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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, C4, C4, C22, S3, C6, C8, C2×C4, Q8, C32, Dic3, C12, D6, M4(2), C2×Q8, C3⋊S3, C3×C6, C3⋊C8, C24, Dic6, C4×S3, C3×Q8, C4.10D4, C3×Dic3, C3⋊Dic3, C3×C12, C2×C3⋊S3, C8⋊S3, S3×Q8, C3×C3⋊C8, C322C8, C6.D6, C322Q8, C3×Dic6, C4×C3⋊S3, C12.31D6, C32⋊M4(2), Dic3.D6, (C3×C12).D4
Quotients: C1, C2, C4, C22, C2×C4, D4, 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 40 10 43 7 46 4 37)(2 39 3 38 8 45 9 44)(5 48 6 47 11 42 12 41)(13 29 24 30 19 35 18 36)(14 28 17 25 20 34 23 31)(15 27 22 32 21 33 16 26)
(1 31 7 25)(2 26 8 32)(3 33 9 27)(4 28 10 34)(5 35 11 29)(6 30 12 36)(13 45 19 39)(14 40 20 46)(15 47 21 41)(16 42 22 48)(17 37 23 43)(18 44 24 38)

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

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

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

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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