direct product, abelian, monomial, 2-elementary
Aliases: C2×C24, SmallGroup(48,23)
Series: Derived ►Chief ►Lower central ►Upper central
| C1 — C2×C24 |
| C1 — C2×C24 |
| C1 — C2×C24 |
Generators and relations for C2×C24
G = < a,b | a2=b24=1, ab=ba >
Smallest permutation representation of C2×C24
►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)]])
C2×C24 is a maximal subgroup of
C12.C8 Dic3⋊C8 C24⋊C4 C2.Dic12 C8⋊Dic3 C24⋊1C4 C24.C4 D6⋊C8 C2.D24 C8○D12 C4○D24 He3⋊2(C2×C8)
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 | C2×C24 | C24 | C2×C12 | C2×C8 | C12 | C2×C6 | C8 | C2×C4 | C6 | C4 | C22 | C2 |
| # reps | 1 | 2 | 1 | 2 | 2 | 2 | 4 | 2 | 8 | 4 | 4 | 16 |
Matrix representation of C2×C24 ►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] >;
C2×C24 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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