Copied to
clipboard

G = C3×D24order 144 = 24·32

Direct product of C3 and D24

direct product, metacyclic, supersoluble, monomial

Aliases: C3×D24, C241C6, C243S3, C324D8, D121C6, C12.61D6, C6.20D12, C81(C3×S3), C31(C3×D8), (C3×C24)⋊2C2, C4.9(S3×C6), C6.2(C3×D4), C12.9(C2×C6), C2.4(C3×D12), (C3×C6).18D4, (C3×D12)⋊10C2, (C3×C12).38C22, SmallGroup(144,72)

Series: Derived Chief Lower central Upper central

C1C12 — C3×D24
C1C3C6C12C3×C12C3×D12 — C3×D24
C3C6C12 — C3×D24
C1C6C12C24

Generators and relations for C3×D24
 G = < a,b,c | a3=b24=c2=1, ab=ba, ac=ca, cbc=b-1 >

12C2
12C2
2C3
6C22
6C22
2C6
4S3
4S3
12C6
12C6
3D4
3D4
2D6
2C12
2D6
6C2×C6
6C2×C6
4C3×S3
4C3×S3
3D8
2C24
3C3×D4
3C3×D4
2S3×C6
2S3×C6
3C3×D8

Smallest permutation representation of C3×D24
On 48 points
Generators in S48
(1 9 17)(2 10 18)(3 11 19)(4 12 20)(5 13 21)(6 14 22)(7 15 23)(8 16 24)(25 41 33)(26 42 34)(27 43 35)(28 44 36)(29 45 37)(30 46 38)(31 47 39)(32 48 40)
(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)
(1 39)(2 38)(3 37)(4 36)(5 35)(6 34)(7 33)(8 32)(9 31)(10 30)(11 29)(12 28)(13 27)(14 26)(15 25)(16 48)(17 47)(18 46)(19 45)(20 44)(21 43)(22 42)(23 41)(24 40)

G:=sub<Sym(48)| (1,9,17)(2,10,18)(3,11,19)(4,12,20)(5,13,21)(6,14,22)(7,15,23)(8,16,24)(25,41,33)(26,42,34)(27,43,35)(28,44,36)(29,45,37)(30,46,38)(31,47,39)(32,48,40), (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), (1,39)(2,38)(3,37)(4,36)(5,35)(6,34)(7,33)(8,32)(9,31)(10,30)(11,29)(12,28)(13,27)(14,26)(15,25)(16,48)(17,47)(18,46)(19,45)(20,44)(21,43)(22,42)(23,41)(24,40)>;

G:=Group( (1,9,17)(2,10,18)(3,11,19)(4,12,20)(5,13,21)(6,14,22)(7,15,23)(8,16,24)(25,41,33)(26,42,34)(27,43,35)(28,44,36)(29,45,37)(30,46,38)(31,47,39)(32,48,40), (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), (1,39)(2,38)(3,37)(4,36)(5,35)(6,34)(7,33)(8,32)(9,31)(10,30)(11,29)(12,28)(13,27)(14,26)(15,25)(16,48)(17,47)(18,46)(19,45)(20,44)(21,43)(22,42)(23,41)(24,40) );

G=PermutationGroup([(1,9,17),(2,10,18),(3,11,19),(4,12,20),(5,13,21),(6,14,22),(7,15,23),(8,16,24),(25,41,33),(26,42,34),(27,43,35),(28,44,36),(29,45,37),(30,46,38),(31,47,39),(32,48,40)], [(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)], [(1,39),(2,38),(3,37),(4,36),(5,35),(6,34),(7,33),(8,32),(9,31),(10,30),(11,29),(12,28),(13,27),(14,26),(15,25),(16,48),(17,47),(18,46),(19,45),(20,44),(21,43),(22,42),(23,41),(24,40)])

C3×D24 is a maximal subgroup of
C322D16  C3⋊D48  D24.S3  C323SD32  D24⋊S3  C244D6  C246D6  D247S3  D245S3  C3×S3×D8  He34D8  D72⋊C3  He35D8
C3×D24 is a maximal quotient of
He34D8  D72⋊C3

45 conjugacy classes

class 1 2A2B2C3A3B3C3D3E 4 6A6B6C6D6E6F6G6H6I8A8B12A···12H24A···24P
order12223333346666666668812···1224···24
size1112121122221122212121212222···22···2

45 irreducible representations

dim111111222222222222
type+++++++++
imageC1C2C2C3C6C6S3D4D6D8C3×S3D12C3×D4S3×C6D24C3×D8C3×D12C3×D24
kernelC3×D24C3×C24C3×D12D24C24D12C24C3×C6C12C32C8C6C6C4C3C3C2C1
# reps112224111222224448

Matrix representation of C3×D24 in GL2(𝔽73) generated by

640
064
,
3012
056
,
3012
6543
G:=sub<GL(2,GF(73))| [64,0,0,64],[30,0,12,56],[30,65,12,43] >;

C3×D24 in GAP, Magma, Sage, TeX

C_3\times D_{24}
% in TeX

G:=Group("C3xD24");
// GroupNames label

G:=SmallGroup(144,72);
// by ID

G=gap.SmallGroup(144,72);
# by ID

G:=PCGroup([6,-2,-2,-3,-2,-2,-3,169,223,867,69,3461]);
// Polycyclic

G:=Group<a,b,c|a^3=b^24=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^-1>;
// generators/relations

Export

Subgroup lattice of C3×D24 in TeX

׿
×
𝔽