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## G = C12.23D4order 96 = 25·3

### 23rd non-split extension by C12 of D4 acting via D4/C2=C22

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C6 — C12.23D4
 Chief series C1 — C3 — C6 — C2×C6 — C22×S3 — C2×D12 — C12.23D4
 Lower central C3 — C2×C6 — C12.23D4
 Upper central C1 — C22 — C2×Q8

Generators and relations for C12.23D4
G = < a,b,c | a12=b4=c2=1, bab-1=a5, cac=a-1, cbc=a6b-1 >

Subgroups: 194 in 76 conjugacy classes, 33 normal (13 characteristic)
C1, C2, C2, C2, C3, C4, C4, C22, C22, S3, C6, C6, C2×C4, C2×C4, C2×C4, D4, Q8, C23, Dic3, C12, C12, D6, C2×C6, C42, C22⋊C4, C2×D4, C2×Q8, D12, C2×Dic3, C2×C12, C2×C12, C3×Q8, C22×S3, C4.4D4, C4×Dic3, D6⋊C4, C2×D12, C6×Q8, C12.23D4
Quotients: C1, C2, C22, S3, D4, C23, D6, C2×D4, C4○D4, C3⋊D4, C22×S3, C4.4D4, Q83S3, C2×C3⋊D4, C12.23D4

Character table of C12.23D4

 class 1 2A 2B 2C 2D 2E 3 4A 4B 4C 4D 4E 4F 4G 4H 6A 6B 6C 12A 12B 12C 12D 12E 12F size 1 1 1 1 12 12 2 2 2 4 4 6 6 6 6 2 2 2 4 4 4 4 4 4 ρ1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 trivial ρ2 1 1 1 1 1 1 1 1 1 -1 -1 -1 -1 -1 -1 1 1 1 1 -1 -1 1 -1 -1 linear of order 2 ρ3 1 1 1 1 -1 1 1 -1 -1 -1 1 -1 1 -1 1 1 1 1 -1 1 -1 -1 -1 1 linear of order 2 ρ4 1 1 1 1 -1 1 1 -1 -1 1 -1 1 -1 1 -1 1 1 1 -1 -1 1 -1 1 -1 linear of order 2 ρ5 1 1 1 1 -1 -1 1 1 1 1 1 -1 -1 -1 -1 1 1 1 1 1 1 1 1 1 linear of order 2 ρ6 1 1 1 1 -1 -1 1 1 1 -1 -1 1 1 1 1 1 1 1 1 -1 -1 1 -1 -1 linear of order 2 ρ7 1 1 1 1 1 -1 1 -1 -1 -1 1 1 -1 1 -1 1 1 1 -1 1 -1 -1 -1 1 linear of order 2 ρ8 1 1 1 1 1 -1 1 -1 -1 1 -1 -1 1 -1 1 1 1 1 -1 -1 1 -1 1 -1 linear of order 2 ρ9 2 2 2 2 0 0 -1 -2 -2 2 -2 0 0 0 0 -1 -1 -1 1 1 -1 1 -1 1 orthogonal lifted from D6 ρ10 2 2 -2 -2 0 0 2 2 -2 0 0 0 0 0 0 -2 2 -2 2 0 0 -2 0 0 orthogonal lifted from D4 ρ11 2 2 2 2 0 0 -1 2 2 -2 -2 0 0 0 0 -1 -1 -1 -1 1 1 -1 1 1 orthogonal lifted from D6 ρ12 2 2 -2 -2 0 0 2 -2 2 0 0 0 0 0 0 -2 2 -2 -2 0 0 2 0 0 orthogonal lifted from D4 ρ13 2 2 2 2 0 0 -1 2 2 2 2 0 0 0 0 -1 -1 -1 -1 -1 -1 -1 -1 -1 orthogonal lifted from S3 ρ14 2 2 2 2 0 0 -1 -2 -2 -2 2 0 0 0 0 -1 -1 -1 1 -1 1 1 1 -1 orthogonal lifted from D6 ρ15 2 2 -2 -2 0 0 -1 -2 2 0 0 0 0 0 0 1 -1 1 1 -√-3 √-3 -1 -√-3 √-3 complex lifted from C3⋊D4 ρ16 2 2 -2 -2 0 0 -1 2 -2 0 0 0 0 0 0 1 -1 1 -1 -√-3 -√-3 1 √-3 √-3 complex lifted from C3⋊D4 ρ17 2 2 -2 -2 0 0 -1 -2 2 0 0 0 0 0 0 1 -1 1 1 √-3 -√-3 -1 √-3 -√-3 complex lifted from C3⋊D4 ρ18 2 2 -2 -2 0 0 -1 2 -2 0 0 0 0 0 0 1 -1 1 -1 √-3 √-3 1 -√-3 -√-3 complex lifted from C3⋊D4 ρ19 2 -2 2 -2 0 0 2 0 0 0 0 0 2i 0 -2i 2 -2 -2 0 0 0 0 0 0 complex lifted from C4○D4 ρ20 2 -2 2 -2 0 0 2 0 0 0 0 0 -2i 0 2i 2 -2 -2 0 0 0 0 0 0 complex lifted from C4○D4 ρ21 2 -2 -2 2 0 0 2 0 0 0 0 -2i 0 2i 0 -2 -2 2 0 0 0 0 0 0 complex lifted from C4○D4 ρ22 2 -2 -2 2 0 0 2 0 0 0 0 2i 0 -2i 0 -2 -2 2 0 0 0 0 0 0 complex lifted from C4○D4 ρ23 4 -4 -4 4 0 0 -2 0 0 0 0 0 0 0 0 2 2 -2 0 0 0 0 0 0 orthogonal lifted from Q8⋊3S3, Schur index 2 ρ24 4 -4 4 -4 0 0 -2 0 0 0 0 0 0 0 0 -2 2 2 0 0 0 0 0 0 orthogonal lifted from Q8⋊3S3, Schur index 2

Smallest permutation representation of C12.23D4
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 31 23 38)(2 36 24 43)(3 29 13 48)(4 34 14 41)(5 27 15 46)(6 32 16 39)(7 25 17 44)(8 30 18 37)(9 35 19 42)(10 28 20 47)(11 33 21 40)(12 26 22 45)
(2 12)(3 11)(4 10)(5 9)(6 8)(13 21)(14 20)(15 19)(16 18)(22 24)(25 38)(26 37)(27 48)(28 47)(29 46)(30 45)(31 44)(32 43)(33 42)(34 41)(35 40)(36 39)```

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

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

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

Matrix representation of C12.23D4 in GL4(𝔽13) generated by

 0 1 0 0 12 12 0 0 0 0 5 3 0 0 0 8
,
 11 9 0 0 11 2 0 0 0 0 12 2 0 0 0 1
,
 12 0 0 0 1 1 0 0 0 0 1 0 0 0 1 12
`G:=sub<GL(4,GF(13))| [0,12,0,0,1,12,0,0,0,0,5,0,0,0,3,8],[11,11,0,0,9,2,0,0,0,0,12,0,0,0,2,1],[12,1,0,0,0,1,0,0,0,0,1,1,0,0,0,12] >;`

C12.23D4 in GAP, Magma, Sage, TeX

`C_{12}._{23}D_4`
`% in TeX`

`G:=Group("C12.23D4");`
`// GroupNames label`

`G:=SmallGroup(96,154);`
`// by ID`

`G=gap.SmallGroup(96,154);`
`# by ID`

`G:=PCGroup([6,-2,-2,-2,-2,-2,-3,217,103,218,188,86,2309]);`
`// Polycyclic`

`G:=Group<a,b,c|a^12=b^4=c^2=1,b*a*b^-1=a^5,c*a*c=a^-1,c*b*c=a^6*b^-1>;`
`// generators/relations`

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