Copied to
clipboard

## G = C62.20C23order 288 = 25·32

### 15th non-split extension by C62 of C23 acting via C23/C2=C22

Series: Derived Chief Lower central Upper central

 Derived series C1 — C62 — C62.20C23
 Chief series C1 — C3 — C32 — C3×C6 — C62 — S3×C2×C6 — D6⋊Dic3 — C62.20C23
 Lower central C32 — C62 — C62.20C23
 Upper central C1 — C22 — C2×C4

Generators and relations for C62.20C23
G = < a,b,c,d,e | a6=b6=c2=1, d2=b3, e2=a3, ab=ba, ac=ca, dad-1=a-1, ae=ea, cbc=b-1, bd=db, be=eb, dcd-1=a3c, ce=ec, ede-1=b3d >

Subgroups: 730 in 173 conjugacy classes, 48 normal (44 characteristic)
C1, C2 [×3], C2 [×3], C3 [×2], C3, C4 [×5], C22, C22 [×7], S3 [×6], C6 [×6], C6 [×5], C2×C4, C2×C4 [×6], D4 [×2], C23 [×2], C32, Dic3 [×6], C12 [×6], D6 [×2], D6 [×12], C2×C6 [×2], C2×C6 [×5], C22⋊C4 [×3], C4⋊C4 [×2], C22×C4, C2×D4, C3×S3 [×2], C3⋊S3, C3×C6 [×3], C4×S3 [×2], D12 [×2], C2×Dic3 [×3], C2×Dic3 [×3], C3⋊D4 [×2], C2×C12 [×2], C2×C12 [×6], C22×S3, C22×S3 [×3], C22×C6, C22.D4, C3×Dic3 [×3], C3⋊Dic3, C3×C12, S3×C6 [×2], S3×C6 [×2], C2×C3⋊S3 [×3], C62, Dic3⋊C4, Dic3⋊C4 [×2], D6⋊C4 [×6], C6.D4, C3×C4⋊C4, S3×C2×C4, C2×D12, C2×C3⋊D4, C22×C12, C3⋊D12 [×2], S3×C12 [×2], C6×Dic3 [×3], C2×C3⋊Dic3, C6×C12, S3×C2×C6, C22×C3⋊S3, D6.D4, C23.28D6, D6⋊Dic3, C6.D12, C62.C22, C3×Dic3⋊C4, C6.11D12, C2×C3⋊D12, S3×C2×C12, C62.20C23
Quotients: C1, C2 [×7], C22 [×7], S3 [×2], D4 [×2], C23, D6 [×6], C2×D4, C4○D4 [×2], C3⋊D4 [×2], C22×S3 [×2], C22.D4, S32, C4○D12 [×3], S3×D4, Q83S3, C2×C3⋊D4, C2×S32, D6.D4, C23.28D6, D6.D6, D6.6D6, S3×C3⋊D4, C62.20C23

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

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

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

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

48 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 3A 3B 3C 4A 4B 4C 4D 4E 4F 4G 6A ··· 6F 6G 6H 6I 6J 6K 6L 6M 12A 12B 12C 12D 12E ··· 12J 12K 12L 12M 12N 12O 12P 12Q 12R order 1 2 2 2 2 2 2 3 3 3 4 4 4 4 4 4 4 6 ··· 6 6 6 6 6 6 6 6 12 12 12 12 12 ··· 12 12 12 12 12 12 12 12 12 size 1 1 1 1 6 6 36 2 2 4 2 2 6 6 12 12 36 2 ··· 2 4 4 4 6 6 6 6 2 2 2 2 4 ··· 4 6 6 6 6 12 12 12 12

48 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 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 Q8⋊3S3 C2×S32 D6.D6 D6.6D6 S3×C3⋊D4 kernel C62.20C23 D6⋊Dic3 C6.D12 C62.C22 C3×Dic3⋊C4 C6.11D12 C2×C3⋊D12 S3×C2×C12 Dic3⋊C4 S3×C2×C4 S3×C6 C2×Dic3 C2×C12 C22×S3 C3×C6 D6 C6 C2×C4 C6 C6 C22 C2 C2 C2 # reps 1 1 1 1 1 1 1 1 1 1 2 3 2 1 4 4 12 1 1 1 1 2 2 2

Matrix representation of C62.20C23 in GL6(𝔽13)

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

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

C62.20C23 in GAP, Magma, Sage, TeX

`C_6^2._{20}C_2^3`
`% in TeX`

`G:=Group("C6^2.20C2^3");`
`// GroupNames label`

`G:=SmallGroup(288,498);`
`// by ID`

`G=gap.SmallGroup(288,498);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,141,120,219,58,1356,9414]);`
`// Polycyclic`

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

׿
×
𝔽