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## G = (C3×C12).D4order 288 = 25·32

### 2nd non-split extension by C3×C12 of D4 acting faithfully

Aliases: C4.10S3≀C2, (C3×C12).2D4, C6.D6.C4, C3⋊Dic3.3D4, C32⋊(C4.10D4), Dic3.D6.1C2, C12.31D6.1C2, C32⋊M4(2).1C2, C2.4(S32⋊C4), (C4×C3⋊S3).2C22, (C3×C6).3(C22⋊C4), (C2×C3⋊S3).7(C2×C4), SmallGroup(288,376)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C32 — C2×C3⋊S3 — (C3×C12).D4
 Chief series C1 — C32 — C3×C6 — C2×C3⋊S3 — C4×C3⋊S3 — Dic3.D6 — (C3×C12).D4
 Lower central C32 — C3×C6 — C2×C3⋊S3 — (C3×C12).D4
 Upper central C1 — C2 — C4

Generators and relations for (C3×C12).D4
G = < a,b,c,d | a3=b12=1, c4=d2=b6, ab=ba, cac-1=b4, dad-1=a-1, cbc-1=a-1b3, dbd-1=b7, dcd-1=b9c3 >

Subgroups: 328 in 67 conjugacy classes, 15 normal (13 characteristic)
C1, C2, C2, C3 [×2], C4, C4 [×3], C22, S3 [×3], C6 [×2], C8 [×2], C2×C4 [×3], Q8 [×2], C32, Dic3 [×4], C12 [×4], D6 [×2], M4(2) [×2], C2×Q8, C3⋊S3, C3×C6, C3⋊C8, C24, Dic6 [×3], C4×S3 [×4], C3×Q8, C4.10D4, C3×Dic3 [×2], C3⋊Dic3, C3×C12, C2×C3⋊S3, C8⋊S3, S3×Q8, C3×C3⋊C8, C322C8, C6.D6 [×2], C322Q8, C3×Dic6, C4×C3⋊S3, C12.31D6, C32⋊M4(2), Dic3.D6, (C3×C12).D4
Quotients: C1, C2 [×3], C4 [×2], C22, C2×C4, D4 [×2], C22⋊C4, C4.10D4, S3≀C2, S32⋊C4, (C3×C12).D4

Character table of (C3×C12).D4

 class 1 2A 2B 3A 3B 4A 4B 4C 4D 6A 6B 8A 8B 8C 8D 12A 12B 12C 12D 12E 24A 24B 24C 24D size 1 1 18 4 4 2 12 12 18 4 4 12 12 36 36 4 4 8 24 24 12 12 12 12 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 -i i -i i -1 -1 -1 -1 1 i -i -i i linear of order 4 ρ6 1 1 1 1 1 -1 -1 1 -1 1 1 i -i -i i -1 -1 -1 1 -1 -i i i -i linear of order 4 ρ7 1 1 1 1 1 -1 1 -1 -1 1 1 i -i i -i -1 -1 -1 -1 1 -i i i -i linear of order 4 ρ8 1 1 1 1 1 -1 -1 1 -1 1 1 -i i i -i -1 -1 -1 1 -1 i -i -i i linear of order 4 ρ9 2 2 -2 2 2 2 0 0 -2 2 2 0 0 0 0 2 2 2 0 0 0 0 0 0 orthogonal lifted from D4 ρ10 2 2 -2 2 2 -2 0 0 2 2 2 0 0 0 0 -2 -2 -2 0 0 0 0 0 0 orthogonal lifted from D4 ρ11 4 4 0 -2 1 4 0 0 0 -2 1 2 2 0 0 1 1 -2 0 0 -1 -1 -1 -1 orthogonal lifted from S3≀C2 ρ12 4 4 0 1 -2 4 -2 -2 0 1 -2 0 0 0 0 -2 -2 1 1 1 0 0 0 0 orthogonal lifted from S3≀C2 ρ13 4 4 0 1 -2 -4 2 -2 0 1 -2 0 0 0 0 2 2 -1 1 -1 0 0 0 0 orthogonal lifted from S32⋊C4 ρ14 4 4 0 -2 1 4 0 0 0 -2 1 -2 -2 0 0 1 1 -2 0 0 1 1 1 1 orthogonal lifted from S3≀C2 ρ15 4 4 0 1 -2 -4 -2 2 0 1 -2 0 0 0 0 2 2 -1 -1 1 0 0 0 0 orthogonal lifted from S32⋊C4 ρ16 4 4 0 1 -2 4 2 2 0 1 -2 0 0 0 0 -2 -2 1 -1 -1 0 0 0 0 orthogonal lifted from S3≀C2 ρ17 4 -4 0 4 4 0 0 0 0 -4 -4 0 0 0 0 0 0 0 0 0 0 0 0 0 symplectic lifted from C4.10D4, Schur index 2 ρ18 4 4 0 -2 1 -4 0 0 0 -2 1 2i -2i 0 0 -1 -1 2 0 0 i -i -i i complex lifted from S32⋊C4 ρ19 4 4 0 -2 1 -4 0 0 0 -2 1 -2i 2i 0 0 -1 -1 2 0 0 -i i i -i complex lifted from S32⋊C4 ρ20 4 -4 0 -2 1 0 0 0 0 2 -1 0 0 0 0 3i -3i 0 0 0 2ζ8ζ3+ζ8 2ζ87ζ3+ζ87 2ζ83ζ3+ζ83 2ζ85ζ3+ζ85 complex faithful ρ21 4 -4 0 -2 1 0 0 0 0 2 -1 0 0 0 0 -3i 3i 0 0 0 2ζ87ζ3+ζ87 2ζ8ζ3+ζ8 2ζ85ζ3+ζ85 2ζ83ζ3+ζ83 complex faithful ρ22 4 -4 0 -2 1 0 0 0 0 2 -1 0 0 0 0 3i -3i 0 0 0 2ζ85ζ3+ζ85 2ζ83ζ3+ζ83 2ζ87ζ3+ζ87 2ζ8ζ3+ζ8 complex faithful ρ23 4 -4 0 -2 1 0 0 0 0 2 -1 0 0 0 0 -3i 3i 0 0 0 2ζ83ζ3+ζ83 2ζ85ζ3+ζ85 2ζ8ζ3+ζ8 2ζ87ζ3+ζ87 complex faithful ρ24 8 -8 0 2 -4 0 0 0 0 -2 4 0 0 0 0 0 0 0 0 0 0 0 0 0 symplectic faithful, Schur index 2

Smallest permutation representation of (C3×C12).D4
On 48 points
Generators in S48
(1 5 9)(2 6 10)(3 7 11)(4 8 12)(13 17 21)(14 18 22)(15 19 23)(16 20 24)(25 33 29)(26 34 30)(27 35 31)(28 36 32)(37 45 41)(38 46 42)(39 47 43)(40 48 44)
(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 46 10 37 7 40 4 43)(2 45 3 44 8 39 9 38)(5 42 6 41 11 48 12 47)(13 34 24 35 19 28 18 29)(14 33 17 30 20 27 23 36)(15 32 22 25 21 26 16 31)
(1 33 7 27)(2 28 8 34)(3 35 9 29)(4 30 10 36)(5 25 11 31)(6 32 12 26)(13 42 19 48)(14 37 20 43)(15 44 21 38)(16 39 22 45)(17 46 23 40)(18 41 24 47)

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

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

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

Matrix representation of (C3×C12).D4 in GL4(𝔽5) generated by

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

(C3×C12).D4 in GAP, Magma, Sage, TeX

(C_3\times C_{12}).D_4
% in TeX

G:=Group("(C3xC12).D4");
// GroupNames label

G:=SmallGroup(288,376);
// by ID

G=gap.SmallGroup(288,376);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,3,112,85,64,219,675,80,2693,2028,691,797,2372]);
// Polycyclic

G:=Group<a,b,c,d|a^3=b^12=1,c^4=d^2=b^6,a*b=b*a,c*a*c^-1=b^4,d*a*d^-1=a^-1,c*b*c^-1=a^-1*b^3,d*b*d^-1=b^7,d*c*d^-1=b^9*c^3>;
// generators/relations

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