direct product, abelian, monomial, 2-elementary
Aliases: C2xC24, SmallGroup(48,23)
Series: Derived ►Chief ►Lower central ►Upper central
| C1 — C2xC24 |
| C1 — C2xC24 |
| C1 — C2xC24 |
Generators and relations for C2xC24
G = < a,b | a2=b24=1, ab=ba >
Quotients: C1, C2, C3, C4, C22, C6, C8, C2xC4, C12, C2xC6, C2xC8, C24, C2xC12, C2xC24
Smallest permutation representation of C2xC24
►Regular action on 48 points
Generators in S48
(1 35)(2 36)(3 37)(4 38)(5 39)(6 40)(7 41)(8 42)(9 43)(10 44)(11 45)(12 46)(13 47)(14 48)(15 25)(16 26)(17 27)(18 28)(19 29)(20 30)(21 31)(22 32)(23 33)(24 34)
(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)| (1,35)(2,36)(3,37)(4,38)(5,39)(6,40)(7,41)(8,42)(9,43)(10,44)(11,45)(12,46)(13,47)(14,48)(15,25)(16,26)(17,27)(18,28)(19,29)(20,30)(21,31)(22,32)(23,33)(24,34), (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( (1,35)(2,36)(3,37)(4,38)(5,39)(6,40)(7,41)(8,42)(9,43)(10,44)(11,45)(12,46)(13,47)(14,48)(15,25)(16,26)(17,27)(18,28)(19,29)(20,30)(21,31)(22,32)(23,33)(24,34), (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([[(1,35),(2,36),(3,37),(4,38),(5,39),(6,40),(7,41),(8,42),(9,43),(10,44),(11,45),(12,46),(13,47),(14,48),(15,25),(16,26),(17,27),(18,28),(19,29),(20,30),(21,31),(22,32),(23,33),(24,34)], [(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)]])
C2xC24 is a maximal subgroup of
C12.C8 Dic3:C8 C24:C4 C2.Dic12 C8:Dic3 C24:1C4 C24.C4 D6:C8 C2.D24 C8oD12 C4oD24 He3:2(C2xC8)
48 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3A | 3B | 4A | 4B | 4C | 4D | 6A | ··· | 6F | 8A | ··· | 8H | 12A | ··· | 12H | 24A | ··· | 24P |
| order | 1 | 2 | 2 | 2 | 3 | 3 | 4 | 4 | 4 | 4 | 6 | ··· | 6 | 8 | ··· | 8 | 12 | ··· | 12 | 24 | ··· | 24 |
| size | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | ··· | 1 | 1 | ··· | 1 | 1 | ··· | 1 | 1 | ··· | 1 |
48 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |
| type | + | + | + | |||||||||
| image | C1 | C2 | C2 | C3 | C4 | C4 | C6 | C6 | C8 | C12 | C12 | C24 |
| kernel | C2xC24 | C24 | C2xC12 | C2xC8 | C12 | C2xC6 | C8 | C2xC4 | C6 | C4 | C22 | C2 |
| # reps | 1 | 2 | 1 | 2 | 2 | 2 | 4 | 2 | 8 | 4 | 4 | 16 |
Matrix representation of C2xC24 ►in GL2(F73) generated by
| 72 | 0 |
| 0 | 1 |
| 51 | 0 |
| 0 | 17 |
G:=sub<GL(2,GF(73))| [72,0,0,1],[51,0,0,17] >;
C2xC24 in GAP, Magma, Sage, TeX
C_2\times C_{24} % in TeX
G:=Group("C2xC24"); // GroupNames label
G:=SmallGroup(48,23);
// by ID
G=gap.SmallGroup(48,23);
# by ID
G:=PCGroup([5,-2,-2,-3,-2,-2,60,58]);
// Polycyclic
G:=Group<a,b|a^2=b^24=1,a*b=b*a>;
// generators/relations
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