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## G = C3×C24⋊C2order 144 = 24·32

### Direct product of C3 and C24⋊C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C12 — C3×C24⋊C2
 Chief series C1 — C3 — C6 — C12 — C3×C12 — C3×D12 — C3×C24⋊C2
 Lower central C3 — C6 — C12 — C3×C24⋊C2
 Upper central C1 — C6 — C12 — C24

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

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 2A 2B 3A 3B 3C 3D 3E 4A 4B 6A 6B 6C 6D 6E 6F 6G 8A 8B 12A ··· 12H 12I 12J 24A ··· 24P order 1 2 2 3 3 3 3 3 4 4 6 6 6 6 6 6 6 8 8 12 ··· 12 12 12 24 ··· 24 size 1 1 12 1 1 2 2 2 2 12 1 1 2 2 2 12 12 2 2 2 ··· 2 12 12 2 ··· 2

45 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 type + + + + + + + + image C1 C2 C2 C2 C3 C6 C6 C6 S3 D4 D6 SD16 C3×S3 D12 C3×D4 S3×C6 C24⋊C2 C3×SD16 C3×D12 C3×C24⋊C2 kernel C3×C24⋊C2 C3×C24 C3×Dic6 C3×D12 C24⋊C2 C24 Dic6 D12 C24 C3×C6 C12 C32 C8 C6 C6 C4 C3 C3 C2 C1 # reps 1 1 1 1 2 2 2 2 1 1 1 2 2 2 2 2 4 4 4 8

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

 64 0 0 64
,
 7 0 0 52
,
 0 1 1 0
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

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