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## G = C6×C4○D12order 288 = 25·32

### Direct product of C6 and C4○D12

Series: Derived Chief Lower central Upper central

 Derived series C1 — C6 — C6×C4○D12
 Chief series C1 — C3 — C6 — C3×C6 — S3×C6 — S3×C2×C6 — S3×C2×C12 — C6×C4○D12
 Lower central C3 — C6 — C6×C4○D12
 Upper central C1 — C2×C12 — C22×C12

Generators and relations for C6×C4○D12
G = < a,b,c,d | a6=b4=d2=1, c6=b2, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd=b2c5 >

Subgroups: 698 in 355 conjugacy classes, 178 normal (34 characteristic)
C1, C2, C2, C2, C3, C3, C4, C4, C22, C22, C22, S3, C6, C6, C6, C2×C4, C2×C4, C2×C4, D4, Q8, C23, C23, C32, Dic3, C12, C12, D6, D6, C2×C6, C2×C6, C2×C6, C22×C4, C22×C4, C2×D4, C2×Q8, C4○D4, C3×S3, C3×C6, C3×C6, C3×C6, Dic6, C4×S3, D12, C2×Dic3, C3⋊D4, C2×C12, C2×C12, C2×C12, C3×D4, C3×Q8, C22×S3, C22×C6, C22×C6, C2×C4○D4, C3×Dic3, C3×C12, S3×C6, S3×C6, C62, C62, C62, C2×Dic6, S3×C2×C4, C2×D12, C4○D12, C2×C3⋊D4, C22×C12, C22×C12, C6×D4, C6×Q8, C3×C4○D4, C3×Dic6, S3×C12, C3×D12, C6×Dic3, C3×C3⋊D4, C6×C12, C6×C12, S3×C2×C6, C2×C62, C2×C4○D12, C6×C4○D4, C6×Dic6, S3×C2×C12, C6×D12, C3×C4○D12, C6×C3⋊D4, C2×C6×C12, C6×C4○D12
Quotients: C1, C2, C3, C22, S3, C6, C23, D6, C2×C6, C4○D4, C24, C3×S3, C22×S3, C22×C6, C2×C4○D4, S3×C6, C4○D12, C3×C4○D4, S3×C23, C23×C6, S3×C2×C6, C2×C4○D12, C6×C4○D4, C3×C4○D12, S3×C22×C6, C6×C4○D12

Smallest permutation representation of C6×C4○D12
On 48 points
Generators in S48
(1 22 5 14 9 18)(2 23 6 15 10 19)(3 24 7 16 11 20)(4 13 8 17 12 21)(25 41 33 37 29 45)(26 42 34 38 30 46)(27 43 35 39 31 47)(28 44 36 40 32 48)
(1 10 7 4)(2 11 8 5)(3 12 9 6)(13 22 19 16)(14 23 20 17)(15 24 21 18)(25 28 31 34)(26 29 32 35)(27 30 33 36)(37 40 43 46)(38 41 44 47)(39 42 45 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 48)(2 47)(3 46)(4 45)(5 44)(6 43)(7 42)(8 41)(9 40)(10 39)(11 38)(12 37)(13 25)(14 36)(15 35)(16 34)(17 33)(18 32)(19 31)(20 30)(21 29)(22 28)(23 27)(24 26)

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

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

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

108 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 2H 2I 3A 3B 3C 3D 3E 4A 4B 4C 4D 4E 4F 4G 4H 4I 4J 6A ··· 6F 6G ··· 6AE 6AF ··· 6AM 12A ··· 12H 12I ··· 12AJ 12AK ··· 12AR order 1 2 2 2 2 2 2 2 2 2 3 3 3 3 3 4 4 4 4 4 4 4 4 4 4 6 ··· 6 6 ··· 6 6 ··· 6 12 ··· 12 12 ··· 12 12 ··· 12 size 1 1 1 1 2 2 6 6 6 6 1 1 2 2 2 1 1 1 1 2 2 6 6 6 6 1 ··· 1 2 ··· 2 6 ··· 6 1 ··· 1 2 ··· 2 6 ··· 6

108 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 type + + + + + + + + + + image C1 C2 C2 C2 C2 C2 C2 C3 C6 C6 C6 C6 C6 C6 S3 D6 D6 C4○D4 C3×S3 S3×C6 S3×C6 C4○D12 C3×C4○D4 C3×C4○D12 kernel C6×C4○D12 C6×Dic6 S3×C2×C12 C6×D12 C3×C4○D12 C6×C3⋊D4 C2×C6×C12 C2×C4○D12 C2×Dic6 S3×C2×C4 C2×D12 C4○D12 C2×C3⋊D4 C22×C12 C22×C12 C2×C12 C22×C6 C3×C6 C22×C4 C2×C4 C23 C6 C6 C2 # reps 1 1 2 1 8 2 1 2 2 4 2 16 4 2 1 6 1 4 2 12 2 8 8 16

Matrix representation of C6×C4○D12 in GL3(𝔽13) generated by

 12 0 0 0 9 0 0 0 9
,
 1 0 0 0 5 0 0 0 5
,
 12 0 0 0 2 0 0 0 7
,
 12 0 0 0 0 7 0 2 0
G:=sub<GL(3,GF(13))| [12,0,0,0,9,0,0,0,9],[1,0,0,0,5,0,0,0,5],[12,0,0,0,2,0,0,0,7],[12,0,0,0,0,2,0,7,0] >;

C6×C4○D12 in GAP, Magma, Sage, TeX

C_6\times C_4\circ D_{12}
% in TeX

G:=Group("C6xC4oD12");
// GroupNames label

G:=SmallGroup(288,991);
// by ID

G=gap.SmallGroup(288,991);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-3,-2,-3,268,1571,9414]);
// Polycyclic

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

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