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G = D4×C3.A4order 288 = 25·32

Direct product of D4 and C3.A4

direct product, metabelian, soluble, monomial

Aliases: D4×C3.A4, C242C18, C3.(D4×A4), C22⋊(D4×C9), (C22×D4)⋊C9, (C22×C4)⋊C18, C12.7(C2×A4), (C3×D4).1A4, (C23×C6).2C6, C23.8(C2×C18), C6.13(C22×A4), (C22×C12).3C6, C4⋊(C2×C3.A4), (D4×C2×C6).C3, (C4×C3.A4)⋊3C2, (C2×C6).14(C2×A4), (C2×C6).12(C3×D4), C222(C2×C3.A4), (C22×C3.A4)⋊1C2, C2.2(C22×C3.A4), (C22×C6).36(C2×C6), (C2×C3.A4).7C22, SmallGroup(288,344)

Series: Derived Chief Lower central Upper central

C1C23 — D4×C3.A4
C1C22C2×C6C22×C6C2×C3.A4C22×C3.A4 — D4×C3.A4
C22C23 — D4×C3.A4
C1C6C3×D4

Generators and relations for D4×C3.A4
 G = < a,b,c,d,e,f | a4=b2=c3=d2=e2=1, f3=c, bab=a-1, ac=ca, ad=da, ae=ea, af=fa, bc=cb, bd=db, be=eb, bf=fb, cd=dc, ce=ec, cf=fc, fdf-1=de=ed, fef-1=d >

Subgroups: 366 in 116 conjugacy classes, 30 normal (20 characteristic)
C1, C2, C2 [×6], C3, C4, C4, C22, C22 [×2], C22 [×12], C6, C6 [×6], C2×C4 [×2], D4, D4 [×5], C23, C23 [×8], C9, C12, C12, C2×C6, C2×C6 [×2], C2×C6 [×12], C22×C4, C2×D4 [×4], C24 [×2], C18 [×3], C2×C12 [×2], C3×D4, C3×D4 [×5], C22×C6, C22×C6 [×8], C22×D4, C36, C3.A4, C2×C18 [×2], C22×C12, C6×D4 [×4], C23×C6 [×2], D4×C9, C2×C3.A4, C2×C3.A4 [×2], D4×C2×C6, C4×C3.A4, C22×C3.A4 [×2], D4×C3.A4
Quotients: C1, C2 [×3], C3, C22, C6 [×3], D4, C9, A4, C2×C6, C18 [×3], C3×D4, C2×A4 [×3], C3.A4, C2×C18, C22×A4, D4×C9, C2×C3.A4 [×3], D4×A4, C22×C3.A4, D4×C3.A4

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

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

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

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

60 conjugacy classes

class 1 2A2B2C2D2E2F2G3A3B4A4B6A6B6C6D6E6F6G6H6I6J6K6L6M6N9A···9F12A12B12C12D18A···18F18G···18R36A···36F
order122222223344666666666666669···91212121218···1818···1836···36
size112233661126112222333366664···422664···48···88···8

60 irreducible representations

dim11111111122233333366
type++++++++
imageC1C2C2C3C6C6C9C18C18D4C3×D4D4×C9A4C2×A4C2×A4C3.A4C2×C3.A4C2×C3.A4D4×A4D4×C3.A4
kernelD4×C3.A4C4×C3.A4C22×C3.A4D4×C2×C6C22×C12C23×C6C22×D4C22×C4C24C3.A4C2×C6C22C3×D4C12C2×C6D4C4C22C3C1
# reps112224661212611222412

Matrix representation of D4×C3.A4 in GL5(𝔽37)

01000
360000
00100
00010
00001
,
01000
10000
003600
000360
000036
,
10000
01000
001000
000100
000010
,
10000
01000
00363636
00001
00010
,
10000
01000
00001
00363636
00100
,
260000
026000
000033
003300
000330

G:=sub<GL(5,GF(37))| [0,36,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1],[0,1,0,0,0,1,0,0,0,0,0,0,36,0,0,0,0,0,36,0,0,0,0,0,36],[1,0,0,0,0,0,1,0,0,0,0,0,10,0,0,0,0,0,10,0,0,0,0,0,10],[1,0,0,0,0,0,1,0,0,0,0,0,36,0,0,0,0,36,0,1,0,0,36,1,0],[1,0,0,0,0,0,1,0,0,0,0,0,0,36,1,0,0,0,36,0,0,0,1,36,0],[26,0,0,0,0,0,26,0,0,0,0,0,0,33,0,0,0,0,0,33,0,0,33,0,0] >;

D4×C3.A4 in GAP, Magma, Sage, TeX

D_4\times C_3.A_4
% in TeX

G:=Group("D4xC3.A4");
// GroupNames label

G:=SmallGroup(288,344);
// by ID

G=gap.SmallGroup(288,344);
# by ID

G:=PCGroup([7,-2,-2,-3,-2,-3,-2,2,197,142,1531,2666]);
// Polycyclic

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