Copied to
clipboard

## G = C12.76C24order 192 = 26·3

### 23rd non-split extension by C12 of C24 acting via C24/C23=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C6 — C12.76C24
 Chief series C1 — C3 — C6 — C12 — C3⋊C8 — C2×C3⋊C8 — D4.Dic3 — C12.76C24
 Lower central C3 — C6 — C12.76C24
 Upper central C1 — C4 — C2×C4○D4

Generators and relations for C12.76C24
G = < a,b,c,d,e | a12=c2=d2=e2=1, b2=a9, bab-1=a5, ac=ca, ad=da, ae=ea, bc=cb, bd=db, ebe=a6b, dcd=a6c, ce=ec, de=ed >

Subgroups: 392 in 258 conjugacy classes, 187 normal (17 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, C22, C6, C6, C8, C2×C4, C2×C4, D4, Q8, C23, C12, C12, C2×C6, C2×C6, C2×C6, C2×C8, M4(2), C22×C4, C2×D4, C2×Q8, C4○D4, C3⋊C8, C2×C12, C2×C12, C3×D4, C3×Q8, C22×C6, C2×M4(2), C8○D4, C2×C4○D4, C2×C3⋊C8, C4.Dic3, C22×C12, C6×D4, C6×Q8, C3×C4○D4, Q8○M4(2), C2×C4.Dic3, D4.Dic3, C6×C4○D4, C12.76C24
Quotients: C1, C2, C4, C22, S3, C2×C4, C23, Dic3, D6, C22×C4, C24, C2×Dic3, C22×S3, C23×C4, C22×Dic3, S3×C23, Q8○M4(2), C23×Dic3, C12.76C24

Smallest permutation representation of C12.76C24
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 25 7 34 4 31)(2 33 11 30 8 27 5 36)(3 26 12 35 9 32 6 29)(13 46 22 43 19 40 16 37)(14 39 23 48 20 45 17 42)(15 44 24 41 21 38 18 47)
(1 13)(2 14)(3 15)(4 16)(5 17)(6 18)(7 19)(8 20)(9 21)(10 22)(11 23)(12 24)(25 43)(26 44)(27 45)(28 46)(29 47)(30 48)(31 37)(32 38)(33 39)(34 40)(35 41)(36 42)
(1 22)(2 23)(3 24)(4 13)(5 14)(6 15)(7 16)(8 17)(9 18)(10 19)(11 20)(12 21)(25 40)(26 41)(27 42)(28 43)(29 44)(30 45)(31 46)(32 47)(33 48)(34 37)(35 38)(36 39)
(25 31)(26 32)(27 33)(28 34)(29 35)(30 36)(37 43)(38 44)(39 45)(40 46)(41 47)(42 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), (1,28,10,25,7,34,4,31)(2,33,11,30,8,27,5,36)(3,26,12,35,9,32,6,29)(13,46,22,43,19,40,16,37)(14,39,23,48,20,45,17,42)(15,44,24,41,21,38,18,47), (1,13)(2,14)(3,15)(4,16)(5,17)(6,18)(7,19)(8,20)(9,21)(10,22)(11,23)(12,24)(25,43)(26,44)(27,45)(28,46)(29,47)(30,48)(31,37)(32,38)(33,39)(34,40)(35,41)(36,42), (1,22)(2,23)(3,24)(4,13)(5,14)(6,15)(7,16)(8,17)(9,18)(10,19)(11,20)(12,21)(25,40)(26,41)(27,42)(28,43)(29,44)(30,45)(31,46)(32,47)(33,48)(34,37)(35,38)(36,39), (25,31)(26,32)(27,33)(28,34)(29,35)(30,36)(37,43)(38,44)(39,45)(40,46)(41,47)(42,48)>;`

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

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

54 conjugacy classes

 class 1 2A 2B ··· 2H 3 4A 4B 4C ··· 4I 6A 6B 6C 6D ··· 6I 8A ··· 8P 12A 12B 12C 12D 12E ··· 12J order 1 2 2 ··· 2 3 4 4 4 ··· 4 6 6 6 6 ··· 6 8 ··· 8 12 12 12 12 12 ··· 12 size 1 1 2 ··· 2 2 1 1 2 ··· 2 2 2 2 4 ··· 4 6 ··· 6 2 2 2 2 4 ··· 4

54 irreducible representations

 dim 1 1 1 1 1 1 1 2 2 2 2 2 2 4 4 type + + + + + + - - - + image C1 C2 C2 C2 C4 C4 C4 S3 D6 Dic3 Dic3 Dic3 D6 Q8○M4(2) C12.76C24 kernel C12.76C24 C2×C4.Dic3 D4.Dic3 C6×C4○D4 C6×D4 C6×Q8 C3×C4○D4 C2×C4○D4 C22×C4 C2×D4 C2×Q8 C4○D4 C4○D4 C3 C1 # reps 1 6 8 1 6 2 8 1 3 3 1 4 4 2 4

Matrix representation of C12.76C24 in GL4(𝔽73) generated by

 24 0 0 0 0 24 0 0 0 0 3 0 0 0 0 3
,
 0 0 1 0 0 0 0 1 46 0 0 0 0 46 0 0
,
 46 43 0 0 34 27 0 0 0 0 46 43 0 0 34 27
,
 72 0 0 0 31 1 0 0 0 0 72 0 0 0 31 1
,
 1 0 0 0 0 1 0 0 0 0 72 0 0 0 0 72
`G:=sub<GL(4,GF(73))| [24,0,0,0,0,24,0,0,0,0,3,0,0,0,0,3],[0,0,46,0,0,0,0,46,1,0,0,0,0,1,0,0],[46,34,0,0,43,27,0,0,0,0,46,34,0,0,43,27],[72,31,0,0,0,1,0,0,0,0,72,31,0,0,0,1],[1,0,0,0,0,1,0,0,0,0,72,0,0,0,0,72] >;`

C12.76C24 in GAP, Magma, Sage, TeX

`C_{12}._{76}C_2^4`
`% in TeX`

`G:=Group("C12.76C2^4");`
`// GroupNames label`

`G:=SmallGroup(192,1378);`
`// by ID`

`G=gap.SmallGroup(192,1378);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,112,387,1123,102,6278]);`
`// Polycyclic`

`G:=Group<a,b,c,d,e|a^12=c^2=d^2=e^2=1,b^2=a^9,b*a*b^-1=a^5,a*c=c*a,a*d=d*a,a*e=e*a,b*c=c*b,b*d=d*b,e*b*e=a^6*b,d*c*d=a^6*c,c*e=e*c,d*e=e*d>;`
`// generators/relations`

׿
×
𝔽