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G = C72⋊C2  order 144 = 24·32

2nd semidirect product of C72 and C2 acting faithfully

metacyclic, supersoluble, monomial, 2-hyperelementary

Aliases: C8⋊2D9, C72⋊2C2, C9⋊1SD16, C24.4S3, C18.2D4, C2.4D36, C4.9D18, C6.2D12, D36.1C2, C12.41D6, Dic18⋊1C2, C36.9C22, C3.(C24⋊C2), SmallGroup(144,7)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C36 — C72⋊C2
Chief series
C1 — C3 — C9 — C18 — C36 — D36 — C72⋊C2
Lower central
C9 — C18 — C36 — C72⋊C2
Upper central
C1 — C2 — C4 — C8

Generators and relations for C72⋊C2
 G = < a,b | a72=b2=1, bab=a35 >

C1
C2
36C2
C3
C4
18C4
18C22
C6
12S3
C9
C8
9Q8
9D4
C12
6Dic3
6D6
C18
4D9
9SD16
C24
3Dic6
3D12
C36
2Dic9
2D18
3C24⋊C2
Dic18
C72
D36
C72⋊C2

Smallest permutation representation of C72⋊C2
►On 72 points
Generators in S72
(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)

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

C72⋊C2 is a maximal subgroup of
 D72⋊7C2  C8⋊D18  C8.D18  D8⋊D9  SD16×D9  SD16⋊3D9  Q16⋊D9  C216⋊C2  D36.S3  C6.D36  C72⋊2C6  C24⋊D9
C72⋊C2 is a maximal quotient of
 C36.45D4  C8⋊Dic9  C2.D72  C216⋊C2  D36.S3  C6.D36  C24⋊D9

39 conjugacy classes

class 1 2A2B 3 4A4B 6 8A8B9A9B9C12A12B18A18B18C24A24B24C24D36A···36F72A···72L
order12234468899912121818182424242436···3672···72
size113622362222222222222222···22···2

39 irreducible representations

dim11112222222222
type+++++++++++
imageC1C2C2C2S3D4D6SD16D9D12D18C24⋊C2D36C72⋊C2
kernelC72⋊C2C72Dic18D36C24C18C12C9C8C6C4C3C2C1
# reps111111123234612

Matrix representation of C72⋊C2 ►in GL2(𝔽73) generated by

6814
5954
,
01
10
G:=sub<GL(2,GF(73))| [68,59,14,54],[0,1,1,0] >;
 

C72⋊C2 in GAP, Magma, Sage, TeX 

C_{72}\rtimes C_2
 
% in TeX
 
G:=Group("C72:C2");
 
// GroupNames label
 
G:=SmallGroup(144,7);
 
// by ID
 
G=gap.SmallGroup(144,7);
 
# by ID
 
G:=PCGroup([6,-2,-2,-2,-2,-3,-3,73,31,218,50,2404,208,3461]);
 
// Polycyclic
 
G:=Group<a,b|a^72=b^2=1,b*a*b=a^35>;
 
// generators/relations
 

Export

Subgroup lattice of C72⋊C2 in TeX

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