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## G = C3×C23.6D6order 288 = 25·32

### Direct product of C3 and C23.6D6

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C6 — C3×C23.6D6
 Chief series C1 — C3 — C6 — C2×C6 — C22×C6 — C2×C62 — C6×C3⋊D4 — C3×C23.6D6
 Lower central C3 — C6 — C2×C6 — C3×C23.6D6
 Upper central C1 — C6 — C22×C6 — C3×C22⋊C4

Generators and relations for C3×C23.6D6
G = < a,b,c,d,e,f | a3=b2=c2=d2=1, e6=b, f2=bcd, ab=ba, ac=ca, ad=da, ae=ea, af=fa, bc=cb, fbf-1=bd=db, be=eb, ece-1=fcf-1=cd=dc, de=ed, df=fd, fef-1=cde5 >

Subgroups: 354 in 121 conjugacy classes, 38 normal (all characteristic)
C1, C2, C2 [×4], C3 [×2], C3, C4 [×3], C22 [×3], C22 [×3], S3, C6 [×2], C6 [×12], C2×C4 [×3], D4 [×2], C23, C23, C32, Dic3 [×2], C12 [×6], D6 [×2], C2×C6 [×6], C2×C6 [×9], C22⋊C4, C22⋊C4, C2×D4, C3×S3, C3×C6, C3×C6 [×3], C2×Dic3, C2×Dic3, C3⋊D4 [×2], C2×C12 [×6], C3×D4 [×2], C22×S3, C22×C6 [×2], C22×C6 [×2], C23⋊C4, C3×Dic3 [×2], C3×C12, S3×C6 [×2], C62 [×3], C62, C6.D4, C3×C22⋊C4 [×2], C3×C22⋊C4 [×2], C2×C3⋊D4, C6×D4, C6×Dic3, C6×Dic3, C3×C3⋊D4 [×2], C6×C12, S3×C2×C6, C2×C62, C23.6D6, C3×C23⋊C4, C3×C6.D4, C32×C22⋊C4, C6×C3⋊D4, C3×C23.6D6
Quotients: C1, C2 [×3], C3, C4 [×2], C22, S3, C6 [×3], C2×C4, D4 [×2], C12 [×2], D6, C2×C6, C22⋊C4, C3×S3, C4×S3, D12, C3⋊D4, C2×C12, C3×D4 [×2], C23⋊C4, S3×C6, D6⋊C4, C3×C22⋊C4, S3×C12, C3×D12, C3×C3⋊D4, C23.6D6, C3×C23⋊C4, C3×D6⋊C4, C3×C23.6D6

Permutation representations of C3×C23.6D6
On 24 points - transitive group 24T587
Generators in S24
(1 3 5)(2 4 6)(7 9 11)(8 10 12)(13 17 21)(14 18 22)(15 19 23)(16 20 24)
(13 19)(14 20)(15 21)(16 22)(17 23)(18 24)
(2 11)(4 7)(6 9)(14 20)(16 22)(18 24)
(1 10)(2 11)(3 12)(4 7)(5 8)(6 9)(13 19)(14 20)(15 21)(16 22)(17 23)(18 24)
(1 2 3 4 5 6)(7 8 9 10 11 12)(13 14 15 16 17 18 19 20 21 22 23 24)
(1 22 10 16)(2 21)(3 14 12 20)(4 13)(5 18 8 24)(6 17)(7 19)(9 23)(11 15)

G:=sub<Sym(24)| (1,3,5)(2,4,6)(7,9,11)(8,10,12)(13,17,21)(14,18,22)(15,19,23)(16,20,24), (13,19)(14,20)(15,21)(16,22)(17,23)(18,24), (2,11)(4,7)(6,9)(14,20)(16,22)(18,24), (1,10)(2,11)(3,12)(4,7)(5,8)(6,9)(13,19)(14,20)(15,21)(16,22)(17,23)(18,24), (1,2,3,4,5,6)(7,8,9,10,11,12)(13,14,15,16,17,18,19,20,21,22,23,24), (1,22,10,16)(2,21)(3,14,12,20)(4,13)(5,18,8,24)(6,17)(7,19)(9,23)(11,15)>;

G:=Group( (1,3,5)(2,4,6)(7,9,11)(8,10,12)(13,17,21)(14,18,22)(15,19,23)(16,20,24), (13,19)(14,20)(15,21)(16,22)(17,23)(18,24), (2,11)(4,7)(6,9)(14,20)(16,22)(18,24), (1,10)(2,11)(3,12)(4,7)(5,8)(6,9)(13,19)(14,20)(15,21)(16,22)(17,23)(18,24), (1,2,3,4,5,6)(7,8,9,10,11,12)(13,14,15,16,17,18,19,20,21,22,23,24), (1,22,10,16)(2,21)(3,14,12,20)(4,13)(5,18,8,24)(6,17)(7,19)(9,23)(11,15) );

G=PermutationGroup([(1,3,5),(2,4,6),(7,9,11),(8,10,12),(13,17,21),(14,18,22),(15,19,23),(16,20,24)], [(13,19),(14,20),(15,21),(16,22),(17,23),(18,24)], [(2,11),(4,7),(6,9),(14,20),(16,22),(18,24)], [(1,10),(2,11),(3,12),(4,7),(5,8),(6,9),(13,19),(14,20),(15,21),(16,22),(17,23),(18,24)], [(1,2,3,4,5,6),(7,8,9,10,11,12),(13,14,15,16,17,18,19,20,21,22,23,24)], [(1,22,10,16),(2,21),(3,14,12,20),(4,13),(5,18,8,24),(6,17),(7,19),(9,23),(11,15)])

G:=TransitiveGroup(24,587);

On 24 points - transitive group 24T629
Generators in S24
(1 3 2)(4 6 5)(7 9 11)(8 10 12)(13 17 21)(14 18 22)(15 19 23)(16 20 24)
(13 19)(14 20)(15 21)(16 22)(17 23)(18 24)
(1 11)(2 9)(3 7)(4 10)(5 8)(6 12)(13 22)(14 17)(15 24)(16 19)(18 21)(20 23)
(1 5)(2 6)(3 4)(7 10)(8 11)(9 12)(13 19)(14 20)(15 21)(16 22)(17 23)(18 24)
(1 2 3)(4 5 6)(7 8 9 10 11 12)(13 14 15 16 17 18 19 20 21 22 23 24)
(1 20 8 23)(2 16 12 19)(3 24 10 15)(4 18 7 21)(5 14 11 17)(6 22 9 13)

G:=sub<Sym(24)| (1,3,2)(4,6,5)(7,9,11)(8,10,12)(13,17,21)(14,18,22)(15,19,23)(16,20,24), (13,19)(14,20)(15,21)(16,22)(17,23)(18,24), (1,11)(2,9)(3,7)(4,10)(5,8)(6,12)(13,22)(14,17)(15,24)(16,19)(18,21)(20,23), (1,5)(2,6)(3,4)(7,10)(8,11)(9,12)(13,19)(14,20)(15,21)(16,22)(17,23)(18,24), (1,2,3)(4,5,6)(7,8,9,10,11,12)(13,14,15,16,17,18,19,20,21,22,23,24), (1,20,8,23)(2,16,12,19)(3,24,10,15)(4,18,7,21)(5,14,11,17)(6,22,9,13)>;

G:=Group( (1,3,2)(4,6,5)(7,9,11)(8,10,12)(13,17,21)(14,18,22)(15,19,23)(16,20,24), (13,19)(14,20)(15,21)(16,22)(17,23)(18,24), (1,11)(2,9)(3,7)(4,10)(5,8)(6,12)(13,22)(14,17)(15,24)(16,19)(18,21)(20,23), (1,5)(2,6)(3,4)(7,10)(8,11)(9,12)(13,19)(14,20)(15,21)(16,22)(17,23)(18,24), (1,2,3)(4,5,6)(7,8,9,10,11,12)(13,14,15,16,17,18,19,20,21,22,23,24), (1,20,8,23)(2,16,12,19)(3,24,10,15)(4,18,7,21)(5,14,11,17)(6,22,9,13) );

G=PermutationGroup([(1,3,2),(4,6,5),(7,9,11),(8,10,12),(13,17,21),(14,18,22),(15,19,23),(16,20,24)], [(13,19),(14,20),(15,21),(16,22),(17,23),(18,24)], [(1,11),(2,9),(3,7),(4,10),(5,8),(6,12),(13,22),(14,17),(15,24),(16,19),(18,21),(20,23)], [(1,5),(2,6),(3,4),(7,10),(8,11),(9,12),(13,19),(14,20),(15,21),(16,22),(17,23),(18,24)], [(1,2,3),(4,5,6),(7,8,9,10,11,12),(13,14,15,16,17,18,19,20,21,22,23,24)], [(1,20,8,23),(2,16,12,19),(3,24,10,15),(4,18,7,21),(5,14,11,17),(6,22,9,13)])

G:=TransitiveGroup(24,629);

63 conjugacy classes

 class 1 2A 2B 2C 2D 2E 3A 3B 3C 3D 3E 4A 4B 4C 4D 4E 6A 6B 6C ··· 6Q 6R ··· 6W 6X 6Y 12A ··· 12P 12Q ··· 12V order 1 2 2 2 2 2 3 3 3 3 3 4 4 4 4 4 6 6 6 ··· 6 6 ··· 6 6 6 12 ··· 12 12 ··· 12 size 1 1 2 2 2 12 1 1 2 2 2 4 4 12 12 12 1 1 2 ··· 2 4 ··· 4 12 12 4 ··· 4 12 ··· 12

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 4 4 4 4 type + + + + + + + + + image C1 C2 C2 C2 C3 C4 C4 C6 C6 C6 C12 C12 S3 D4 D6 C3×S3 C4×S3 D12 C3⋊D4 C3×D4 S3×C6 S3×C12 C3×D12 C3×C3⋊D4 C23⋊C4 C23.6D6 C3×C23⋊C4 C3×C23.6D6 kernel C3×C23.6D6 C3×C6.D4 C32×C22⋊C4 C6×C3⋊D4 C23.6D6 C6×Dic3 S3×C2×C6 C6.D4 C3×C22⋊C4 C2×C3⋊D4 C2×Dic3 C22×S3 C3×C22⋊C4 C62 C22×C6 C22⋊C4 C2×C6 C2×C6 C2×C6 C2×C6 C23 C22 C22 C22 C32 C3 C3 C1 # reps 1 1 1 1 2 2 2 2 2 2 4 4 1 2 1 2 2 2 2 4 2 4 4 4 1 2 2 4

Matrix representation of C3×C23.6D6 in GL4(𝔽7) generated by

 2 0 0 0 0 2 0 0 0 0 2 0 0 0 0 2
,
 0 6 3 2 6 0 4 2 0 0 6 0 0 0 0 1
,
 1 5 2 6 1 5 5 3 0 0 1 0 5 2 1 0
,
 6 0 0 0 0 6 0 0 0 0 6 0 0 0 0 6
,
 3 0 4 1 3 6 5 2 1 1 1 5 0 0 0 4
,
 2 4 3 3 0 3 3 4 1 6 3 4 4 4 5 6
G:=sub<GL(4,GF(7))| [2,0,0,0,0,2,0,0,0,0,2,0,0,0,0,2],[0,6,0,0,6,0,0,0,3,4,6,0,2,2,0,1],[1,1,0,5,5,5,0,2,2,5,1,1,6,3,0,0],[6,0,0,0,0,6,0,0,0,0,6,0,0,0,0,6],[3,3,1,0,0,6,1,0,4,5,1,0,1,2,5,4],[2,0,1,4,4,3,6,4,3,3,3,5,3,4,4,6] >;

C3×C23.6D6 in GAP, Magma, Sage, TeX

C_3\times C_2^3._6D_6
% in TeX

G:=Group("C3xC2^3.6D6");
// GroupNames label

G:=SmallGroup(288,240);
// by ID

G=gap.SmallGroup(288,240);
# by ID

G:=PCGroup([7,-2,-2,-3,-2,-2,-2,-3,365,92,1683,1271,9414]);
// Polycyclic

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

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