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G = C36.C12order 432 = 24·33

1st non-split extension by C36 of C12 acting via C12/C2=C6

metacyclic, supersoluble, monomial

Aliases: C36.1C12, C62.4Dic3, 3- 1+2:2M4(2), C9:C8:5C6, C9:C24:5C2, C4.(C9:C12), C4.Dic9:C3, (C6xC12).6S3, (C2xC36).1C6, C12.95(S3xC6), C18.6(C2xC12), C36.16(C2xC6), (C2xC18).2C12, C22.(C9:C12), (C3xC12).63D6, C9:2(C3xM4(2)), C6.15(C6xDic3), C12.5(C3xDic3), (C3xC12).2Dic3, C32.(C4.Dic3), (C4x3- 1+2).1C4, (C22x3- 1+2).2C4, (C4x3- 1+2).15C22, C4.15(C2xC9:C6), C2.3(C2xC9:C12), (C2xC4).2(C9:C6), (C2xC12).24(C3xS3), C3.3(C3xC4.Dic3), (C2xC6).17(C3xDic3), (C3xC6).11(C2xDic3), (C2xC4x3- 1+2).1C2, (C2x3- 1+2).6(C2xC4), SmallGroup(432,143)

Series: Derived Chief Lower central Upper central

C1C18 — C36.C12
C1C3C9C18C36C4x3- 1+2C9:C24 — C36.C12
C9C18 — C36.C12
C1C4C2xC4

Generators and relations for C36.C12
 G = < a,b | a36=1, b12=a18, bab-1=a23 >

Subgroups: 158 in 68 conjugacy classes, 36 normal (32 characteristic)
C1, C2, C2, C3, C3, C4, C22, C6, C6, C8, C2xC4, C9, C9, C32, C12, C12, C2xC6, C2xC6, M4(2), C18, C18, C3xC6, C3xC6, C3:C8, C24, C2xC12, C2xC12, 3- 1+2, C36, C36, C2xC18, C2xC18, C3xC12, C62, C4.Dic3, C3xM4(2), C2x3- 1+2, C2x3- 1+2, C9:C8, C2xC36, C2xC36, C3xC3:C8, C6xC12, C4x3- 1+2, C22x3- 1+2, C4.Dic9, C3xC4.Dic3, C9:C24, C2xC4x3- 1+2, C36.C12
Quotients: C1, C2, C3, C4, C22, S3, C6, C2xC4, Dic3, C12, D6, C2xC6, M4(2), C3xS3, C2xDic3, C2xC12, C3xDic3, S3xC6, C4.Dic3, C3xM4(2), C9:C6, C6xDic3, C9:C12, C2xC9:C6, C3xC4.Dic3, C2xC9:C12, C36.C12

Smallest permutation representation of C36.C12
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)
(1 47 10 38 19 65 28 56)(2 58 23 37 8 52 29 67 14 46 35 61 20 40 5 55 26 70 11 49 32 64 17 43)(3 69 36 72 33 39 30 42 27 45 24 48 21 51 18 54 15 57 12 60 9 63 6 66)(4 44 13 71 22 62 31 53)(7 41 16 68 25 59 34 50)

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), (1,47,10,38,19,65,28,56)(2,58,23,37,8,52,29,67,14,46,35,61,20,40,5,55,26,70,11,49,32,64,17,43)(3,69,36,72,33,39,30,42,27,45,24,48,21,51,18,54,15,57,12,60,9,63,6,66)(4,44,13,71,22,62,31,53)(7,41,16,68,25,59,34,50)>;

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), (1,47,10,38,19,65,28,56)(2,58,23,37,8,52,29,67,14,46,35,61,20,40,5,55,26,70,11,49,32,64,17,43)(3,69,36,72,33,39,30,42,27,45,24,48,21,51,18,54,15,57,12,60,9,63,6,66)(4,44,13,71,22,62,31,53)(7,41,16,68,25,59,34,50) );

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)], [(1,47,10,38,19,65,28,56),(2,58,23,37,8,52,29,67,14,46,35,61,20,40,5,55,26,70,11,49,32,64,17,43),(3,69,36,72,33,39,30,42,27,45,24,48,21,51,18,54,15,57,12,60,9,63,6,66),(4,44,13,71,22,62,31,53),(7,41,16,68,25,59,34,50)]])

62 conjugacy classes

class 1 2A2B3A3B3C4A4B4C6A6B6C6D6E6F6G8A8B8C8D9A9B9C12A12B12C12D12E12F12G12H12I12J18A···18I24A···24H36A···36L
order122333444666666688889991212121212121212121218···1824···2436···36
size11223311222233661818181866622223333666···618···186···6

62 irreducible representations

dim111111111122222222222266666
type++++-+-+-+-
imageC1C2C2C3C4C4C6C6C12C12S3Dic3D6Dic3M4(2)C3xS3C3xDic3S3xC6C3xDic3C3xM4(2)C4.Dic3C3xC4.Dic3C9:C6C9:C12C2xC9:C6C9:C12C36.C12
kernelC36.C12C9:C24C2xC4x3- 1+2C4.Dic9C4x3- 1+2C22x3- 1+2C9:C8C2xC36C36C2xC18C6xC12C3xC12C3xC12C623- 1+2C2xC12C12C12C2xC6C9C32C3C2xC4C4C4C22C1
# reps121222424411112222244811114

Matrix representation of C36.C12 in GL6(F73)

70300000
0370000
2430000
2190003
22002400
21090240
,
20007100
002029600
5914029042
4005300
192900059
370680530

G:=sub<GL(6,GF(73))| [70,0,24,21,22,21,30,3,3,9,0,0,0,70,0,0,0,9,0,0,0,0,24,0,0,0,0,0,0,24,0,0,0,3,0,0],[20,0,59,4,19,37,0,0,14,0,29,0,0,20,0,0,0,68,71,29,29,53,0,0,0,60,0,0,0,53,0,0,42,0,59,0] >;

C36.C12 in GAP, Magma, Sage, TeX

C_{36}.C_{12}
% in TeX

G:=Group("C36.C12");
// GroupNames label

G:=SmallGroup(432,143);
// by ID

G=gap.SmallGroup(432,143);
# by ID

G:=PCGroup([7,-2,-2,-3,-2,-2,-3,-3,84,365,80,10085,2035,292,14118]);
// Polycyclic

G:=Group<a,b|a^36=1,b^12=a^18,b*a*b^-1=a^23>;
// generators/relations

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