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G = C6.562+ 1+4order 192 = 26·3

56th 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 — C6.562+ 1+4
 Chief series C1 — C3 — C6 — C2×C6 — C22×S3 — S3×C23 — D6⋊D4 — C6.562+ 1+4
 Lower central C3 — C2×C6 — C6.562+ 1+4
 Upper central C1 — C22 — C22⋊Q8

Generators and relations for C6.562+ 1+4
G = < a,b,c,d,e | a6=b4=c2=1, d2=b2, e2=a3, ab=ba, ac=ca, dad-1=a-1, ae=ea, cbc=a3b-1, bd=db, ebe-1=a3b, cd=dc, ce=ec, ede-1=a3b2d >

Subgroups: 768 in 250 conjugacy classes, 95 normal (31 characteristic)
C1, C2, C2, C3, C4, C22, C22, C22, S3, C6, C6, C2×C4, C2×C4, C2×C4, D4, Q8, C23, C23, Dic3, C12, D6, C2×C6, C2×C6, C2×C6, C42, C22⋊C4, C22⋊C4, C4⋊C4, C4⋊C4, C4⋊C4, C22×C4, C22×C4, C2×D4, C2×Q8, C24, C4×S3, D12, C2×Dic3, C2×Dic3, C3⋊D4, C2×C12, C2×C12, C2×C12, C3×Q8, C22×S3, C22×S3, C22×C6, C2×C22⋊C4, C4×D4, C22≀C2, C4⋊D4, C22⋊Q8, C22.D4, C4.4D4, C422C2, C4×Dic3, Dic3⋊C4, C4⋊Dic3, D6⋊C4, C3×C22⋊C4, C3×C4⋊C4, C3×C4⋊C4, S3×C2×C4, C2×D12, C2×D12, C22×Dic3, C2×C3⋊D4, C22×C12, C6×Q8, S3×C23, C22.32C24, Dic34D4, D6⋊D4, D6.D4, C12⋊D4, C4⋊C4⋊S3, C2×D6⋊C4, C127D4, C12.23D4, C3×C22⋊Q8, C6.562+ 1+4
Quotients: C1, C2, C22, S3, C23, D6, C4○D4, C24, C22×S3, C2×C4○D4, 2+ 1+4, Q83S3, S3×C23, C22.32C24, D46D6, C2×Q83S3, D4○D12, C6.562+ 1+4

Smallest permutation representation of C6.562+ 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 28 10 31)(2 29 11 32)(3 30 12 33)(4 25 7 34)(5 26 8 35)(6 27 9 36)(13 40 22 43)(14 41 23 44)(15 42 24 45)(16 37 19 46)(17 38 20 47)(18 39 21 48)
(25 31)(26 32)(27 33)(28 34)(29 35)(30 36)(37 43)(38 44)(39 45)(40 46)(41 47)(42 48)
(1 22 10 13)(2 21 11 18)(3 20 12 17)(4 19 7 16)(5 24 8 15)(6 23 9 14)(25 46 34 37)(26 45 35 42)(27 44 36 41)(28 43 31 40)(29 48 32 39)(30 47 33 38)
(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 37 28 40)(26 38 29 41)(27 39 30 42)(31 43 34 46)(32 44 35 47)(33 45 36 48)```

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

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

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

36 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 2H 2I 3 4A ··· 4F 4G 4H 4I 4J 4K 4L 6A 6B 6C 6D 6E 12A 12B 12C 12D 12E 12F 12G 12H order 1 2 2 2 2 2 2 2 2 2 3 4 ··· 4 4 4 4 4 4 4 6 6 6 6 6 12 12 12 12 12 12 12 12 size 1 1 1 1 2 2 12 12 12 12 2 4 ··· 4 6 6 6 6 12 12 2 2 2 4 4 4 4 4 4 8 8 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 D6 D6 D6 D6 C4○D4 2+ 1+4 Q8⋊3S3 D4⋊6D6 D4○D12 kernel C6.562+ 1+4 Dic3⋊4D4 D6⋊D4 D6.D4 C12⋊D4 C4⋊C4⋊S3 C2×D6⋊C4 C12⋊7D4 C12.23D4 C3×C22⋊Q8 C22⋊Q8 C22⋊C4 C4⋊C4 C22×C4 C2×Q8 C2×C6 C6 C22 C2 C2 # reps 1 2 2 2 2 2 1 1 2 1 1 2 3 1 1 4 2 2 2 2

Matrix representation of C6.562+ 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 0 12 0 0 0 0 1 12
,
 0 1 0 0 0 0 12 0 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
,
 12 0 0 0 0 0 0 12 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 12 0 0 0 0 0 0 12
,
 8 0 0 0 0 0 0 8 0 0 0 0 0 0 4 11 0 0 0 0 2 9 0 0 0 0 0 0 4 11 0 0 0 0 2 9
,
 5 0 0 0 0 0 0 8 0 0 0 0 0 0 2 9 0 0 0 0 4 11 0 0 0 0 0 0 2 9 0 0 0 0 4 11

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

C6.562+ 1+4 in GAP, Magma, Sage, TeX

`C_6._{56}2_+^{1+4}`
`% in TeX`

`G:=Group("C6.56ES+(2,2)");`
`// GroupNames label`

`G:=SmallGroup(192,1203);`
`// by ID`

`G=gap.SmallGroup(192,1203);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,758,219,184,675,297,136,6278]);`
`// 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,d*a*d^-1=a^-1,a*e=e*a,c*b*c=a^3*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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