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## G = C62.44C23order 288 = 25·32

### 39th non-split extension by C62 of C23 acting via C23/C2=C22

Series: Derived Chief Lower central Upper central

 Derived series C1 — C3×C6 — C62.44C23
 Chief series C1 — C3 — C32 — C3×C6 — C62 — C6×Dic3 — C62.C22 — C62.44C23
 Lower central C32 — C3×C6 — C62.44C23
 Upper central C1 — C2×C4

Generators and relations for C62.44C23
G = < a,b,c,d,e | a6=b6=1, c2=d2=a3, e2=b3, ab=ba, ac=ca, dad-1=a-1, ae=ea, cbc-1=b-1, bd=db, be=eb, dcd-1=b3c, ce=ec, de=ed >

Subgroups: 658 in 179 conjugacy classes, 60 normal (12 characteristic)
C1, C2, C2 [×2], C2 [×2], C3 [×2], C3, C4 [×2], C4 [×6], C22, C22 [×4], S3 [×8], C6 [×6], C6 [×3], C2×C4, C2×C4 [×9], C23, C32, Dic3 [×10], C12 [×4], C12 [×6], D6 [×14], C2×C6 [×2], C2×C6, C42 [×2], C22⋊C4 [×2], C4⋊C4 [×2], C22×C4, C3⋊S3 [×2], C3×C6, C3×C6 [×2], C4×S3 [×12], C2×Dic3 [×4], C2×Dic3 [×3], C2×C12 [×2], C2×C12 [×5], C22×S3 [×3], C42⋊C2, C3×Dic3 [×4], C3⋊Dic3 [×2], C3×C12 [×2], C2×C3⋊S3 [×2], C2×C3⋊S3 [×2], C62, C4×Dic3 [×2], Dic3⋊C4 [×4], D6⋊C4 [×4], C4×C12 [×2], S3×C2×C4 [×3], C6×Dic3 [×4], C4×C3⋊S3 [×4], C2×C3⋊Dic3, C6×C12, C22×C3⋊S3, C422S3 [×2], C6.D12 [×2], C62.C22 [×2], Dic3×C12 [×2], C2×C4×C3⋊S3, C62.44C23
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], S3 [×2], C2×C4 [×6], C23, D6 [×6], C22×C4, C4○D4 [×2], C4×S3 [×4], C22×S3 [×2], C42⋊C2, S32, S3×C2×C4 [×2], C4○D12 [×4], C6.D6 [×2], C2×S32, C422S3 [×2], D6.D6 [×2], C2×C6.D6, C62.44C23

Smallest permutation representation of C62.44C23
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 18 5 16 3 14)(2 13 6 17 4 15)(7 43 9 45 11 47)(8 44 10 46 12 48)(19 30 23 28 21 26)(20 25 24 29 22 27)(31 40 33 42 35 38)(32 41 34 37 36 39)
(1 45 4 48)(2 46 5 43)(3 47 6 44)(7 13 10 16)(8 14 11 17)(9 15 12 18)(19 39 22 42)(20 40 23 37)(21 41 24 38)(25 31 28 34)(26 32 29 35)(27 33 30 36)
(1 33 4 36)(2 32 5 35)(3 31 6 34)(7 30 10 27)(8 29 11 26)(9 28 12 25)(13 41 16 38)(14 40 17 37)(15 39 18 42)(19 44 22 47)(20 43 23 46)(21 48 24 45)
(1 27 16 24)(2 28 17 19)(3 29 18 20)(4 30 13 21)(5 25 14 22)(6 26 15 23)(7 38 45 33)(8 39 46 34)(9 40 47 35)(10 41 48 36)(11 42 43 31)(12 37 44 32)```

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

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

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

60 conjugacy classes

 class 1 2A 2B 2C 2D 2E 3A 3B 3C 4A 4B 4C 4D 4E ··· 4L 4M 4N 6A ··· 6F 6G 6H 6I 12A ··· 12H 12I 12J 12K 12L 12M ··· 12AB order 1 2 2 2 2 2 3 3 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 18 18 2 2 4 1 1 1 1 6 ··· 6 18 18 2 ··· 2 4 4 4 2 ··· 2 4 4 4 4 6 ··· 6

60 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 2 2 4 4 4 4 type + + + + + + + + + + + image C1 C2 C2 C2 C2 C4 S3 D6 D6 C4○D4 C4×S3 C4○D12 S32 C6.D6 C2×S32 D6.D6 kernel C62.44C23 C6.D12 C62.C22 Dic3×C12 C2×C4×C3⋊S3 C4×C3⋊S3 C4×Dic3 C2×Dic3 C2×C12 C3×C6 C12 C6 C2×C4 C4 C22 C2 # reps 1 2 2 2 1 8 2 4 2 4 8 16 1 2 1 4

Matrix representation of C62.44C23 in GL8(𝔽13)

 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 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 12
,
 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 1 0 0 0 0 0 0 0 0 1
,
 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 8 0 0 0 0 0 0 5 0 0 0 0 0 0 0 0 0 1 3 0 0 0 0 0 0 8 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 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 8 11 0 0 0 0 0 0 0 5 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 12 12
,
 12 0 0 0 0 0 0 0 0 12 0 0 0 0 0 0 0 0 8 0 0 0 0 0 0 0 0 8 0 0 0 0 0 0 0 0 5 0 0 0 0 0 0 0 0 5 0 0 0 0 0 0 0 0 12 0 0 0 0 0 0 0 0 12

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

C62.44C23 in GAP, Magma, Sage, TeX

`C_6^2._{44}C_2^3`
`% in TeX`

`G:=Group("C6^2.44C2^3");`
`// GroupNames label`

`G:=SmallGroup(288,522);`
`// by ID`

`G=gap.SmallGroup(288,522);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,56,253,64,100,1356,9414]);`
`// Polycyclic`

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

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