direct product, abelian, monomial, 2-elementary
Aliases: C4×C12, SmallGroup(48,20)
Series: Derived ►Chief ►Lower central ►Upper central
| C1 — C4×C12 |
| C1 — C4×C12 |
| C1 — C4×C12 |
Generators and relations for C4×C12
G = < a,b | a4=b12=1, ab=ba >
Smallest permutation representation of C4×C12
►Regular action on 48 points
Generators in S48
(1 39 36 24)(2 40 25 13)(3 41 26 14)(4 42 27 15)(5 43 28 16)(6 44 29 17)(7 45 30 18)(8 46 31 19)(9 47 32 20)(10 48 33 21)(11 37 34 22)(12 38 35 23)
(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)(37 38 39 40 41 42 43 44 45 46 47 48)
G:=sub<Sym(48)| (1,39,36,24)(2,40,25,13)(3,41,26,14)(4,42,27,15)(5,43,28,16)(6,44,29,17)(7,45,30,18)(8,46,31,19)(9,47,32,20)(10,48,33,21)(11,37,34,22)(12,38,35,23), (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)(37,38,39,40,41,42,43,44,45,46,47,48)>;
G:=Group( (1,39,36,24)(2,40,25,13)(3,41,26,14)(4,42,27,15)(5,43,28,16)(6,44,29,17)(7,45,30,18)(8,46,31,19)(9,47,32,20)(10,48,33,21)(11,37,34,22)(12,38,35,23), (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)(37,38,39,40,41,42,43,44,45,46,47,48) );
G=PermutationGroup([[(1,39,36,24),(2,40,25,13),(3,41,26,14),(4,42,27,15),(5,43,28,16),(6,44,29,17),(7,45,30,18),(8,46,31,19),(9,47,32,20),(10,48,33,21),(11,37,34,22),(12,38,35,23)], [(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),(37,38,39,40,41,42,43,44,45,46,47,48)]])
C4×C12 is a maximal subgroup of
C42.S3 C12⋊C8 C42⋊4S3 C12⋊2Q8 C12.6Q8 C42⋊2S3 C4⋊D12 C42⋊7S3 C42⋊3S3 C42⋊C9
48 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3A | 3B | 4A | ··· | 4L | 6A | ··· | 6F | 12A | ··· | 12X |
| order | 1 | 2 | 2 | 2 | 3 | 3 | 4 | ··· | 4 | 6 | ··· | 6 | 12 | ··· | 12 |
| size | 1 | 1 | 1 | 1 | 1 | 1 | 1 | ··· | 1 | 1 | ··· | 1 | 1 | ··· | 1 |
48 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 |
| type | + | + | ||||
| image | C1 | C2 | C3 | C4 | C6 | C12 |
| kernel | C4×C12 | C2×C12 | C42 | C12 | C2×C4 | C4 |
| # reps | 1 | 3 | 2 | 12 | 6 | 24 |
Matrix representation of C4×C12 ►in GL2(𝔽13) generated by
| 12 | 0 |
| 0 | 5 |
| 5 | 0 |
| 0 | 4 |
G:=sub<GL(2,GF(13))| [12,0,0,5],[5,0,0,4] >;
C4×C12 in GAP, Magma, Sage, TeX
C_4\times C_{12} % in TeX
G:=Group("C4xC12"); // GroupNames label
G:=SmallGroup(48,20);
// by ID
G=gap.SmallGroup(48,20);
# by ID
G:=PCGroup([5,-2,-2,-3,-2,-2,60,126]);
// Polycyclic
G:=Group<a,b|a^4=b^12=1,a*b=b*a>;
// generators/relations
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