direct product, abelian, monomial, 2-elementary
Aliases: C22×C12, SmallGroup(48,44)
Series: Derived ►Chief ►Lower central ►Upper central
| C1 — C22×C12 |
| C1 — C22×C12 |
| C1 — C22×C12 |
Generators and relations for C22×C12
G = < a,b,c | a2=b2=c12=1, ab=ba, ac=ca, bc=cb >
Subgroups: 54, all normal (8 characteristic)
C1, C2, C2, C3, C4, C22, C6, C6, C2×C4, C23, C12, C2×C6, C22×C4, C2×C12, C22×C6, C22×C12
Quotients: C1, C2, C3, C4, C22, C6, C2×C4, C23, C12, C2×C6, C22×C4, C2×C12, C22×C6, C22×C12
►Regular action on 48 points
Generators in S48
(1 32)(2 33)(3 34)(4 35)(5 36)(6 25)(7 26)(8 27)(9 28)(10 29)(11 30)(12 31)(13 39)(14 40)(15 41)(16 42)(17 43)(18 44)(19 45)(20 46)(21 47)(22 48)(23 37)(24 38)
(1 16)(2 17)(3 18)(4 19)(5 20)(6 21)(7 22)(8 23)(9 24)(10 13)(11 14)(12 15)(25 47)(26 48)(27 37)(28 38)(29 39)(30 40)(31 41)(32 42)(33 43)(34 44)(35 45)(36 46)
(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)| (1,32)(2,33)(3,34)(4,35)(5,36)(6,25)(7,26)(8,27)(9,28)(10,29)(11,30)(12,31)(13,39)(14,40)(15,41)(16,42)(17,43)(18,44)(19,45)(20,46)(21,47)(22,48)(23,37)(24,38), (1,16)(2,17)(3,18)(4,19)(5,20)(6,21)(7,22)(8,23)(9,24)(10,13)(11,14)(12,15)(25,47)(26,48)(27,37)(28,38)(29,39)(30,40)(31,41)(32,42)(33,43)(34,44)(35,45)(36,46), (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( (1,32)(2,33)(3,34)(4,35)(5,36)(6,25)(7,26)(8,27)(9,28)(10,29)(11,30)(12,31)(13,39)(14,40)(15,41)(16,42)(17,43)(18,44)(19,45)(20,46)(21,47)(22,48)(23,37)(24,38), (1,16)(2,17)(3,18)(4,19)(5,20)(6,21)(7,22)(8,23)(9,24)(10,13)(11,14)(12,15)(25,47)(26,48)(27,37)(28,38)(29,39)(30,40)(31,41)(32,42)(33,43)(34,44)(35,45)(36,46), (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([[(1,32),(2,33),(3,34),(4,35),(5,36),(6,25),(7,26),(8,27),(9,28),(10,29),(11,30),(12,31),(13,39),(14,40),(15,41),(16,42),(17,43),(18,44),(19,45),(20,46),(21,47),(22,48),(23,37),(24,38)], [(1,16),(2,17),(3,18),(4,19),(5,20),(6,21),(7,22),(8,23),(9,24),(10,13),(11,14),(12,15),(25,47),(26,48),(27,37),(28,38),(29,39),(30,40),(31,41),(32,42),(33,43),(34,44),(35,45),(36,46)], [(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)]])
C22×C12 is a maximal subgroup of
C12.55D4 C6.C42 C12.48D4 C23.26D6 C23.28D6 C12⋊7D4
48 conjugacy classes
| class | 1 | 2A | ··· | 2G | 3A | 3B | 4A | ··· | 4H | 6A | ··· | 6N | 12A | ··· | 12P |
| order | 1 | 2 | ··· | 2 | 3 | 3 | 4 | ··· | 4 | 6 | ··· | 6 | 12 | ··· | 12 |
| size | 1 | 1 | ··· | 1 | 1 | 1 | 1 | ··· | 1 | 1 | ··· | 1 | 1 | ··· | 1 |
48 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |
| type | + | + | + | |||||
| image | C1 | C2 | C2 | C3 | C4 | C6 | C6 | C12 |
| kernel | C22×C12 | C2×C12 | C22×C6 | C22×C4 | C2×C6 | C2×C4 | C23 | C22 |
| # reps | 1 | 6 | 1 | 2 | 8 | 12 | 2 | 16 |
Matrix representation of C22×C12 ►in GL3(𝔽13) generated by
| 12 | 0 | 0 |
| 0 | 12 | 0 |
| 0 | 0 | 1 |
| 1 | 0 | 0 |
| 0 | 1 | 0 |
| 0 | 0 | 12 |
| 12 | 0 | 0 |
| 0 | 11 | 0 |
| 0 | 0 | 8 |
G:=sub<GL(3,GF(13))| [12,0,0,0,12,0,0,0,1],[1,0,0,0,1,0,0,0,12],[12,0,0,0,11,0,0,0,8] >;
C22×C12 in GAP, Magma, Sage, TeX
C_2^2\times C_{12} % in TeX
G:=Group("C2^2xC12"); // GroupNames label
G:=SmallGroup(48,44);
// by ID
G=gap.SmallGroup(48,44);
# by ID
G:=PCGroup([5,-2,-2,-2,-3,-2,120]);
// Polycyclic
G:=Group<a,b,c|a^2=b^2=c^12=1,a*b=b*a,a*c=c*a,b*c=c*b>;
// generators/relations