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## G = C62.77C23order 288 = 25·32

### 72nd non-split extension by C62 of C23 acting via C23/C2=C22

Series: Derived Chief Lower central Upper central

 Derived series C1 — C62 — C62.77C23
 Chief series C1 — C3 — C32 — C3×C6 — C62 — C6×Dic3 — C2×C3⋊D12 — C62.77C23
 Lower central C32 — C62 — C62.77C23
 Upper central C1 — C22 — C2×C4

Generators and relations for C62.77C23
G = < a,b,c,d,e | a6=b6=d2=1, c2=e2=b3, ab=ba, ac=ca, dad=a-1, ae=ea, cbc-1=b-1, bd=db, be=eb, dcd=ece-1=b3c, ede-1=a3d >

Subgroups: 730 in 169 conjugacy classes, 48 normal (44 characteristic)
C1, C2 [×3], C2 [×2], C3 [×2], C3, C4 [×6], C22, C22 [×6], S3 [×5], C6 [×6], C6 [×4], C2×C4, C2×C4 [×4], D4 [×2], Q8 [×2], C23 [×2], C32, Dic3 [×2], Dic3 [×5], C12 [×7], D6 [×13], C2×C6 [×2], C2×C6 [×4], C42, C22⋊C4 [×4], C2×D4, C2×Q8, C3×S3, C3⋊S3, C3×C6 [×3], Dic6 [×2], D12 [×2], C2×Dic3 [×3], C2×Dic3 [×3], C3⋊D4 [×2], C2×C12 [×2], C2×C12 [×4], C3×Q8 [×2], C22×S3, C22×S3 [×3], C22×C6, C4.4D4, C3×Dic3 [×2], C3×Dic3 [×2], C3⋊Dic3, C3×C12, S3×C6 [×3], C2×C3⋊S3 [×3], C62, C4×Dic3 [×2], D6⋊C4, D6⋊C4 [×6], C6.D4, C3×C22⋊C4, C2×Dic6, C2×D12, C2×C3⋊D4, C6×Q8, C3⋊D12 [×2], C3×Dic6 [×2], C6×Dic3 [×3], C2×C3⋊Dic3, C6×C12, S3×C2×C6, C22×C3⋊S3, C23.11D6, C12.23D4, Dic32, D6⋊Dic3, C6.D12, C3×D6⋊C4, C6.11D12, C2×C3⋊D12, C6×Dic6, C62.77C23
Quotients: C1, C2 [×7], C22 [×7], S3 [×2], D4 [×2], C23, D6 [×6], C2×D4, C4○D4 [×2], C3⋊D4 [×2], C22×S3 [×2], C4.4D4, S32, C4○D12, S3×D4, D42S3, Q83S3 [×2], C2×C3⋊D4, C2×S32, C23.11D6, C12.23D4, D12⋊S3, D6.6D6, S3×C3⋊D4, C62.77C23

Smallest permutation representation of C62.77C23
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 15 3 17 5 13)(2 16 4 18 6 14)(7 46 11 44 9 48)(8 47 12 45 10 43)(19 25 21 27 23 29)(20 26 22 28 24 30)(31 41 35 39 33 37)(32 42 36 40 34 38)
(1 32 17 40)(2 33 18 41)(3 34 13 42)(4 35 14 37)(5 36 15 38)(6 31 16 39)(7 27 44 19)(8 28 45 20)(9 29 46 21)(10 30 47 22)(11 25 48 23)(12 26 43 24)
(1 32)(2 31)(3 36)(4 35)(5 34)(6 33)(7 30)(8 29)(9 28)(10 27)(11 26)(12 25)(13 38)(14 37)(15 42)(16 41)(17 40)(18 39)(19 47)(20 46)(21 45)(22 44)(23 43)(24 48)
(1 30 17 22)(2 25 18 23)(3 26 13 24)(4 27 14 19)(5 28 15 20)(6 29 16 21)(7 37 44 35)(8 38 45 36)(9 39 46 31)(10 40 47 32)(11 41 48 33)(12 42 43 34)```

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

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

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

42 conjugacy classes

 class 1 2A 2B 2C 2D 2E 3A 3B 3C 4A 4B 4C 4D 4E 4F 4G 4H 6A ··· 6F 6G 6H 6I 6J 6K 12A ··· 12H 12I ··· 12N order 1 2 2 2 2 2 3 3 3 4 4 4 4 4 4 4 4 6 ··· 6 6 6 6 6 6 12 ··· 12 12 ··· 12 size 1 1 1 1 12 36 2 2 4 4 6 6 6 6 12 18 18 2 ··· 2 4 4 4 12 12 4 ··· 4 12 ··· 12

42 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 4 4 4 4 4 4 4 4 type + + + + + + + + + + + + + + + + - + + + image C1 C2 C2 C2 C2 C2 C2 C2 S3 S3 D4 D6 D6 D6 C4○D4 C3⋊D4 C4○D12 S32 S3×D4 D4⋊2S3 Q8⋊3S3 C2×S32 D12⋊S3 D6.6D6 S3×C3⋊D4 kernel C62.77C23 Dic32 D6⋊Dic3 C6.D12 C3×D6⋊C4 C6.11D12 C2×C3⋊D12 C6×Dic6 D6⋊C4 C2×Dic6 C3×Dic3 C2×Dic3 C2×C12 C22×S3 C3×C6 Dic3 C6 C2×C4 C6 C6 C6 C22 C2 C2 C2 # reps 1 1 1 1 1 1 1 1 1 1 2 3 2 1 4 4 4 1 1 1 2 1 2 2 2

Matrix representation of C62.77C23 in GL8(𝔽13)

 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 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 1 0 0 0 0 0 0 12 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 12 1 0 0 0 0 0 0 12 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1
,
 12 0 0 0 0 0 0 0 0 12 0 0 0 0 0 0 0 0 0 5 0 0 0 0 0 0 5 0 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 1 0 0 0 0 0 0 0 0 1
,
 12 0 0 0 0 0 0 0 12 1 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 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
,
 1 11 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 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

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

C62.77C23 in GAP, Magma, Sage, TeX

`C_6^2._{77}C_2^3`
`% in TeX`

`G:=Group("C6^2.77C2^3");`
`// GroupNames label`

`G:=SmallGroup(288,555);`
`// by ID`

`G=gap.SmallGroup(288,555);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,112,253,422,135,100,1356,9414]);`
`// Polycyclic`

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

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