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G = C72:2C6  order 432 = 24·33

2nd semidirect product of C72 and C6 acting faithfully

metacyclic, supersoluble, monomial

Aliases: C72:2C6, D36.2C6, Dic18:4C6, 3- 1+2:1SD16, C72:C2:C3, C8:2(C9:C6), C36.9(C2xC6), (C3xC24).5S3, C9:1(C3xSD16), C18.2(C3xD4), C6.8(C3xD12), C12.71(S3xC6), C24.10(C3xS3), D36:C3.2C2, (C3xC6).17D12, (C3xC12).44D6, C36.C6:4C2, C2.4(D36:C3), C32.(C24:C2), (C8x3- 1+2):2C2, (C2x3- 1+2).2D4, (C4x3- 1+2).9C22, C4.9(C2xC9:C6), C3.3(C3xC24:C2), SmallGroup(432,122)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C36 — C72:2C6
Chief series
C1 — C3 — C9 — C18 — C36 — C4x3- 1+2 — D36:C3 — C72:2C6
Lower central
C9 — C18 — C36 — C72:2C6
Upper central
C1 — C2 — C4 — C8

Generators and relations for C72:2C6
 G = < a,b | a72=b6=1, bab-1=a11 >

Subgroups: 326 in 64 conjugacy classes, 26 normal (all characteristic)
C1, C2, C2, C3, C3, C4, C4, C22, S3, C6, C6, C8, D4, Q8, C9, C9, C32, Dic3, C12, C12, D6, C2xC6, SD16, D9, C18, C18, C3xS3, C3xC6, C24, C24, Dic6, D12, C3xD4, C3xQ8, 3- 1+2, Dic9, C36, C36, D18, C3xDic3, C3xC12, S3xC6, C24:C2, C3xSD16, C9:C6, C2x3- 1+2, C72, C72, Dic18, D36, C3xC24, C3xDic6, C3xD12, C9:C12, C4x3- 1+2, C2xC9:C6, C72:C2, C3xC24:C2, C8x3- 1+2, C36.C6, D36:C3, C72:2C6
Quotients: C1, C2, C3, C22, S3, C6, D4, D6, C2xC6, SD16, C3xS3, D12, C3xD4, S3xC6, C24:C2, C3xSD16, C9:C6, C3xD12, C2xC9:C6, C3xC24:C2, D36:C3, C72:2C6

Smallest permutation representation of C72:2C6
►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 60 26 36 50 12)(3 47 51 71 27 23)(4 34)(5 21 29 69 53 45)(6 8 54 32 30 56)(7 67)(9 41 57 65 33 17)(10 28)(11 15 35 63 59 39)(13 61)(14 48 38 24 62 72)(16 22)(18 68 66 20 42 44)(19 55)(25 49)(31 43)(40 70)(46 64)(52 58)
 
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), (2,60,26,36,50,12)(3,47,51,71,27,23)(4,34)(5,21,29,69,53,45)(6,8,54,32,30,56)(7,67)(9,41,57,65,33,17)(10,28)(11,15,35,63,59,39)(13,61)(14,48,38,24,62,72)(16,22)(18,68,66,20,42,44)(19,55)(25,49)(31,43)(40,70)(46,64)(52,58)>;
 
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), (2,60,26,36,50,12)(3,47,51,71,27,23)(4,34)(5,21,29,69,53,45)(6,8,54,32,30,56)(7,67)(9,41,57,65,33,17)(10,28)(11,15,35,63,59,39)(13,61)(14,48,38,24,62,72)(16,22)(18,68,66,20,42,44)(19,55)(25,49)(31,43)(40,70)(46,64)(52,58) );
 
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)], [(2,60,26,36,50,12),(3,47,51,71,27,23),(4,34),(5,21,29,69,53,45),(6,8,54,32,30,56),(7,67),(9,41,57,65,33,17),(10,28),(11,15,35,63,59,39),(13,61),(14,48,38,24,62,72),(16,22),(18,68,66,20,42,44),(19,55),(25,49),(31,43),(40,70),(46,64),(52,58)]])
 

53 conjugacy classes

class 1 2A2B3A3B3C4A4B6A6B6C6D6E8A8B9A9B9C12A12B12C12D12E12F18A18B18C24A24B24C24D24E24F24G24H36A···36F72A···72L
order122333446666688999121212121212181818242424242424242436···3672···72
size113623323623336362266622663636666222266666···66···6

53 irreducible representations

dim111111112222222222226666
type+++++++++++
imageC1C2C2C2C3C6C6C6S3D4D6SD16C3xS3C3xD4D12S3xC6C3xSD16C24:C2C3xD12C3xC24:C2C9:C6C2xC9:C6D36:C3C72:2C6
kernelC72:2C6C8x3- 1+2C36.C6D36:C3C72:C2C72Dic18D36C3xC24C2x3- 1+2C3xC123- 1+2C24C18C3xC6C12C9C32C6C3C8C4C2C1
# reps111122221112222244481124

Matrix representation of C72:2C6 ►in GL8(F73)

1125000000
4836000000
000000714
0000005966
007660000
007140000
000076600
000071400
,
09000000
90000000
00100000
0072720000
00000010
0000007272
0000727200
00000100

G:=sub<GL(8,GF(73))| [11,48,0,0,0,0,0,0,25,36,0,0,0,0,0,0,0,0,0,0,7,7,0,0,0,0,0,0,66,14,0,0,0,0,0,0,0,0,7,7,0,0,0,0,0,0,66,14,0,0,7,59,0,0,0,0,0,0,14,66,0,0,0,0],[0,9,0,0,0,0,0,0,9,0,0,0,0,0,0,0,0,0,1,72,0,0,0,0,0,0,0,72,0,0,0,0,0,0,0,0,0,0,72,0,0,0,0,0,0,0,72,1,0,0,0,0,1,72,0,0,0,0,0,0,0,72,0,0] >;
 

C72:2C6 in GAP, Magma, Sage, TeX 

C_{72}\rtimes_2C_6
 
% in TeX
 
G:=Group("C72:2C6");
 
// GroupNames label
 
G:=SmallGroup(432,122);
 
// by ID
 
G=gap.SmallGroup(432,122);
 
# by ID
 
G:=PCGroup([7,-2,-2,-3,-2,-2,-3,-3,197,92,1011,80,10085,2035,292,14118]);
 
// Polycyclic
 
G:=Group<a,b|a^72=b^6=1,b*a*b^-1=a^11>;
 
// generators/relations
 

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