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G = A4×C3⋊D4order 288 = 25·32

Direct product of A4 and C3⋊D4

direct product, metabelian, soluble, monomial

Aliases: A4×C3⋊D4, C32(D4×A4), D62(C2×A4), (C3×A4)⋊9D4, Dic3⋊(C2×A4), C223(S3×A4), (C23×C6)⋊4C6, C247(C3×S3), (S3×C23)⋊2C6, (C22×A4)⋊3S3, (Dic3×A4)⋊4C2, (C2×A4).17D6, (C22×Dic3)⋊C6, C6.11(C22×A4), C23.20(S3×C6), (C6×A4).22C22, (C2×S3×A4)⋊4C2, (A4×C2×C6)⋊4C2, (C2×C6)⋊4(C2×A4), (C2×C6)⋊2(C3×D4), C2.11(C2×S3×A4), (C22×C3⋊D4)⋊C3, C222(C3×C3⋊D4), (C22×C6).5(C2×C6), SmallGroup(288,928)

Series: Derived Chief Lower central Upper central

C1C22×C6 — A4×C3⋊D4
C1C3C2×C6C22×C6C6×A4C2×S3×A4 — A4×C3⋊D4
C2×C6C22×C6 — A4×C3⋊D4
C1C2C22

Generators and relations for A4×C3⋊D4
 G = < a,b,c,d,e,f | a2=b2=c3=d3=e4=f2=1, cac-1=ab=ba, ad=da, ae=ea, af=fa, cbc-1=a, bd=db, be=eb, bf=fb, cd=dc, ce=ec, cf=fc, ede-1=fdf=d-1, fef=e-1 >

Subgroups: 698 in 142 conjugacy classes, 27 normal (all characteristic)
C1, C2, C2 [×6], C3, C3 [×2], C4 [×2], C22 [×2], C22 [×13], S3 [×2], C6, C6 [×10], C2×C4 [×2], D4 [×6], C23, C23 [×8], C32, Dic3, Dic3, C12, A4, A4, D6, D6 [×5], C2×C6 [×2], C2×C6 [×10], C22×C4, C2×D4 [×4], C24, C24, C3×S3, C3×C6 [×2], C2×Dic3 [×2], C3⋊D4, C3⋊D4 [×5], C3×D4, C2×A4, C2×A4 [×5], C22×S3 [×4], C22×C6, C22×C6 [×4], C22×D4, C3×Dic3, C3×A4, S3×C6, C62, C4×A4, C22×Dic3, C2×C3⋊D4 [×4], C22×A4, C22×A4 [×2], S3×C23, C23×C6, C3×C3⋊D4, S3×A4, C6×A4, C6×A4, D4×A4, C22×C3⋊D4, Dic3×A4, C2×S3×A4, A4×C2×C6, A4×C3⋊D4
Quotients: C1, C2 [×3], C3, C22, S3, C6 [×3], D4, A4, D6, C2×C6, C3×S3, C3⋊D4, C3×D4, C2×A4 [×3], S3×C6, C22×A4, C3×C3⋊D4, S3×A4, D4×A4, C2×S3×A4, A4×C3⋊D4

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

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

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

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

36 conjugacy classes

class 1 2A2B2C2D2E2F2G3A3B3C3D3E4A4B6A6B6C6D6E6F6G6H6I6J···6Q6R6S12A12B
order1222222233333446666666666···6661212
size112336618244886182224466668···824242424

36 irreducible representations

dim111111112222222233336666
type++++++++++++++
imageC1C2C2C2C3C6C6C6S3D4D6C3×S3C3⋊D4C3×D4S3×C6C3×C3⋊D4A4C2×A4C2×A4C2×A4S3×A4D4×A4C2×S3×A4A4×C3⋊D4
kernelA4×C3⋊D4Dic3×A4C2×S3×A4A4×C2×C6C22×C3⋊D4C22×Dic3S3×C23C23×C6C22×A4C3×A4C2×A4C24A4C2×C6C23C22C3⋊D4Dic3D6C2×C6C22C3C2C1
# reps111122221112222411111112

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

10000
01000
000121
000120
001120
,
10000
01000
001200
001201
001210
,
10000
01000
00010
00001
00100
,
90000
03000
00100
00010
00001
,
012000
10000
00100
00010
00001
,
01000
10000
00100
00010
00001

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

A4×C3⋊D4 in GAP, Magma, Sage, TeX

A_4\times C_3\rtimes D_4
% in TeX

G:=Group("A4xC3:D4");
// GroupNames label

G:=SmallGroup(288,928);
// by ID

G=gap.SmallGroup(288,928);
# by ID

G:=PCGroup([7,-2,-2,-3,-2,-2,2,-3,197,648,271,9414]);
// Polycyclic

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