direct product, metabelian, nilpotent (class 3), monomial, 3-elementary
Aliases: C4×C3≀C3, He3⋊3C12, C33⋊7C12, C12.2He3, 3- 1+2⋊1C12, (C4×He3)⋊1C3, C3.2(C4×He3), C6.3(C2×He3), (C32×C12)⋊1C3, (C2×He3).6C6, (C32×C6).10C6, (C3×C12).1C32, C32.1(C3×C12), (C4×3- 1+2)⋊1C3, (C2×3- 1+2).1C6, C2.(C2×C3≀C3), (C3×C6).2(C3×C6), (C2×C3≀C3).4C2, SmallGroup(324,31)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C4×C3≀C3
G = < a,b,c,d,e | a4=b3=c3=d3=e3=1, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, dbd-1=bc-1, be=eb, cd=dc, ce=ec, ede-1=bc-1d >
Subgroups: 150 in 60 conjugacy classes, 24 normal (18 characteristic)
C1, C2, C3, C3, C4, C6, C6, C9, C32, C32, C12, C12, C18, C3×C6, C3×C6, He3, 3- 1+2, C33, C36, C3×C12, C3×C12, C2×He3, C2×3- 1+2, C32×C6, C3≀C3, C4×He3, C4×3- 1+2, C32×C12, C2×C3≀C3, C4×C3≀C3
Quotients: C1, C2, C3, C4, C6, C32, C12, C3×C6, He3, C3×C12, C2×He3, C3≀C3, C4×He3, C2×C3≀C3, C4×C3≀C3
►On 36 points
(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)
(5 33 10)(6 34 11)(7 35 12)(8 36 9)(17 22 27)(18 23 28)(19 24 25)(20 21 26)
(1 14 29)(2 15 30)(3 16 31)(4 13 32)(5 33 10)(6 34 11)(7 35 12)(8 36 9)(17 27 22)(18 28 23)(19 25 24)(20 26 21)
(1 26 35)(2 27 36)(3 28 33)(4 25 34)(5 31 18)(6 32 19)(7 29 20)(8 30 17)(9 15 22)(10 16 23)(11 13 24)(12 14 21)
(1 14 29)(2 15 30)(3 16 31)(4 13 32)(5 33 10)(6 34 11)(7 35 12)(8 36 9)(17 22 27)(18 23 28)(19 24 25)(20 21 26)
G:=sub<Sym(36)| (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), (5,33,10)(6,34,11)(7,35,12)(8,36,9)(17,22,27)(18,23,28)(19,24,25)(20,21,26), (1,14,29)(2,15,30)(3,16,31)(4,13,32)(5,33,10)(6,34,11)(7,35,12)(8,36,9)(17,27,22)(18,28,23)(19,25,24)(20,26,21), (1,26,35)(2,27,36)(3,28,33)(4,25,34)(5,31,18)(6,32,19)(7,29,20)(8,30,17)(9,15,22)(10,16,23)(11,13,24)(12,14,21), (1,14,29)(2,15,30)(3,16,31)(4,13,32)(5,33,10)(6,34,11)(7,35,12)(8,36,9)(17,22,27)(18,23,28)(19,24,25)(20,21,26)>;
G:=Group( (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), (5,33,10)(6,34,11)(7,35,12)(8,36,9)(17,22,27)(18,23,28)(19,24,25)(20,21,26), (1,14,29)(2,15,30)(3,16,31)(4,13,32)(5,33,10)(6,34,11)(7,35,12)(8,36,9)(17,27,22)(18,28,23)(19,25,24)(20,26,21), (1,26,35)(2,27,36)(3,28,33)(4,25,34)(5,31,18)(6,32,19)(7,29,20)(8,30,17)(9,15,22)(10,16,23)(11,13,24)(12,14,21), (1,14,29)(2,15,30)(3,16,31)(4,13,32)(5,33,10)(6,34,11)(7,35,12)(8,36,9)(17,22,27)(18,23,28)(19,24,25)(20,21,26) );
G=PermutationGroup([[(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)], [(5,33,10),(6,34,11),(7,35,12),(8,36,9),(17,22,27),(18,23,28),(19,24,25),(20,21,26)], [(1,14,29),(2,15,30),(3,16,31),(4,13,32),(5,33,10),(6,34,11),(7,35,12),(8,36,9),(17,27,22),(18,28,23),(19,25,24),(20,26,21)], [(1,26,35),(2,27,36),(3,28,33),(4,25,34),(5,31,18),(6,32,19),(7,29,20),(8,30,17),(9,15,22),(10,16,23),(11,13,24),(12,14,21)], [(1,14,29),(2,15,30),(3,16,31),(4,13,32),(5,33,10),(6,34,11),(7,35,12),(8,36,9),(17,22,27),(18,23,28),(19,24,25),(20,21,26)]])
68 conjugacy classes
| class | 1 | 2 | 3A | 3B | 3C | ··· | 3J | 3K | 3L | 4A | 4B | 6A | 6B | 6C | ··· | 6J | 6K | 6L | 9A | 9B | 9C | 9D | 12A | 12B | 12C | 12D | 12E | ··· | 12T | 12U | 12V | 12W | 12X | 18A | 18B | 18C | 18D | 36A | ··· | 36H |
| order | 1 | 2 | 3 | 3 | 3 | ··· | 3 | 3 | 3 | 4 | 4 | 6 | 6 | 6 | ··· | 6 | 6 | 6 | 9 | 9 | 9 | 9 | 12 | 12 | 12 | 12 | 12 | ··· | 12 | 12 | 12 | 12 | 12 | 18 | 18 | 18 | 18 | 36 | ··· | 36 |
| size | 1 | 1 | 1 | 1 | 3 | ··· | 3 | 9 | 9 | 1 | 1 | 1 | 1 | 3 | ··· | 3 | 9 | 9 | 9 | 9 | 9 | 9 | 1 | 1 | 1 | 1 | 3 | ··· | 3 | 9 | 9 | 9 | 9 | 9 | 9 | 9 | 9 | 9 | ··· | 9 |
68 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 3 | 3 | 3 | 3 | 3 | 3 |
| type | + | + | ||||||||||||||||
| image | C1 | C2 | C3 | C3 | C3 | C4 | C6 | C6 | C6 | C12 | C12 | C12 | He3 | C2×He3 | C3≀C3 | C4×He3 | C2×C3≀C3 | C4×C3≀C3 |
| kernel | C4×C3≀C3 | C2×C3≀C3 | C4×He3 | C4×3- 1+2 | C32×C12 | C3≀C3 | C2×He3 | C2×3- 1+2 | C32×C6 | He3 | 3- 1+2 | C33 | C12 | C6 | C4 | C3 | C2 | C1 |
| # reps | 1 | 1 | 2 | 4 | 2 | 2 | 2 | 4 | 2 | 4 | 8 | 4 | 2 | 2 | 6 | 4 | 6 | 12 |
Matrix representation of C4×C3≀C3 ►in GL3(𝔽13) generated by
| 8 | 0 | 0 |
| 0 | 8 | 0 |
| 0 | 0 | 8 |
| 3 | 0 | 0 |
| 0 | 1 | 0 |
| 0 | 0 | 9 |
| 9 | 0 | 0 |
| 0 | 9 | 0 |
| 0 | 0 | 9 |
| 0 | 7 | 0 |
| 0 | 0 | 7 |
| 4 | 0 | 0 |
| 9 | 0 | 0 |
| 0 | 1 | 0 |
| 0 | 0 | 1 |
G:=sub<GL(3,GF(13))| [8,0,0,0,8,0,0,0,8],[3,0,0,0,1,0,0,0,9],[9,0,0,0,9,0,0,0,9],[0,0,4,7,0,0,0,7,0],[9,0,0,0,1,0,0,0,1] >;
C4×C3≀C3 in GAP, Magma, Sage, TeX
C_4\times C_3\wr C_3
% in TeX
G:=Group("C4xC3wrC3"); // GroupNames label
G:=SmallGroup(324,31);
// by ID
G=gap.SmallGroup(324,31);
# by ID
G:=PCGroup([6,-2,-3,-3,-2,-3,-3,108,386,2170]);
// Polycyclic
G:=Group<a,b,c,d,e|a^4=b^3=c^3=d^3=e^3=1,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,b*c=c*b,d*b*d^-1=b*c^-1,b*e=e*b,c*d=d*c,c*e=e*c,e*d*e^-1=b*c^-1*d>;
// generators/relations