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G = C3×C24⋊C2order 144 = 24·32

Direct product of C3 and C24⋊C2

direct product, metacyclic, supersoluble, monomial

Aliases: C3×C24⋊C2, C242C6, C246S3, Dic61C6, D12.1C6, C6.19D12, C12.60D6, C326SD16, C82(C3×S3), (C3×C24)⋊4C2, C4.8(S3×C6), C6.1(C3×D4), C12.8(C2×C6), C31(C3×SD16), (C3×C6).17D4, C2.3(C3×D12), (C3×D12).4C2, (C3×Dic6)⋊10C2, (C3×C12).37C22, SmallGroup(144,71)

Series: Derived Chief Lower central Upper central

C1C12 — C3×C24⋊C2
C1C3C6C12C3×C12C3×D12 — C3×C24⋊C2
C3C6C12 — C3×C24⋊C2
C1C6C12C24

Generators and relations for C3×C24⋊C2
 G = < a,b,c | a3=b24=c2=1, ab=ba, ac=ca, cbc=b11 >

12C2
2C3
6C22
6C4
2C6
4S3
12C6
3Q8
3D4
2C12
2Dic3
2D6
6C2×C6
6C12
4C3×S3
3SD16
2C24
3C3×D4
3C3×Q8
2S3×C6
2C3×Dic3
3C3×SD16

Smallest permutation representation of C3×C24⋊C2
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 31)(2 42)(3 29)(4 40)(5 27)(6 38)(7 25)(8 36)(9 47)(10 34)(11 45)(12 32)(13 43)(14 30)(15 41)(16 28)(17 39)(18 26)(19 37)(20 48)(21 35)(22 46)(23 33)(24 44)

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

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

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

C3×C24⋊C2 is a maximal subgroup of
C241D6  C249D6  C246D6  C24.3D6  D6.1D12  D12.2D6  D12.4D6  C3×S3×SD16  He36SD16  C722C6  He37SD16
C3×C24⋊C2 is a maximal quotient of
He36SD16  C722C6

45 conjugacy classes

class 1 2A2B3A3B3C3D3E4A4B6A6B6C6D6E6F6G8A8B12A···12H12I12J24A···24P
order122333334466666668812···12121224···24
size111211222212112221212222···212122···2

45 irreducible representations

dim11111111222222222222
type++++++++
imageC1C2C2C2C3C6C6C6S3D4D6SD16C3×S3D12C3×D4S3×C6C24⋊C2C3×SD16C3×D12C3×C24⋊C2
kernelC3×C24⋊C2C3×C24C3×Dic6C3×D12C24⋊C2C24Dic6D12C24C3×C6C12C32C8C6C6C4C3C3C2C1
# reps11112222111222224448

Matrix representation of C3×C24⋊C2 in GL2(𝔽73) generated by

640
064
,
70
052
,
01
10
G:=sub<GL(2,GF(73))| [64,0,0,64],[7,0,0,52],[0,1,1,0] >;

C3×C24⋊C2 in GAP, Magma, Sage, TeX

C_3\times C_{24}\rtimes C_2
% in TeX

G:=Group("C3xC24:C2");
// GroupNames label

G:=SmallGroup(144,71);
// by ID

G=gap.SmallGroup(144,71);
# by ID

G:=PCGroup([6,-2,-2,-3,-2,-2,-3,169,79,867,69,3461]);
// 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^11>;
// generators/relations

Export

Subgroup lattice of C3×C24⋊C2 in TeX

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