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## G = C3×C8⋊D6order 288 = 25·32

### Direct product of C3 and C8⋊D6

Series: Derived Chief Lower central Upper central

 Derived series C1 — C12 — C3×C8⋊D6
 Chief series C1 — C3 — C6 — C12 — C3×C12 — C3×D12 — C6×D12 — C3×C8⋊D6
 Lower central C3 — C6 — C12 — C3×C8⋊D6
 Upper central C1 — C6 — C2×C12 — C3×M4(2)

Generators and relations for C3×C8⋊D6
G = < a,b,c,d | a3=b8=c6=d2=1, ab=ba, ac=ca, ad=da, cbc-1=b5, dbd=b-1, dcd=c-1 >

Subgroups: 442 in 146 conjugacy classes, 58 normal (38 characteristic)
C1, C2, C2 [×4], C3 [×2], C3, C4 [×2], C4, C22, C22 [×5], S3 [×3], C6 [×2], C6 [×7], C8 [×2], C2×C4, C2×C4, D4 [×5], Q8, C23, C32, Dic3, C12 [×4], C12 [×3], D6 [×5], C2×C6 [×2], C2×C6 [×6], M4(2), D8 [×2], SD16 [×2], C2×D4, C4○D4, C3×S3 [×3], C3×C6, C3×C6, C24 [×4], C24 [×2], Dic6, C4×S3, D12, D12 [×2], D12, C3⋊D4, C2×C12 [×2], C2×C12 [×2], C3×D4 [×5], C3×Q8, C22×S3, C22×C6, C8⋊C22, C3×Dic3, C3×C12 [×2], S3×C6 [×5], C62, C24⋊C2 [×2], D24 [×2], C3×M4(2) [×2], C3×M4(2), C3×D8 [×2], C3×SD16 [×2], C2×D12, C4○D12, C6×D4, C3×C4○D4, C3×C24 [×2], C3×Dic6, S3×C12, C3×D12, C3×D12 [×2], C3×D12, C3×C3⋊D4, C6×C12, S3×C2×C6, C8⋊D6, C3×C8⋊C22, C3×C24⋊C2 [×2], C3×D24 [×2], C32×M4(2), C6×D12, C3×C4○D12, C3×C8⋊D6
Quotients: C1, C2 [×7], C3, C22 [×7], S3, C6 [×7], D4 [×2], C23, D6 [×3], C2×C6 [×7], C2×D4, C3×S3, D12 [×2], C3×D4 [×2], C22×S3, C22×C6, C8⋊C22, S3×C6 [×3], C2×D12, C6×D4, C3×D12 [×2], S3×C2×C6, C8⋊D6, C3×C8⋊C22, C6×D12, C3×C8⋊D6

Smallest permutation representation of C3×C8⋊D6
On 48 points
Generators in S48
(1 36 29)(2 37 30)(3 38 31)(4 39 32)(5 40 25)(6 33 26)(7 34 27)(8 35 28)(9 43 24)(10 44 17)(11 45 18)(12 46 19)(13 47 20)(14 48 21)(15 41 22)(16 42 23)
(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 29 36)(2 26 37 6 30 33)(3 31 38)(4 28 39 8 32 35)(5 25 40)(7 27 34)(9 47 24 13 43 20)(10 44 17)(11 41 18 15 45 22)(12 46 19)(14 48 21)(16 42 23)
(1 19)(2 18)(3 17)(4 24)(5 23)(6 22)(7 21)(8 20)(9 39)(10 38)(11 37)(12 36)(13 35)(14 34)(15 33)(16 40)(25 42)(26 41)(27 48)(28 47)(29 46)(30 45)(31 44)(32 43)

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

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

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

63 conjugacy classes

 class 1 2A 2B 2C 2D 2E 3A 3B 3C 3D 3E 4A 4B 4C 6A 6B 6C ··· 6G 6H 6I 6J 6K ··· 6P 8A 8B 12A ··· 12J 12K 12L 12M 12N 12O 24A ··· 24P order 1 2 2 2 2 2 3 3 3 3 3 4 4 4 6 6 6 ··· 6 6 6 6 6 ··· 6 8 8 12 ··· 12 12 12 12 12 12 24 ··· 24 size 1 1 2 12 12 12 1 1 2 2 2 2 2 12 1 1 2 ··· 2 4 4 4 12 ··· 12 4 4 2 ··· 2 4 4 4 12 12 4 ··· 4

63 irreducible representations

 dim 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 4 4 4 4 type + + + + + + + + + + + + + + + image C1 C2 C2 C2 C2 C2 C3 C6 C6 C6 C6 C6 S3 D4 D4 D6 D6 C3×S3 D12 C3×D4 D12 C3×D4 S3×C6 S3×C6 C3×D12 C3×D12 C8⋊C22 C8⋊D6 C3×C8⋊C22 C3×C8⋊D6 kernel C3×C8⋊D6 C3×C24⋊C2 C3×D24 C32×M4(2) C6×D12 C3×C4○D12 C8⋊D6 C24⋊C2 D24 C3×M4(2) C2×D12 C4○D12 C3×M4(2) C3×C12 C62 C24 C2×C12 M4(2) C12 C12 C2×C6 C2×C6 C8 C2×C4 C4 C22 C32 C3 C3 C1 # reps 1 2 2 1 1 1 2 4 4 2 2 2 1 1 1 2 1 2 2 2 2 2 4 2 4 4 1 2 2 4

Matrix representation of C3×C8⋊D6 in GL4(𝔽73) generated by

 8 0 0 0 0 8 0 0 0 0 8 0 0 0 0 8
,
 0 1 0 0 27 0 0 0 0 0 0 27 0 0 72 0
,
 64 0 0 0 0 9 0 0 0 0 8 0 0 0 0 65
,
 0 0 8 0 0 0 0 65 64 0 0 0 0 9 0 0
G:=sub<GL(4,GF(73))| [8,0,0,0,0,8,0,0,0,0,8,0,0,0,0,8],[0,27,0,0,1,0,0,0,0,0,0,72,0,0,27,0],[64,0,0,0,0,9,0,0,0,0,8,0,0,0,0,65],[0,0,64,0,0,0,0,9,8,0,0,0,0,65,0,0] >;

C3×C8⋊D6 in GAP, Magma, Sage, TeX

C_3\times C_8\rtimes D_6
% in TeX

G:=Group("C3xC8:D6");
// GroupNames label

G:=SmallGroup(288,679);
// by ID

G=gap.SmallGroup(288,679);
# by ID

G:=PCGroup([7,-2,-2,-2,-3,-2,-2,-3,590,555,142,2524,102,9414]);
// Polycyclic

G:=Group<a,b,c,d|a^3=b^8=c^6=d^2=1,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c^-1=b^5,d*b*d=b^-1,d*c*d=c^-1>;
// generators/relations

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