Copied to
clipboard

## G = C23⋊4D12order 192 = 26·3

### 2nd semidirect product of C23 and D12 acting via D12/C6=C22

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C6 — C23⋊4D12
 Chief series C1 — C3 — C6 — C2×C6 — C22×S3 — S3×C23 — C22×C3⋊D4 — C23⋊4D12
 Lower central C3 — C2×C6 — C23⋊4D12
 Upper central C1 — C22 — C2×C22⋊C4

Generators and relations for C234D12
G = < a,b,c,d,e | a2=b2=c2=d12=e2=1, ab=ba, dad-1=eae=ac=ca, ebe=bc=cb, bd=db, cd=dc, ce=ec, ede=d-1 >

Subgroups: 1096 in 346 conjugacy classes, 111 normal (13 characteristic)
C1, C2, C2 [×2], C2 [×10], C3, C4 [×8], C22, C22 [×6], C22 [×30], S3 [×4], C6, C6 [×2], C6 [×6], C2×C4 [×4], C2×C4 [×10], D4 [×20], C23, C23 [×6], C23 [×14], Dic3 [×4], C12 [×4], D6 [×20], C2×C6, C2×C6 [×6], C2×C6 [×10], C22⋊C4 [×4], C22⋊C4 [×8], C4⋊C4 [×4], C22×C4 [×2], C22×C4 [×2], C2×D4 [×20], C24, C24 [×2], D12 [×4], C2×Dic3 [×4], C2×Dic3 [×4], C3⋊D4 [×16], C2×C12 [×4], C2×C12 [×2], C22×S3 [×4], C22×S3 [×8], C22×C6, C22×C6 [×6], C22×C6 [×2], C2×C22⋊C4, C22≀C2 [×4], C4⋊D4 [×4], C22.D4 [×4], C22×D4 [×2], C4⋊Dic3 [×4], D6⋊C4 [×8], C3×C22⋊C4 [×4], C2×D12 [×4], C22×Dic3 [×2], C2×C3⋊D4 [×8], C2×C3⋊D4 [×8], C22×C12 [×2], S3×C23 [×2], C23×C6, C233D4, D6⋊D4 [×4], C23.21D6 [×4], C127D4 [×4], C6×C22⋊C4, C22×C3⋊D4 [×2], C234D12
Quotients: C1, C2 [×15], C22 [×35], S3, D4 [×4], C23 [×15], D6 [×7], C2×D4 [×6], C24, D12 [×4], C22×S3 [×7], C22×D4, 2+ 1+4 [×2], C2×D12 [×6], S3×C23, C233D4, C22×D12, D46D6 [×2], C234D12

Smallest permutation representation of C234D12
On 48 points
Generators in S48
```(1 30)(2 8)(3 32)(4 10)(5 34)(6 12)(7 36)(9 26)(11 28)(13 43)(14 20)(15 45)(16 22)(17 47)(18 24)(19 37)(21 39)(23 41)(25 31)(27 33)(29 35)(38 44)(40 46)(42 48)
(1 15)(2 16)(3 17)(4 18)(5 19)(6 20)(7 21)(8 22)(9 23)(10 24)(11 13)(12 14)(25 40)(26 41)(27 42)(28 43)(29 44)(30 45)(31 46)(32 47)(33 48)(34 37)(35 38)(36 39)
(1 36)(2 25)(3 26)(4 27)(5 28)(6 29)(7 30)(8 31)(9 32)(10 33)(11 34)(12 35)(13 37)(14 38)(15 39)(16 40)(17 41)(18 42)(19 43)(20 44)(21 45)(22 46)(23 47)(24 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 12)(2 11)(3 10)(4 9)(5 8)(6 7)(13 40)(14 39)(15 38)(16 37)(17 48)(18 47)(19 46)(20 45)(21 44)(22 43)(23 42)(24 41)(25 34)(26 33)(27 32)(28 31)(29 30)(35 36)```

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

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

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

42 conjugacy classes

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

42 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 2 2 4 4 type + + + + + + + + + + + + + image C1 C2 C2 C2 C2 C2 S3 D4 D6 D6 D6 D12 2+ 1+4 D4⋊6D6 kernel C23⋊4D12 D6⋊D4 C23.21D6 C12⋊7D4 C6×C22⋊C4 C22×C3⋊D4 C2×C22⋊C4 C22×C6 C22⋊C4 C22×C4 C24 C23 C6 C2 # reps 1 4 4 4 1 2 1 4 4 2 1 8 2 4

Matrix representation of C234D12 in GL8(𝔽13)

 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 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 12 0 0 0 0 0 0 0 1 1
,
 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 6 0 3 0 0 0 0 0 0 6 0 3 0 0 0 0 10 0 7 0 0 0 0 0 0 10 0 7
,
 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 12 0 0 0 0 0 0 0 0 12
,
 0 1 0 0 0 0 0 0 12 0 0 0 0 0 0 0 0 0 0 12 0 0 0 0 0 0 1 12 0 0 0 0 0 0 0 0 12 11 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 12 11 0 0 0 0 0 0 0 1
,
 0 12 0 0 0 0 0 0 12 0 0 0 0 0 0 0 0 0 12 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 12 11 0 0 0 0 0 0 0 1 0 0 0 0 0 0 4 8 1 2 0 0 0 0 0 9 0 12

`G:=sub<GL(8,GF(13))| [1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,12,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,12,1,0,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,6,0,10,0,0,0,0,0,0,6,0,10,0,0,0,0,3,0,7,0,0,0,0,0,0,3,0,7],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,12],[0,12,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,12,12,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,11,1,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,11,1],[0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,1,1,0,0,0,0,0,0,0,0,12,0,4,0,0,0,0,0,11,1,8,9,0,0,0,0,0,0,1,0,0,0,0,0,0,0,2,12] >;`

C234D12 in GAP, Magma, Sage, TeX

`C_2^3\rtimes_4D_{12}`
`% in TeX`

`G:=Group("C2^3:4D12");`
`// GroupNames label`

`G:=SmallGroup(192,1052);`
`// by ID`

`G=gap.SmallGroup(192,1052);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,758,675,570,80,6278]);`
`// Polycyclic`

`G:=Group<a,b,c,d,e|a^2=b^2=c^2=d^12=e^2=1,a*b=b*a,d*a*d^-1=e*a*e=a*c=c*a,e*b*e=b*c=c*b,b*d=d*b,c*d=d*c,c*e=e*c,e*d*e=d^-1>;`
`// generators/relations`

׿
×
𝔽