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## G = C6.1202+ 1+4order 192 = 26·3

### 29th 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 — C6.1202+ 1+4
 Chief series C1 — C3 — C6 — C2×C6 — C22×S3 — S3×C23 — C2×S3×D4 — C6.1202+ 1+4
 Lower central C3 — C2×C6 — C6.1202+ 1+4
 Upper central C1 — C22 — C22.D4

Generators and relations for C6.1202+ 1+4
G = < a,b,c,d,e | a6=b4=c2=e2=1, d2=a3b2, ab=ba, ac=ca, ad=da, eae=a-1, cbc=a3b-1, dbd-1=ebe=a3b, cd=dc, ce=ec, ede=a3b2d >

Subgroups: 1168 in 346 conjugacy classes, 103 normal (39 characteristic)
C1, C2, C2, C2, C3, C4, C22, C22, C22, S3, C6, C6, C6, C2×C4, C2×C4, C2×C4, D4, C23, C23, Dic3, C12, D6, D6, C2×C6, C2×C6, C2×C6, C22⋊C4, C22⋊C4, C22⋊C4, C4⋊C4, C4⋊C4, C22×C4, C22×C4, 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×S3, C22×C6, C2×C22⋊C4, C22≀C2, C4⋊D4, C22.D4, C22.D4, C22×D4, Dic3⋊C4, D6⋊C4, C6.D4, C3×C22⋊C4, C3×C22⋊C4, C3×C4⋊C4, S3×C2×C4, S3×C2×C4, C2×D12, C2×D12, C2×D12, S3×D4, C2×C3⋊D4, C2×C3⋊D4, C22×C12, C6×D4, S3×C23, C233D4, S3×C22⋊C4, D6⋊D4, D6⋊D4, Dic3⋊D4, D6.D4, C12⋊D4, C23.28D6, C232D6, C3×C22.D4, C22×D12, C2×S3×D4, C6.1202+ 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, D4○D12, C6.1202+ 1+4

Smallest permutation representation of C6.1202+ 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 40 10 43)(2 41 11 44)(3 42 12 45)(4 37 7 46)(5 38 8 47)(6 39 9 48)(13 34 22 25)(14 35 23 26)(15 36 24 27)(16 31 19 28)(17 32 20 29)(18 33 21 30)
(1 4)(2 5)(3 6)(7 10)(8 11)(9 12)(13 16)(14 17)(15 18)(19 22)(20 23)(21 24)(25 34)(26 35)(27 36)(28 31)(29 32)(30 33)(37 46)(38 47)(39 48)(40 43)(41 44)(42 45)
(1 19 7 13)(2 20 8 14)(3 21 9 15)(4 22 10 16)(5 23 11 17)(6 24 12 18)(25 46 31 40)(26 47 32 41)(27 48 33 42)(28 43 34 37)(29 44 35 38)(30 45 36 39)
(1 16)(2 15)(3 14)(4 13)(5 18)(6 17)(7 22)(8 21)(9 20)(10 19)(11 24)(12 23)(25 43)(26 48)(27 47)(28 46)(29 45)(30 44)(31 37)(32 42)(33 41)(34 40)(35 39)(36 38)```

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

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

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

36 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 2H 2I 2J 2K 2L 2M 3 4A ··· 4E 4F 4G 4H 6A 6B 6C 6D 6E 6F 12A 12B 12C 12D 12E 12F 12G order 1 2 2 2 2 2 2 2 2 2 2 2 2 2 3 4 ··· 4 4 4 4 6 6 6 6 6 6 12 12 12 12 12 12 12 size 1 1 1 1 2 2 4 6 6 6 6 12 12 12 2 4 ··· 4 12 12 12 2 2 2 4 4 8 4 4 4 4 8 8 8

36 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 4 4 4 type + + + + + + + + + + + + + + + + + + + + image C1 C2 C2 C2 C2 C2 C2 C2 C2 C2 C2 S3 D4 D6 D6 D6 D6 2+ 1+4 S3×D4 D4○D12 kernel C6.1202+ 1+4 S3×C22⋊C4 D6⋊D4 Dic3⋊D4 D6.D4 C12⋊D4 C23.28D6 C23⋊2D6 C3×C22.D4 C22×D12 C2×S3×D4 C22.D4 C22×S3 C22⋊C4 C4⋊C4 C22×C4 C2×D4 C6 C22 C2 # reps 1 1 3 2 2 2 1 1 1 1 1 1 4 3 2 1 1 2 2 4

Matrix representation of C6.1202+ 1+4 in GL6(𝔽13)

 12 0 0 0 0 0 0 12 0 0 0 0 0 0 0 12 0 0 0 0 1 12 0 0 0 0 0 0 12 12 0 0 0 0 1 0
,
 0 12 0 0 0 0 1 0 0 0 0 0 0 0 10 6 9 9 0 0 7 3 0 9 0 0 0 0 10 7 0 0 0 0 6 3
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 8 8 12 0 0 0 10 8 0 12
,
 1 0 0 0 0 0 0 12 0 0 0 0 0 0 3 7 0 0 0 0 6 10 0 0 0 0 0 0 10 7 0 0 0 0 6 3
,
 1 0 0 0 0 0 0 12 0 0 0 0 0 0 6 10 0 0 0 0 3 7 0 0 0 0 7 6 3 6 0 0 6 0 3 10

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

C6.1202+ 1+4 in GAP, Magma, Sage, TeX

`C_6._{120}2_+^{1+4}`
`% in TeX`

`G:=Group("C6.120ES+(2,2)");`
`// GroupNames label`

`G:=SmallGroup(192,1212);`
`// by ID`

`G=gap.SmallGroup(192,1212);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,219,675,297,80,6278]);`
`// Polycyclic`

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

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