direct product, metabelian, soluble, monomial, A-group
Aliases: C22×C3.A4, C23⋊C18, C24⋊1C9, C22⋊(C2×C18), C6.7(C2×A4), (C2×C6).5A4, C3.(C22×A4), (C22×C6).3C6, (C23×C6).1C3, (C2×C6).2(C2×C6), SmallGroup(144,110)
Series: Derived ►Chief ►Lower central ►Upper central
| C22 — C22×C3.A4 |
Generators and relations for C22×C3.A4
G = < a,b,c,d,e,f | a2=b2=c3=d2=e2=1, f3=c, ab=ba, 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: 159 in 68 conjugacy classes, 25 normal (10 characteristic)
C1, C2, C2, C3, C22, C22, C6, C6, C23, C23, C9, C2×C6, C2×C6, C24, C18, C22×C6, C22×C6, C3.A4, C2×C18, C23×C6, C2×C3.A4, C22×C3.A4
Quotients: C1, C2, C3, C22, C6, C9, A4, C2×C6, C18, C2×A4, C3.A4, C2×C18, C22×A4, C2×C3.A4, C22×C3.A4
►On 36 points
(1 32)(2 33)(3 34)(4 35)(5 36)(6 28)(7 29)(8 30)(9 31)(10 21)(11 22)(12 23)(13 24)(14 25)(15 26)(16 27)(17 19)(18 20)
(1 23)(2 24)(3 25)(4 26)(5 27)(6 19)(7 20)(8 21)(9 22)(10 30)(11 31)(12 32)(13 33)(14 34)(15 35)(16 36)(17 28)(18 29)
(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 23)(2 33)(3 14)(4 26)(5 36)(6 17)(7 20)(8 30)(9 11)(10 21)(12 32)(13 24)(15 35)(16 27)(18 29)(19 28)(22 31)(25 34)
(1 12)(2 24)(3 34)(4 15)(5 27)(6 28)(7 18)(8 21)(9 31)(10 30)(11 22)(13 33)(14 25)(16 36)(17 19)(20 29)(23 32)(26 35)
(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)| (1,32)(2,33)(3,34)(4,35)(5,36)(6,28)(7,29)(8,30)(9,31)(10,21)(11,22)(12,23)(13,24)(14,25)(15,26)(16,27)(17,19)(18,20), (1,23)(2,24)(3,25)(4,26)(5,27)(6,19)(7,20)(8,21)(9,22)(10,30)(11,31)(12,32)(13,33)(14,34)(15,35)(16,36)(17,28)(18,29), (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), (1,23)(2,33)(3,14)(4,26)(5,36)(6,17)(7,20)(8,30)(9,11)(10,21)(12,32)(13,24)(15,35)(16,27)(18,29)(19,28)(22,31)(25,34), (1,12)(2,24)(3,34)(4,15)(5,27)(6,28)(7,18)(8,21)(9,31)(10,30)(11,22)(13,33)(14,25)(16,36)(17,19)(20,29)(23,32)(26,35), (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( (1,32)(2,33)(3,34)(4,35)(5,36)(6,28)(7,29)(8,30)(9,31)(10,21)(11,22)(12,23)(13,24)(14,25)(15,26)(16,27)(17,19)(18,20), (1,23)(2,24)(3,25)(4,26)(5,27)(6,19)(7,20)(8,21)(9,22)(10,30)(11,31)(12,32)(13,33)(14,34)(15,35)(16,36)(17,28)(18,29), (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), (1,23)(2,33)(3,14)(4,26)(5,36)(6,17)(7,20)(8,30)(9,11)(10,21)(12,32)(13,24)(15,35)(16,27)(18,29)(19,28)(22,31)(25,34), (1,12)(2,24)(3,34)(4,15)(5,27)(6,28)(7,18)(8,21)(9,31)(10,30)(11,22)(13,33)(14,25)(16,36)(17,19)(20,29)(23,32)(26,35), (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([[(1,32),(2,33),(3,34),(4,35),(5,36),(6,28),(7,29),(8,30),(9,31),(10,21),(11,22),(12,23),(13,24),(14,25),(15,26),(16,27),(17,19),(18,20)], [(1,23),(2,24),(3,25),(4,26),(5,27),(6,19),(7,20),(8,21),(9,22),(10,30),(11,31),(12,32),(13,33),(14,34),(15,35),(16,36),(17,28),(18,29)], [(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)], [(1,23),(2,33),(3,14),(4,26),(5,36),(6,17),(7,20),(8,30),(9,11),(10,21),(12,32),(13,24),(15,35),(16,27),(18,29),(19,28),(22,31),(25,34)], [(1,12),(2,24),(3,34),(4,15),(5,27),(6,28),(7,18),(8,21),(9,31),(10,30),(11,22),(13,33),(14,25),(16,36),(17,19),(20,29),(23,32),(26,35)], [(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)]])
C22×C3.A4 is a maximal subgroup of
C23.D18 C3.A42 C24⋊3- 1+2 C24⋊23- 1+2 A4×C2×C18
C22×C3.A4 is a maximal quotient of 2+ 1+4⋊C9 2- 1+4⋊C9
48 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 3A | 3B | 6A | ··· | 6F | 6G | ··· | 6N | 9A | ··· | 9F | 18A | ··· | 18R |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 3 | 6 | ··· | 6 | 6 | ··· | 6 | 9 | ··· | 9 | 18 | ··· | 18 |
| size | 1 | 1 | 1 | 1 | 3 | 3 | 3 | 3 | 1 | 1 | 1 | ··· | 1 | 3 | ··· | 3 | 4 | ··· | 4 | 4 | ··· | 4 |
48 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 3 | 3 | 3 | 3 |
| type | + | + | + | + | ||||||
| image | C1 | C2 | C3 | C6 | C9 | C18 | A4 | C2×A4 | C3.A4 | C2×C3.A4 |
| kernel | C22×C3.A4 | C2×C3.A4 | C23×C6 | C22×C6 | C24 | C23 | C2×C6 | C6 | C22 | C2 |
| # reps | 1 | 3 | 2 | 6 | 6 | 18 | 1 | 3 | 2 | 6 |
Matrix representation of C22×C3.A4 ►in GL4(𝔽19) generated by
| 18 | 0 | 0 | 0 |
| 0 | 18 | 0 | 0 |
| 0 | 0 | 18 | 0 |
| 0 | 0 | 0 | 18 |
| 18 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 7 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 18 | 0 |
| 0 | 0 | 0 | 18 |
| 1 | 0 | 0 | 0 |
| 0 | 18 | 0 | 0 |
| 0 | 0 | 18 | 0 |
| 0 | 0 | 0 | 1 |
| 9 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 1 | 0 | 0 |
G:=sub<GL(4,GF(19))| [18,0,0,0,0,18,0,0,0,0,18,0,0,0,0,18],[18,0,0,0,0,1,0,0,0,0,1,0,0,0,0,1],[7,0,0,0,0,1,0,0,0,0,1,0,0,0,0,1],[1,0,0,0,0,1,0,0,0,0,18,0,0,0,0,18],[1,0,0,0,0,18,0,0,0,0,18,0,0,0,0,1],[9,0,0,0,0,0,0,1,0,1,0,0,0,0,1,0] >;
C22×C3.A4 in GAP, Magma, Sage, TeX
C_2^2\times C_3.A_4
% in TeX
G:=Group("C2^2xC3.A4"); // GroupNames label
G:=SmallGroup(144,110);
// by ID
G=gap.SmallGroup(144,110);
# by ID
G:=PCGroup([6,-2,-2,-3,-3,-2,2,68,556,989]);
// Polycyclic
G:=Group<a,b,c,d,e,f|a^2=b^2=c^3=d^2=e^2=1,f^3=c,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,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