direct product, non-abelian, soluble, monomial
Aliases: C4×C3.S4, C12.10S4, C23.2D18, C3.(C4×S4), C22⋊(C4×D9), C6.17(C2×S4), (C22×C4)⋊1D9, C6.S4⋊2C2, (C22×C12).6S3, (C22×C6).14D6, (C2×C6).(C4×S3), (C2×C3.S4).C2, (C4×C3.A4)⋊2C2, C3.A4⋊1(C2×C4), C2.1(C2×C3.S4), (C2×C3.A4).2C22, SmallGroup(288,333)
Series: Derived ►Chief ►Lower central ►Upper central
C3.A4 — C4×C3.S4 |
Generators and relations for C4×C3.S4
G = < a,b,c,d,e,f | a4=b3=c2=d2=f2=1, e3=b, ab=ba, ac=ca, ad=da, ae=ea, af=fa, bc=cb, bd=db, be=eb, fbf=b-1, ece-1=fcf=cd=dc, ede-1=c, df=fd, fef=b-1e2 >
Subgroups: 514 in 96 conjugacy classes, 20 normal (18 characteristic)
C1, C2, C2 [×4], C3, C4, C4 [×5], C22, C22 [×6], S3 [×2], C6, C6 [×2], C2×C4 [×7], D4 [×4], C23, C23, C9, Dic3 [×4], C12, C12, D6 [×4], C2×C6, C2×C6 [×2], C42, C22⋊C4 [×2], C4⋊C4, C22×C4, C22×C4, C2×D4, D9 [×2], C18, C4×S3 [×2], C2×Dic3 [×3], C3⋊D4 [×4], C2×C12 [×2], C22×S3, C22×C6, C4×D4, Dic9, C36, C3.A4, D18, C4×Dic3, Dic3⋊C4, D6⋊C4, C6.D4, S3×C2×C4, C2×C3⋊D4, C22×C12, C4×D9, C3.S4 [×2], C2×C3.A4, C4×C3⋊D4, C6.S4, C4×C3.A4, C2×C3.S4, C4×C3.S4
Quotients: C1, C2 [×3], C4 [×2], C22, S3, C2×C4, D6, D9, C4×S3, S4, D18, C2×S4, C4×D9, C3.S4, C4×S4, C2×C3.S4, C4×C3.S4
(1 11 22 34)(2 12 23 35)(3 13 24 36)(4 14 25 28)(5 15 26 29)(6 16 27 30)(7 17 19 31)(8 18 20 32)(9 10 21 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)
(1 22)(2 23)(4 25)(5 26)(7 19)(8 20)(11 34)(12 35)(14 28)(15 29)(17 31)(18 32)
(2 23)(3 24)(5 26)(6 27)(8 20)(9 21)(10 33)(12 35)(13 36)(15 29)(16 30)(18 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 22)(2 21)(3 20)(4 19)(5 27)(6 26)(7 25)(8 24)(9 23)(10 35)(11 34)(12 33)(13 32)(14 31)(15 30)(16 29)(17 28)(18 36)
G:=sub<Sym(36)| (1,11,22,34)(2,12,23,35)(3,13,24,36)(4,14,25,28)(5,15,26,29)(6,16,27,30)(7,17,19,31)(8,18,20,32)(9,10,21,33), (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), (1,22)(2,23)(4,25)(5,26)(7,19)(8,20)(11,34)(12,35)(14,28)(15,29)(17,31)(18,32), (2,23)(3,24)(5,26)(6,27)(8,20)(9,21)(10,33)(12,35)(13,36)(15,29)(16,30)(18,32), (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), (1,22)(2,21)(3,20)(4,19)(5,27)(6,26)(7,25)(8,24)(9,23)(10,35)(11,34)(12,33)(13,32)(14,31)(15,30)(16,29)(17,28)(18,36)>;
G:=Group( (1,11,22,34)(2,12,23,35)(3,13,24,36)(4,14,25,28)(5,15,26,29)(6,16,27,30)(7,17,19,31)(8,18,20,32)(9,10,21,33), (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), (1,22)(2,23)(4,25)(5,26)(7,19)(8,20)(11,34)(12,35)(14,28)(15,29)(17,31)(18,32), (2,23)(3,24)(5,26)(6,27)(8,20)(9,21)(10,33)(12,35)(13,36)(15,29)(16,30)(18,32), (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), (1,22)(2,21)(3,20)(4,19)(5,27)(6,26)(7,25)(8,24)(9,23)(10,35)(11,34)(12,33)(13,32)(14,31)(15,30)(16,29)(17,28)(18,36) );
G=PermutationGroup([(1,11,22,34),(2,12,23,35),(3,13,24,36),(4,14,25,28),(5,15,26,29),(6,16,27,30),(7,17,19,31),(8,18,20,32),(9,10,21,33)], [(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)], [(1,22),(2,23),(4,25),(5,26),(7,19),(8,20),(11,34),(12,35),(14,28),(15,29),(17,31),(18,32)], [(2,23),(3,24),(5,26),(6,27),(8,20),(9,21),(10,33),(12,35),(13,36),(15,29),(16,30),(18,32)], [(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)], [(1,22),(2,21),(3,20),(4,19),(5,27),(6,26),(7,25),(8,24),(9,23),(10,35),(11,34),(12,33),(13,32),(14,31),(15,30),(16,29),(17,28),(18,36)])
36 conjugacy classes
class | 1 | 2A | 2B | 2C | 2D | 2E | 3 | 4A | 4B | 4C | 4D | 4E | ··· | 4J | 6A | 6B | 6C | 9A | 9B | 9C | 12A | 12B | 12C | 12D | 18A | 18B | 18C | 36A | ··· | 36F |
order | 1 | 2 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 6 | 6 | 6 | 9 | 9 | 9 | 12 | 12 | 12 | 12 | 18 | 18 | 18 | 36 | ··· | 36 |
size | 1 | 1 | 3 | 3 | 18 | 18 | 2 | 1 | 1 | 3 | 3 | 18 | ··· | 18 | 2 | 6 | 6 | 8 | 8 | 8 | 2 | 2 | 6 | 6 | 8 | 8 | 8 | 8 | ··· | 8 |
36 irreducible representations
dim | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 3 | 3 | 6 | 6 | 6 |
type | + | + | + | + | + | + | + | + | + | + | + | + | |||||
image | C1 | C2 | C2 | C2 | C4 | S3 | D6 | D9 | C4×S3 | D18 | C4×D9 | S4 | C2×S4 | C4×S4 | C3.S4 | C2×C3.S4 | C4×C3.S4 |
kernel | C4×C3.S4 | C6.S4 | C4×C3.A4 | C2×C3.S4 | C3.S4 | C22×C12 | C22×C6 | C22×C4 | C2×C6 | C23 | C22 | C12 | C6 | C3 | C4 | C2 | C1 |
# reps | 1 | 1 | 1 | 1 | 4 | 1 | 1 | 3 | 2 | 3 | 6 | 2 | 2 | 4 | 1 | 1 | 2 |
Matrix representation of C4×C3.S4 ►in GL5(𝔽37)
6 | 0 | 0 | 0 | 0 |
0 | 6 | 0 | 0 | 0 |
0 | 0 | 36 | 0 | 0 |
0 | 0 | 0 | 36 | 0 |
0 | 0 | 0 | 0 | 36 |
36 | 36 | 0 | 0 | 0 |
1 | 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 | 36 | 0 | 0 |
0 | 0 | 1 | 1 | 0 |
0 | 0 | 0 | 0 | 36 |
1 | 0 | 0 | 0 | 0 |
0 | 1 | 0 | 0 | 0 |
0 | 0 | 36 | 0 | 0 |
0 | 0 | 0 | 36 | 0 |
0 | 0 | 36 | 0 | 1 |
26 | 6 | 0 | 0 | 0 |
31 | 20 | 0 | 0 | 0 |
0 | 0 | 1 | 0 | 35 |
0 | 0 | 0 | 0 | 1 |
0 | 0 | 0 | 36 | 36 |
1 | 0 | 0 | 0 | 0 |
36 | 36 | 0 | 0 | 0 |
0 | 0 | 36 | 35 | 0 |
0 | 0 | 0 | 1 | 0 |
0 | 0 | 0 | 36 | 36 |
G:=sub<GL(5,GF(37))| [6,0,0,0,0,0,6,0,0,0,0,0,36,0,0,0,0,0,36,0,0,0,0,0,36],[36,1,0,0,0,36,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1],[1,0,0,0,0,0,1,0,0,0,0,0,36,1,0,0,0,0,1,0,0,0,0,0,36],[1,0,0,0,0,0,1,0,0,0,0,0,36,0,36,0,0,0,36,0,0,0,0,0,1],[26,31,0,0,0,6,20,0,0,0,0,0,1,0,0,0,0,0,0,36,0,0,35,1,36],[1,36,0,0,0,0,36,0,0,0,0,0,36,0,0,0,0,35,1,36,0,0,0,0,36] >;
C4×C3.S4 in GAP, Magma, Sage, TeX
C_4\times C_3.S_4
% in TeX
G:=Group("C4xC3.S4");
// GroupNames label
G:=SmallGroup(288,333);
// by ID
G=gap.SmallGroup(288,333);
# by ID
G:=PCGroup([7,-2,-2,-2,-3,-3,-2,2,36,1123,192,1684,6053,782,3534,1350]);
// Polycyclic
G:=Group<a,b,c,d,e,f|a^4=b^3=c^2=d^2=f^2=1,e^3=b,a*b=b*a,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,f*b*f=b^-1,e*c*e^-1=f*c*f=c*d=d*c,e*d*e^-1=c,d*f=f*d,f*e*f=b^-1*e^2>;
// generators/relations