # reps 1 2 2 2 1 1 2 2 1 2 1 4 2 1 1 3 2 2 2 2

Matrix representation of C6.372+ 1+4 in GL8(𝔽13)

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

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

C6.372+ 1+4 in GAP, Magma, Sage, TeX

`C_6._{37}2_+^{1+4}`
`% in TeX`

`G:=Group("C6.37ES+(2,2)");`
`// GroupNames label`

`G:=SmallGroup(192,1164);`
`// by ID`

`G=gap.SmallGroup(192,1164);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,219,675,297,136,6278]);`
`// Polycyclic`

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

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