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G = C3×D4×A4order 288 = 25·32

Direct product of C3, D4 and A4

direct product, metabelian, soluble, monomial

Aliases: C3×D4×A4, C23.7C62, C4⋊(C6×A4), (C4×A4)⋊3C6, C123(C2×A4), (C12×A4)⋊7C2, C223(C6×A4), (C23×C6)⋊2C6, C243(C3×C6), (C22×D4)⋊C32, C22⋊(D4×C32), (C22×A4)⋊3C6, (C22×C12)⋊3C6, C6.21(C22×A4), (C6×A4).24C22, (D4×C2×C6)⋊C3, (A4×C2×C6)⋊1C2, C2.2(A4×C2×C6), (C2×C6)⋊3(C2×A4), (C2×C6)⋊6(C3×D4), (C22×C4)⋊(C3×C6), (C2×A4).7(C2×C6), (C22×C6).40(C2×C6), SmallGroup(288,980)

Series: Derived Chief Lower central Upper central

C1C23 — C3×D4×A4
C1C22C23C22×C6C6×A4A4×C2×C6 — C3×D4×A4
C22C23 — C3×D4×A4
C1C6C3×D4

Generators and relations for C3×D4×A4
 G = < a,b,c,d,e,f | a3=b4=c2=d2=e2=f3=1, ab=ba, ac=ca, ad=da, ae=ea, af=fa, cbc=b-1, bd=db, be=eb, bf=fb, cd=dc, ce=ec, cf=fc, fdf-1=de=ed, fef-1=d >

Subgroups: 516 in 164 conjugacy classes, 48 normal (20 characteristic)
C1, C2, C2 [×6], C3, C3 [×3], C4, C4, C22, C22 [×2], C22 [×12], C6, C6 [×15], C2×C4 [×2], D4, D4 [×5], C23, C23 [×8], C32, C12, C12 [×4], A4 [×3], C2×C6, C2×C6 [×2], C2×C6 [×18], C22×C4, C2×D4 [×4], C24 [×2], C3×C6 [×3], C2×C12 [×2], C3×D4, C3×D4 [×8], C2×A4 [×3], C2×A4 [×6], C22×C6, C22×C6 [×8], C22×D4, C3×C12, C3×A4, C62 [×2], C4×A4 [×3], C22×C12, C6×D4 [×4], C22×A4 [×6], C23×C6 [×2], D4×C32, C6×A4, C6×A4 [×2], D4×A4 [×3], D4×C2×C6, C12×A4, A4×C2×C6 [×2], C3×D4×A4
Quotients: C1, C2 [×3], C3 [×4], C22, C6 [×12], D4, C32, A4, C2×C6 [×4], C3×C6 [×3], C3×D4 [×4], C2×A4 [×3], C3×A4, C62, C22×A4, D4×C32, C6×A4 [×3], D4×A4, A4×C2×C6, C3×D4×A4

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

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

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

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

60 conjugacy classes

class 1 2A2B2C2D2E2F2G3A3B3C···3H4A4B6A6B6C6D6E6F6G6H6I6J6K···6P6Q6R6S6T6U···6AF12A12B12C12D12E···12J
order12222222333···34466666666666···666666···61212121212···12
size11223366114···42611222233334···466668···822668···8

60 irreducible representations

dim11111111122233333366
type++++++++
imageC1C2C2C3C3C6C6C6C6D4C3×D4C3×D4A4C2×A4C2×A4C3×A4C6×A4C6×A4D4×A4C3×D4×A4
kernelC3×D4×A4C12×A4A4×C2×C6D4×A4D4×C2×C6C4×A4C22×C12C22×A4C23×C6C3×A4A4C2×C6C3×D4C12C2×C6D4C4C22C3C1
# reps112626212416211222412

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

30000
03000
00900
00090
00009
,
012000
10000
001200
000120
000012
,
012000
120000
00100
00010
00001
,
10000
01000
00121212
00001
00010
,
10000
01000
00001
00121212
00100
,
10000
01000
00001
00100
00010

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

C3×D4×A4 in GAP, Magma, Sage, TeX

C_3\times D_4\times A_4
% in TeX

G:=Group("C3xD4xA4");
// GroupNames label

G:=SmallGroup(288,980);
// by ID

G=gap.SmallGroup(288,980);
# by ID

G:=PCGroup([7,-2,-2,-3,-3,-2,-2,2,533,1531,2666]);
// Polycyclic

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

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