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## G = C3×A4⋊D4order 288 = 25·32

### Direct product of C3 and A4⋊D4

Series: Derived Chief Lower central Upper central

 Derived series C1 — C22 — C2×A4 — C3×A4⋊D4
 Chief series C1 — C22 — A4 — C2×A4 — C6×A4 — C6×S4 — C3×A4⋊D4
 Lower central A4 — C2×A4 — C3×A4⋊D4
 Upper central C1 — C6 — C2×C6

Generators and relations for C3×A4⋊D4
G = < a,b,c,d,e,f | a3=b2=c2=d3=e4=f2=1, ab=ba, ac=ca, ad=da, ae=ea, af=fa, dbd-1=ebe-1=fbf=bc=cb, dcd-1=b, ce=ec, cf=fc, ede-1=fdf=d-1, fef=e-1 >

Subgroups: 518 in 134 conjugacy classes, 24 normal (all characteristic)
C1, C2, C2 [×5], C3, C3 [×2], C4 [×3], C22 [×2], C22 [×11], S3, C6, C6 [×10], C2×C4 [×3], D4 [×6], C23, C23 [×5], C32, Dic3, C12 [×3], A4, A4, D6, C2×C6 [×2], C2×C6 [×13], C22⋊C4 [×3], C2×D4 [×3], C24, C3×S3, C3×C6 [×2], C3⋊D4, C2×C12 [×3], C3×D4 [×6], S4, C2×A4, C2×A4 [×4], C22×C6, C22×C6 [×5], C22≀C2, C3×Dic3, C3×A4, S3×C6, C62, C3×C22⋊C4 [×3], A4⋊C4, C6×D4 [×3], C2×S4, C22×A4, C22×A4, C23×C6, C3×C3⋊D4, C3×S4, C6×A4, C6×A4, C3×C22≀C2, A4⋊D4, C3×A4⋊C4, C6×S4, A4×C2×C6, C3×A4⋊D4
Quotients: C1, C2 [×3], C3, C22, S3, C6 [×3], D4, D6, C2×C6, C3×S3, C3⋊D4, C3×D4, S4, S3×C6, C2×S4, C3×C3⋊D4, C3×S4, A4⋊D4, C6×S4, C3×A4⋊D4

Smallest permutation representation of C3×A4⋊D4
On 36 points
Generators in S36
(1 26 14)(2 27 15)(3 28 16)(4 25 13)(5 29 17)(6 30 18)(7 31 19)(8 32 20)(9 33 21)(10 34 22)(11 35 23)(12 36 24)
(1 4)(2 3)(5 11)(6 8)(7 9)(10 12)(13 14)(15 16)(17 23)(18 20)(19 21)(22 24)(25 26)(27 28)(29 35)(30 32)(31 33)(34 36)
(1 3)(2 4)(5 9)(6 10)(7 11)(8 12)(13 15)(14 16)(17 21)(18 22)(19 23)(20 24)(25 27)(26 28)(29 33)(30 34)(31 35)(32 36)
(1 5 10)(2 11 6)(3 7 12)(4 9 8)(13 21 20)(14 17 22)(15 23 18)(16 19 24)(25 33 32)(26 29 34)(27 35 30)(28 31 36)
(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)
(2 4)(5 10)(6 9)(7 12)(8 11)(13 15)(17 22)(18 21)(19 24)(20 23)(25 27)(29 34)(30 33)(31 36)(32 35)

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

G:=Group( (1,26,14)(2,27,15)(3,28,16)(4,25,13)(5,29,17)(6,30,18)(7,31,19)(8,32,20)(9,33,21)(10,34,22)(11,35,23)(12,36,24), (1,4)(2,3)(5,11)(6,8)(7,9)(10,12)(13,14)(15,16)(17,23)(18,20)(19,21)(22,24)(25,26)(27,28)(29,35)(30,32)(31,33)(34,36), (1,3)(2,4)(5,9)(6,10)(7,11)(8,12)(13,15)(14,16)(17,21)(18,22)(19,23)(20,24)(25,27)(26,28)(29,33)(30,34)(31,35)(32,36), (1,5,10)(2,11,6)(3,7,12)(4,9,8)(13,21,20)(14,17,22)(15,23,18)(16,19,24)(25,33,32)(26,29,34)(27,35,30)(28,31,36), (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), (2,4)(5,10)(6,9)(7,12)(8,11)(13,15)(17,22)(18,21)(19,24)(20,23)(25,27)(29,34)(30,33)(31,36)(32,35) );

G=PermutationGroup([(1,26,14),(2,27,15),(3,28,16),(4,25,13),(5,29,17),(6,30,18),(7,31,19),(8,32,20),(9,33,21),(10,34,22),(11,35,23),(12,36,24)], [(1,4),(2,3),(5,11),(6,8),(7,9),(10,12),(13,14),(15,16),(17,23),(18,20),(19,21),(22,24),(25,26),(27,28),(29,35),(30,32),(31,33),(34,36)], [(1,3),(2,4),(5,9),(6,10),(7,11),(8,12),(13,15),(14,16),(17,21),(18,22),(19,23),(20,24),(25,27),(26,28),(29,33),(30,34),(31,35),(32,36)], [(1,5,10),(2,11,6),(3,7,12),(4,9,8),(13,21,20),(14,17,22),(15,23,18),(16,19,24),(25,33,32),(26,29,34),(27,35,30),(28,31,36)], [(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)], [(2,4),(5,10),(6,9),(7,12),(8,11),(13,15),(17,22),(18,21),(19,24),(20,23),(25,27),(29,34),(30,33),(31,36),(32,35)])

42 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 3A 3B 3C 3D 3E 4A 4B 4C 6A 6B 6C 6D 6E 6F 6G 6H 6I 6J 6K ··· 6S 6T 6U 12A ··· 12F order 1 2 2 2 2 2 2 3 3 3 3 3 4 4 4 6 6 6 6 6 6 6 6 6 6 6 ··· 6 6 6 12 ··· 12 size 1 1 2 3 3 6 12 1 1 8 8 8 12 12 12 1 1 2 2 3 3 3 3 6 6 8 ··· 8 12 12 12 ··· 12

42 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 3 3 3 3 6 6 type + + + + + + + + + + image C1 C2 C2 C2 C3 C6 C6 C6 S3 D4 D6 C3×S3 C3×D4 C3⋊D4 S3×C6 C3×C3⋊D4 S4 C2×S4 C3×S4 C6×S4 A4⋊D4 C3×A4⋊D4 kernel C3×A4⋊D4 C3×A4⋊C4 C6×S4 A4×C2×C6 A4⋊D4 A4⋊C4 C2×S4 C22×A4 C23×C6 C3×A4 C22×C6 C24 A4 C2×C6 C23 C22 C2×C6 C6 C22 C2 C3 C1 # reps 1 1 1 1 2 2 2 2 1 1 1 2 2 2 2 4 2 2 4 4 1 2

Matrix representation of C3×A4⋊D4 in GL5(𝔽13)

 1 0 0 0 0 0 1 0 0 0 0 0 9 0 0 0 0 0 9 0 0 0 0 0 9
,
 1 0 0 0 0 0 1 0 0 0 0 0 12 0 0 0 0 0 12 0 0 0 12 0 1
,
 1 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 1 12 0 0 0 1 0 12
,
 3 7 0 0 0 0 9 0 0 0 0 0 1 0 11 0 0 0 0 12 0 0 0 1 12
,
 3 12 0 0 0 10 10 0 0 0 0 0 12 0 0 0 0 0 0 12 0 0 0 12 0
,
 3 7 0 0 0 10 10 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 1 0

G:=sub<GL(5,GF(13))| [1,0,0,0,0,0,1,0,0,0,0,0,9,0,0,0,0,0,9,0,0,0,0,0,9],[1,0,0,0,0,0,1,0,0,0,0,0,12,0,12,0,0,0,12,0,0,0,0,0,1],[1,0,0,0,0,0,1,0,0,0,0,0,1,1,1,0,0,0,12,0,0,0,0,0,12],[3,0,0,0,0,7,9,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,11,12,12],[3,10,0,0,0,12,10,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,12,0],[3,10,0,0,0,7,10,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,1,0] >;

C3×A4⋊D4 in GAP, Magma, Sage, TeX

C_3\times A_4\rtimes D_4
% in TeX

G:=Group("C3xA4:D4");
// GroupNames label

G:=SmallGroup(288,906);
// by ID

G=gap.SmallGroup(288,906);
# by ID

G:=PCGroup([7,-2,-2,-3,-2,-3,-2,2,197,1684,6053,285,3534,475]);
// Polycyclic

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

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