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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 — C3×C4○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=1, c6=d2=b4, ab=ba, ac=ca, ad=da, cbc-1=b5, dbd-1=b-1, dcd-1=c5 >

Subgroups: 314 in 130 conjugacy classes, 58 normal (38 characteristic)
C1, C2, C2, C3, C3, C4, C4, C22, C22, S3, C6, C6, C8, C2×C4, C2×C4, D4, Q8, C32, Dic3, C12, C12, D6, C2×C6, C2×C6, M4(2), SD16, Q16, C2×Q8, C4○D4, C3×S3, C3×C6, C3×C6, C24, C24, Dic6, Dic6, Dic6, C4×S3, D12, C2×Dic3, C3⋊D4, C2×C12, C2×C12, C3×D4, C3×Q8, C8.C22, C3×Dic3, C3×C12, S3×C6, C62, C24⋊C2, Dic12, C3×M4(2), C3×M4(2), C3×SD16, C3×Q16, C2×Dic6, C4○D12, C6×Q8, C3×C4○D4, C3×C24, C3×Dic6, C3×Dic6, C3×Dic6, S3×C12, C3×D12, C6×Dic3, C3×C3⋊D4, C6×C12, C8.D6, C3×C8.C22, C3×C24⋊C2, C3×Dic12, C32×M4(2), C6×Dic6, C3×C4○D12, C3×C8.D6
Quotients: C1, C2, C3, C22, S3, C6, D4, C23, D6, C2×C6, C2×D4, C3×S3, D12, C3×D4, C22×S3, C22×C6, C8.C22, S3×C6, C2×D12, C6×D4, C3×D12, 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 9 5)(2 10 6)(3 11 7)(4 12 8)(13 21 17)(14 22 18)(15 23 19)(16 24 20)(25 29 33)(26 30 34)(27 31 35)(28 32 36)(37 41 45)(38 42 46)(39 43 47)(40 44 48)
(1 13 4 22 7 19 10 16)(2 20 5 17 8 14 11 23)(3 15 6 24 9 21 12 18)(25 37 34 40 31 43 28 46)(26 44 35 47 32 38 29 41)(27 39 36 42 33 45 30 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 33 7 27)(2 26 8 32)(3 31 9 25)(4 36 10 30)(5 29 11 35)(6 34 12 28)(13 42 19 48)(14 47 20 41)(15 40 21 46)(16 45 22 39)(17 38 23 44)(18 43 24 37)

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

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

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

63 conjugacy classes

 class 1 2A 2B 2C 3A 3B 3C 3D 3E 4A 4B 4C 4D 4E 6A 6B 6C ··· 6G 6H 6I 6J 6K 6L 8A 8B 12A ··· 12J 12K 12L 12M 12N ··· 12S 24A ··· 24P order 1 2 2 2 3 3 3 3 3 4 4 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 1 1 2 2 2 2 2 12 12 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×Dic12 C32×M4(2) C6×Dic6 C3×C4○D12 C8.D6 C24⋊C2 Dic12 C3×M4(2) C2×Dic6 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

 64 0 0 0 0 64 0 0 0 0 64 0 0 0 0 64
,
 0 1 0 0 46 0 0 0 0 28 0 46 26 69 72 0
,
 70 0 0 0 0 3 0 0 56 19 49 0 54 56 0 24
,
 10 29 51 0 44 10 0 22 46 33 63 29 33 27 44 63
G:=sub<GL(4,GF(73))| [64,0,0,0,0,64,0,0,0,0,64,0,0,0,0,64],[0,46,0,26,1,0,28,69,0,0,0,72,0,0,46,0],[70,0,56,54,0,3,19,56,0,0,49,0,0,0,0,24],[10,44,46,33,29,10,33,27,51,0,63,44,0,22,29,63] >;

C3×C8.D6 in GAP, Magma, Sage, TeX

C_3\times C_8.D_6
% in TeX

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

G:=SmallGroup(288,680);
// by ID

G=gap.SmallGroup(288,680);
# by ID

G:=PCGroup([7,-2,-2,-2,-3,-2,-2,-3,344,590,555,142,2524,102,9414]);
// Polycyclic

G:=Group<a,b,c,d|a^3=b^8=1,c^6=d^2=b^4,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c^-1=b^5,d*b*d^-1=b^-1,d*c*d^-1=c^5>;
// generators/relations

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