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G = C3×C24order 72 = 23·32

Abelian group of type [3,24]

direct product, abelian, monomial, 3-elementary

Aliases: C3×C24, SmallGroup(72,14)

Series: Derived Chief Lower central Upper central

C1 — C3×C24
C1C2C4C12C3×C12 — C3×C24
C1 — C3×C24
C1 — C3×C24

Generators and relations for C3×C24
 G = < a,b | a3=b24=1, ab=ba >


Smallest permutation representation of C3×C24
Regular action on 72 points
Generators in S72
(1 66 28)(2 67 29)(3 68 30)(4 69 31)(5 70 32)(6 71 33)(7 72 34)(8 49 35)(9 50 36)(10 51 37)(11 52 38)(12 53 39)(13 54 40)(14 55 41)(15 56 42)(16 57 43)(17 58 44)(18 59 45)(19 60 46)(20 61 47)(21 62 48)(22 63 25)(23 64 26)(24 65 27)
(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)(49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72)

G:=sub<Sym(72)| (1,66,28)(2,67,29)(3,68,30)(4,69,31)(5,70,32)(6,71,33)(7,72,34)(8,49,35)(9,50,36)(10,51,37)(11,52,38)(12,53,39)(13,54,40)(14,55,41)(15,56,42)(16,57,43)(17,58,44)(18,59,45)(19,60,46)(20,61,47)(21,62,48)(22,63,25)(23,64,26)(24,65,27), (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)(49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72)>;

G:=Group( (1,66,28)(2,67,29)(3,68,30)(4,69,31)(5,70,32)(6,71,33)(7,72,34)(8,49,35)(9,50,36)(10,51,37)(11,52,38)(12,53,39)(13,54,40)(14,55,41)(15,56,42)(16,57,43)(17,58,44)(18,59,45)(19,60,46)(20,61,47)(21,62,48)(22,63,25)(23,64,26)(24,65,27), (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)(49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72) );

G=PermutationGroup([(1,66,28),(2,67,29),(3,68,30),(4,69,31),(5,70,32),(6,71,33),(7,72,34),(8,49,35),(9,50,36),(10,51,37),(11,52,38),(12,53,39),(13,54,40),(14,55,41),(15,56,42),(16,57,43),(17,58,44),(18,59,45),(19,60,46),(20,61,47),(21,62,48),(22,63,25),(23,64,26),(24,65,27)], [(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),(49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72)])

C3×C24 is a maximal subgroup of   C24.S3  C24⋊S3  C242S3  C325D8  C325Q16

72 conjugacy classes

class 1  2 3A···3H4A4B6A···6H8A8B8C8D12A···12P24A···24AF
order123···3446···6888812···1224···24
size111···1111···111111···11···1

72 irreducible representations

dim11111111
type++
imageC1C2C3C4C6C8C12C24
kernelC3×C24C3×C12C24C3×C6C12C32C6C3
# reps1182841632

Matrix representation of C3×C24 in GL3(𝔽73) generated by

6400
080
001
,
900
090
0063
G:=sub<GL(3,GF(73))| [64,0,0,0,8,0,0,0,1],[9,0,0,0,9,0,0,0,63] >;

C3×C24 in GAP, Magma, Sage, TeX

C_3\times C_{24}
% in TeX

G:=Group("C3xC24");
// GroupNames label

G:=SmallGroup(72,14);
// by ID

G=gap.SmallGroup(72,14);
# by ID

G:=PCGroup([5,-2,-3,-3,-2,-2,90,58]);
// Polycyclic

G:=Group<a,b|a^3=b^24=1,a*b=b*a>;
// generators/relations

Export

Subgroup lattice of C3×C24 in TeX

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