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## G = D12.C4order 96 = 25·3

### The non-split extension by D12 of C4 acting via C4/C2=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C6 — D12.C4
 Chief series C1 — C3 — C6 — C12 — C4×S3 — C4○D12 — D12.C4
 Lower central C3 — C6 — D12.C4
 Upper central C1 — C4 — M4(2)

Generators and relations for D12.C4
G = < a,b,c | a12=b2=1, c4=a6, bab=a-1, cac-1=a7, cbc-1=a6b >

Subgroups: 114 in 62 conjugacy classes, 37 normal (21 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, S3, C6, C6, C8, C8, C2×C4, C2×C4, D4, Q8, Dic3, C12, D6, C2×C6, C2×C8, M4(2), M4(2), C4○D4, C3⋊C8, C24, Dic6, C4×S3, D12, C3⋊D4, C2×C12, C8○D4, S3×C8, C8⋊S3, C2×C3⋊C8, C3×M4(2), C4○D12, D12.C4
Quotients: C1, C2, C4, C22, S3, C2×C4, C23, D6, C22×C4, C4×S3, C22×S3, C8○D4, S3×C2×C4, D12.C4

Character table of D12.C4

 class 1 2A 2B 2C 2D 3 4A 4B 4C 4D 4E 6A 6B 8A 8B 8C 8D 8E 8F 8G 8H 8I 8J 12A 12B 12C 24A 24B 24C 24D size 1 1 2 6 6 2 1 1 2 6 6 2 4 2 2 2 2 3 3 3 3 6 6 2 2 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 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 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 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 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 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 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 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 1 -1 -1 1 -1 1 linear of order 2 ρ9 1 1 1 -1 1 1 -1 -1 -1 1 -1 1 1 -i i -i i i i -i -i i -i -1 -1 -1 i -i -i i linear of order 4 ρ10 1 1 1 1 -1 1 -1 -1 -1 -1 1 1 1 i -i i -i i i -i -i i -i -1 -1 -1 -i i i -i linear of order 4 ρ11 1 1 -1 1 1 1 -1 -1 1 -1 -1 1 -1 -i -i i i i i -i -i -i i -1 -1 1 -i -i i i linear of order 4 ρ12 1 1 -1 -1 -1 1 -1 -1 1 1 1 1 -1 i i -i -i i i -i -i -i i -1 -1 1 i i -i -i linear of order 4 ρ13 1 1 1 1 -1 1 -1 -1 -1 -1 1 1 1 -i i -i i -i -i i i -i i -1 -1 -1 i -i -i i linear of order 4 ρ14 1 1 1 -1 1 1 -1 -1 -1 1 -1 1 1 i -i i -i -i -i i i -i i -1 -1 -1 -i i i -i linear of order 4 ρ15 1 1 -1 -1 -1 1 -1 -1 1 1 1 1 -1 -i -i i i -i -i i i i -i -1 -1 1 -i -i i i linear of order 4 ρ16 1 1 -1 1 1 1 -1 -1 1 -1 -1 1 -1 i i -i -i -i -i i i i -i -1 -1 1 i i -i -i linear of order 4 ρ17 2 2 -2 0 0 -1 2 2 -2 0 0 -1 1 -2 2 2 -2 0 0 0 0 0 0 -1 -1 1 -1 1 -1 1 orthogonal lifted from D6 ρ18 2 2 2 0 0 -1 2 2 2 0 0 -1 -1 -2 -2 -2 -2 0 0 0 0 0 0 -1 -1 -1 1 1 1 1 orthogonal lifted from D6 ρ19 2 2 2 0 0 -1 2 2 2 0 0 -1 -1 2 2 2 2 0 0 0 0 0 0 -1 -1 -1 -1 -1 -1 -1 orthogonal lifted from S3 ρ20 2 2 -2 0 0 -1 2 2 -2 0 0 -1 1 2 -2 -2 2 0 0 0 0 0 0 -1 -1 1 1 -1 1 -1 orthogonal lifted from D6 ρ21 2 2 -2 0 0 -1 -2 -2 2 0 0 -1 1 2i 2i -2i -2i 0 0 0 0 0 0 1 1 -1 -i -i i i complex lifted from C4×S3 ρ22 2 2 2 0 0 -1 -2 -2 -2 0 0 -1 -1 2i -2i 2i -2i 0 0 0 0 0 0 1 1 1 i -i -i i complex lifted from C4×S3 ρ23 2 2 2 0 0 -1 -2 -2 -2 0 0 -1 -1 -2i 2i -2i 2i 0 0 0 0 0 0 1 1 1 -i i i -i complex lifted from C4×S3 ρ24 2 2 -2 0 0 -1 -2 -2 2 0 0 -1 1 -2i -2i 2i 2i 0 0 0 0 0 0 1 1 -1 i i -i -i complex lifted from C4×S3 ρ25 2 -2 0 0 0 2 -2i 2i 0 0 0 -2 0 0 0 0 0 2ζ87 2ζ83 2ζ85 2ζ8 0 0 2i -2i 0 0 0 0 0 complex lifted from C8○D4 ρ26 2 -2 0 0 0 2 2i -2i 0 0 0 -2 0 0 0 0 0 2ζ85 2ζ8 2ζ87 2ζ83 0 0 -2i 2i 0 0 0 0 0 complex lifted from C8○D4 ρ27 2 -2 0 0 0 2 2i -2i 0 0 0 -2 0 0 0 0 0 2ζ8 2ζ85 2ζ83 2ζ87 0 0 -2i 2i 0 0 0 0 0 complex lifted from C8○D4 ρ28 2 -2 0 0 0 2 -2i 2i 0 0 0 -2 0 0 0 0 0 2ζ83 2ζ87 2ζ8 2ζ85 0 0 2i -2i 0 0 0 0 0 complex lifted from C8○D4 ρ29 4 -4 0 0 0 -2 4i -4i 0 0 0 2 0 0 0 0 0 0 0 0 0 0 0 2i -2i 0 0 0 0 0 complex faithful ρ30 4 -4 0 0 0 -2 -4i 4i 0 0 0 2 0 0 0 0 0 0 0 0 0 0 0 -2i 2i 0 0 0 0 0 complex faithful

Smallest permutation representation of D12.C4
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 3)(4 12)(5 11)(6 10)(7 9)(13 15)(16 24)(17 23)(18 22)(19 21)(25 31)(26 30)(27 29)(32 36)(33 35)(38 48)(39 47)(40 46)(41 45)(42 44)
(1 39 13 36 7 45 19 30)(2 46 14 31 8 40 20 25)(3 41 15 26 9 47 21 32)(4 48 16 33 10 42 22 27)(5 43 17 28 11 37 23 34)(6 38 18 35 12 44 24 29)```

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

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

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

Matrix representation of D12.C4 in GL4(𝔽5) generated by

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

D12.C4 in GAP, Magma, Sage, TeX

`D_{12}.C_4`
`% in TeX`

`G:=Group("D12.C4");`
`// GroupNames label`

`G:=SmallGroup(96,114);`
`// by ID`

`G=gap.SmallGroup(96,114);`
`# by ID`

`G:=PCGroup([6,-2,-2,-2,-2,-2,-3,217,188,50,69,2309]);`
`// Polycyclic`

`G:=Group<a,b,c|a^12=b^2=1,c^4=a^6,b*a*b=a^-1,c*a*c^-1=a^7,c*b*c^-1=a^6*b>;`
`// generators/relations`

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