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G = C36.C6order 216 = 23·33

1st non-split extension by C36 of C6 acting faithfully

metacyclic, supersoluble, monomial

Aliases: C36.1C6, Dic18⋊C3, Dic9.C6, 3- 1+2⋊Q8, C32.Dic6, C9⋊(C3×Q8), C9⋊C12.C2, C4.(C9⋊C6), (C3×C6).9D6, C12.5(C3×S3), C6.12(S3×C6), C18.1(C2×C6), (C3×C12).2S3, C3.3(C3×Dic6), (C4×3- 1+2).1C2, (C2×3- 1+2).1C22, C2.3(C2×C9⋊C6), SmallGroup(216,52)

Series: Derived Chief Lower central Upper central

C1C18 — C36.C6
C1C3C9C18C2×3- 1+2C9⋊C12 — C36.C6
C9C18 — C36.C6
C1C2C4

Generators and relations for C36.C6
 G = < a,b | a36=1, b6=a18, bab-1=a11 >

3C3
9C4
9C4
3C6
2C9
9Q8
3Dic3
3C12
3Dic3
9C12
9C12
2C18
3Dic6
9C3×Q8
2C36
3C3×Dic3
3C3×Dic3
3C3×Dic6

Smallest permutation representation of C36.C6
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 46 19 64)(2 69 8 63 14 57 20 51 26 45 32 39)(3 56 33 62 27 68 21 38 15 44 9 50)(4 43 22 61)(5 66 11 60 17 54 23 48 29 42 35 72)(6 53 36 59 30 65 24 71 18 41 12 47)(7 40 25 58)(10 37 28 55)(13 70 31 52)(16 67 34 49)

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

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

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

C36.C6 is a maximal subgroup of   C72.C6  C722C6  Dic18⋊C6  Dic18.C6  D366C6  Dic182C6  Q8×C9⋊C6
C36.C6 is a maximal quotient of   Dic9⋊C12  C36⋊C12

31 conjugacy classes

class 1  2 3A3B3C4A4B4C6A6B6C9A9B9C12A12B12C12D12E12F12G12H18A18B18C36A···36F
order12333444666999121212121212121218181836···36
size11233218182336662266181818186666···6

31 irreducible representations

dim11111122222222666
type++++-+-++-
imageC1C2C2C3C6C6S3Q8D6C3×S3C3×Q8Dic6S3×C6C3×Dic6C9⋊C6C2×C9⋊C6C36.C6
kernelC36.C6C9⋊C12C4×3- 1+2Dic18Dic9C36C3×C123- 1+2C3×C6C12C9C32C6C3C4C2C1
# reps12124211122224112

Matrix representation of C36.C6 in GL8(𝔽37)

01000000
360000000
0011353600
0011363500
0000363611
00113636360
0001363600
0010363600
,
08000000
80000000
004250000
0029330000
0040002533
00292500812
00292581200
0003342900

G:=sub<GL(8,GF(37))| [0,36,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,1,0,1,0,1,0,0,1,1,0,1,1,0,0,0,35,36,36,36,36,36,0,0,36,35,36,36,36,36,0,0,0,0,1,36,0,0,0,0,0,0,1,0,0,0],[0,8,0,0,0,0,0,0,8,0,0,0,0,0,0,0,0,0,4,29,4,29,29,0,0,0,25,33,0,25,25,33,0,0,0,0,0,0,8,4,0,0,0,0,0,0,12,29,0,0,0,0,25,8,0,0,0,0,0,0,33,12,0,0] >;

C36.C6 in GAP, Magma, Sage, TeX

C_{36}.C_6
% in TeX

G:=Group("C36.C6");
// GroupNames label

G:=SmallGroup(216,52);
// by ID

G=gap.SmallGroup(216,52);
# by ID

G:=PCGroup([6,-2,-2,-3,-2,-3,-3,72,169,79,3604,736,208,5189]);
// Polycyclic

G:=Group<a,b|a^36=1,b^6=a^18,b*a*b^-1=a^11>;
// generators/relations

Export

Subgroup lattice of C36.C6 in TeX

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