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## G = C3×C12.C8order 288 = 25·32

### Direct product of C3 and C12.C8

Series: Derived Chief Lower central Upper central

 Derived series C1 — C6 — C3×C12.C8
 Chief series C1 — C3 — C6 — C12 — C24 — C3×C24 — C3×C3⋊C16 — C3×C12.C8
 Lower central C3 — C6 — C3×C12.C8
 Upper central C1 — C24 — C2×C24

Generators and relations for C3×C12.C8
G = < a,b,c | a3=b24=1, c4=b18, ab=ba, ac=ca, cbc-1=b5 >

Subgroups: 90 in 63 conjugacy classes, 42 normal (38 characteristic)
C1, C2, C2, C3 [×2], C3, C4 [×2], C22, C6 [×2], C6 [×5], C8 [×2], C2×C4, C32, C12 [×4], C12 [×2], C2×C6 [×2], C2×C6, C16 [×2], C2×C8, C3×C6, C3×C6, C24 [×4], C24 [×2], C2×C12 [×2], C2×C12, M5(2), C3×C12 [×2], C62, C3⋊C16 [×2], C48 [×2], C2×C24 [×2], C2×C24, C3×C24 [×2], C6×C12, C12.C8, C3×M5(2), C3×C3⋊C16 [×2], C6×C24, C3×C12.C8
Quotients: C1, C2 [×3], C3, C4 [×2], C22, S3, C6 [×3], C8 [×2], C2×C4, Dic3 [×2], C12 [×2], D6, C2×C6, C2×C8, C3×S3, C3⋊C8 [×2], C24 [×2], C2×Dic3, C2×C12, M5(2), C3×Dic3 [×2], S3×C6, C2×C3⋊C8, C2×C24, C3×C3⋊C8 [×2], C6×Dic3, C12.C8, C3×M5(2), C6×C3⋊C8, C3×C12.C8

Smallest permutation representation of C3×C12.C8
On 48 points
Generators in S48
(1 17 9)(2 18 10)(3 19 11)(4 20 12)(5 21 13)(6 22 14)(7 23 15)(8 24 16)(25 33 41)(26 34 42)(27 35 43)(28 36 44)(29 37 45)(30 38 46)(31 39 47)(32 40 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 25 10 46 19 43 4 40 13 37 22 34 7 31 16 28)(2 30 11 27 20 48 5 45 14 42 23 39 8 36 17 33)(3 35 12 32 21 29 6 26 15 47 24 44 9 41 18 38)

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

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

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

108 conjugacy classes

 class 1 2A 2B 3A 3B 3C 3D 3E 4A 4B 4C 6A 6B 6C ··· 6M 8A 8B 8C 8D 8E 8F 12A 12B 12C 12D 12E ··· 12R 16A ··· 16H 24A ··· 24H 24I ··· 24AJ 48A ··· 48P order 1 2 2 3 3 3 3 3 4 4 4 6 6 6 ··· 6 8 8 8 8 8 8 12 12 12 12 12 ··· 12 16 ··· 16 24 ··· 24 24 ··· 24 48 ··· 48 size 1 1 2 1 1 2 2 2 1 1 2 1 1 2 ··· 2 1 1 1 1 2 2 1 1 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 2 2 2 2 2 2 type + + + + - + - image C1 C2 C2 C3 C4 C4 C6 C6 C8 C8 C12 C12 C24 C24 S3 Dic3 D6 Dic3 C3×S3 C3⋊C8 C3⋊C8 M5(2) C3×Dic3 S3×C6 C3×Dic3 C3×C3⋊C8 C3×C3⋊C8 C12.C8 C3×M5(2) C3×C12.C8 kernel C3×C12.C8 C3×C3⋊C16 C6×C24 C12.C8 C3×C24 C6×C12 C3⋊C16 C2×C24 C3×C12 C62 C24 C2×C12 C12 C2×C6 C2×C24 C24 C24 C2×C12 C2×C8 C12 C2×C6 C32 C8 C8 C2×C4 C4 C22 C3 C3 C1 # reps 1 2 1 2 2 2 4 2 4 4 4 4 8 8 1 1 1 1 2 2 2 4 2 2 2 4 4 8 8 16

Matrix representation of C3×C12.C8 in GL2(𝔽97) generated by

 61 0 0 61
,
 43 0 0 93
,
 0 1 50 0
G:=sub<GL(2,GF(97))| [61,0,0,61],[43,0,0,93],[0,50,1,0] >;

C3×C12.C8 in GAP, Magma, Sage, TeX

C_3\times C_{12}.C_8
% in TeX

G:=Group("C3xC12.C8");
// GroupNames label

G:=SmallGroup(288,246);
// by ID

G=gap.SmallGroup(288,246);
# by ID

G:=PCGroup([7,-2,-2,-3,-2,-2,-2,-3,84,701,80,102,9414]);
// Polycyclic

G:=Group<a,b,c|a^3=b^24=1,c^4=b^18,a*b=b*a,a*c=c*a,c*b*c^-1=b^5>;
// generators/relations

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