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## G = C3×C32⋊2Q8order 216 = 23·33

### Direct product of C3 and C32⋊2Q8

Series: Derived Chief Lower central Upper central

 Derived series C1 — C3×C6 — C3×C32⋊2Q8
 Chief series C1 — C3 — C32 — C3×C6 — C32×C6 — C32×Dic3 — C3×C32⋊2Q8
 Lower central C32 — C3×C6 — C3×C32⋊2Q8
 Upper central C1 — C6

Generators and relations for C3×C322Q8
G = < a,b,c,d,e | a3=b3=c3=d4=1, e2=d2, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, dbd-1=b-1, be=eb, cd=dc, ece-1=c-1, ede-1=d-1 >

Subgroups: 180 in 70 conjugacy classes, 28 normal (12 characteristic)
C1, C2, C3, C3 [×2], C3 [×4], C4 [×3], C6, C6 [×2], C6 [×4], Q8, C32, C32 [×2], C32 [×4], Dic3 [×2], Dic3 [×3], C12 [×7], C3×C6, C3×C6 [×2], C3×C6 [×4], Dic6 [×2], C3×Q8, C33, C3×Dic3 [×4], C3×Dic3 [×5], C3⋊Dic3, C3×C12 [×2], C32×C6, C322Q8, C3×Dic6 [×2], C32×Dic3 [×2], C3×C3⋊Dic3, C3×C322Q8
Quotients: C1, C2 [×3], C3, C22, S3 [×2], C6 [×3], Q8, D6 [×2], C2×C6, C3×S3 [×2], Dic6 [×2], C3×Q8, S32, S3×C6 [×2], C322Q8, C3×Dic6 [×2], C3×S32, C3×C322Q8

Permutation representations of C3×C322Q8
On 24 points - transitive group 24T542
Generators in S24
(1 19 14)(2 20 15)(3 17 16)(4 18 13)(5 10 21)(6 11 22)(7 12 23)(8 9 24)
(1 14 19)(2 20 15)(3 16 17)(4 18 13)(5 10 21)(6 22 11)(7 12 23)(8 24 9)
(1 14 19)(2 15 20)(3 16 17)(4 13 18)(5 10 21)(6 11 22)(7 12 23)(8 9 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 24 3 22)(2 23 4 21)(5 20 7 18)(6 19 8 17)(9 16 11 14)(10 15 12 13)

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

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

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

G:=TransitiveGroup(24,542);

C3×C322Q8 is a maximal subgroup of
C336SD16  C33⋊Q16  C335(C2×Q8)  C336(C2×Q8)  D6.3S32  D6.6S32  Dic3.S32  C3×S3×Dic6

45 conjugacy classes

 class 1 2 3A 3B 3C ··· 3H 3I 3J 3K 4A 4B 4C 6A 6B 6C ··· 6H 6I 6J 6K 12A ··· 12P 12Q 12R order 1 2 3 3 3 ··· 3 3 3 3 4 4 4 6 6 6 ··· 6 6 6 6 12 ··· 12 12 12 size 1 1 1 1 2 ··· 2 4 4 4 6 6 18 1 1 2 ··· 2 4 4 4 6 ··· 6 18 18

45 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 2 2 2 2 4 4 4 4 type + + + + - + - + - image C1 C2 C2 C3 C6 C6 S3 Q8 D6 C3×S3 Dic6 C3×Q8 S3×C6 C3×Dic6 S32 C32⋊2Q8 C3×S32 C3×C32⋊2Q8 kernel C3×C32⋊2Q8 C32×Dic3 C3×C3⋊Dic3 C32⋊2Q8 C3×Dic3 C3⋊Dic3 C3×Dic3 C33 C3×C6 Dic3 C32 C32 C6 C3 C6 C3 C2 C1 # reps 1 2 1 2 4 2 2 1 2 4 4 2 4 8 1 1 2 2

Matrix representation of C3×C322Q8 in GL4(𝔽7) generated by

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

C3×C322Q8 in GAP, Magma, Sage, TeX

C_3\times C_3^2\rtimes_2Q_8
% in TeX

G:=Group("C3xC3^2:2Q8");
// GroupNames label

G:=SmallGroup(216,123);
// by ID

G=gap.SmallGroup(216,123);
# by ID

G:=PCGroup([6,-2,-2,-3,-2,-3,-3,72,169,79,730,5189]);
// Polycyclic

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

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