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G = C3×A4⋊D4order 288 = 25·32

Direct product of C3 and A4⋊D4

direct product, non-abelian, soluble, monomial

Aliases: C3×A4⋊D4, A4⋊C4⋊C6, (C2×C6)⋊3S4, (C2×S4)⋊2C6, (C6×S4)⋊5C2, (C3×A4)⋊7D4, A42(C3×D4), C2.11(C6×S4), C6.49(C2×S4), C243(C3×S3), (C23×C6)⋊1S3, C223(C3×S4), (C22×A4)⋊4C6, C23.5(S3×C6), (C22×C6).12D6, (C6×A4).12C22, (A4×C2×C6)⋊2C2, (C3×A4⋊C4)⋊4C2, C22⋊(C3×C3⋊D4), (C2×C6)⋊3(C3⋊D4), (C2×A4).5(C2×C6), SmallGroup(288,906)

Series: Derived Chief Lower central Upper central

C1C22C2×A4 — C3×A4⋊D4
C1C22A4C2×A4C6×A4C6×S4 — C3×A4⋊D4
A4C2×A4 — C3×A4⋊D4
C1C6C2×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 2A2B2C2D2E2F3A3B3C3D3E4A4B4C6A6B6C6D6E6F6G6H6I6J6K···6S6T6U12A···12F
order12222223333344466666666666···66612···12
size112336121188812121211223333668···8121212···12

42 irreducible representations

dim1111111122222222333366
type++++++++++
imageC1C2C2C2C3C6C6C6S3D4D6C3×S3C3×D4C3⋊D4S3×C6C3×C3⋊D4S4C2×S4C3×S4C6×S4A4⋊D4C3×A4⋊D4
kernelC3×A4⋊D4C3×A4⋊C4C6×S4A4×C2×C6A4⋊D4A4⋊C4C2×S4C22×A4C23×C6C3×A4C22×C6C24A4C2×C6C23C22C2×C6C6C22C2C3C1
# reps1111222211122224224412

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

10000
01000
00900
00090
00009
,
10000
01000
001200
000120
001201
,
10000
01000
00100
001120
001012
,
37000
09000
001011
000012
000112
,
312000
1010000
001200
000012
000120
,
37000
1010000
00100
00001
00010

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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