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G = C9⋊Dic6order 216 = 23·33

The semidirect product of C9 and Dic6 acting via Dic6/Dic3=C2

metabelian, supersoluble, monomial

Aliases: C91Dic6, C6.1D18, C18.1D6, Dic9.S3, Dic3.D9, C31Dic18, C32.2Dic6, (C3×C9)⋊Q8, C6.1S32, C2.4(S3×D9), (C3×C6).22D6, C9⋊Dic3.1C2, (C3×C18).1C22, (C3×Dic9).1C2, (C3×Dic3).1S3, (C9×Dic3).1C2, C3.1(C322Q8), SmallGroup(216,26)

Series: Derived Chief Lower central Upper central

C1C3×C18 — C9⋊Dic6
C1C3C32C3×C9C3×C18C9×Dic3 — C9⋊Dic6
C3×C9C3×C18 — C9⋊Dic6
C1C2

Generators and relations for C9⋊Dic6
 G = < a,b,c | a9=b12=1, c2=b6, bab-1=a-1, ac=ca, cbc-1=b-1 >

2C3
3C4
9C4
27C4
2C6
2C9
27Q8
3C12
3Dic3
9Dic3
9Dic3
9C12
18Dic3
2C18
9Dic6
9Dic6
3C3×Dic3
3Dic9
3C3⋊Dic3
3C36
6Dic9
3C322Q8
3Dic18

Smallest permutation representation of C9⋊Dic6
On 72 points
Generators in S72
(1 35 60 5 27 52 9 31 56)(2 57 32 10 53 28 6 49 36)(3 25 50 7 29 54 11 33 58)(4 59 34 12 55 30 8 51 26)(13 40 68 17 44 72 21 48 64)(14 65 37 22 61 45 18 69 41)(15 42 70 19 46 62 23 38 66)(16 67 39 24 63 47 20 71 43)
(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 20 7 14)(2 19 8 13)(3 18 9 24)(4 17 10 23)(5 16 11 22)(6 15 12 21)(25 69 31 63)(26 68 32 62)(27 67 33 61)(28 66 34 72)(29 65 35 71)(30 64 36 70)(37 60 43 54)(38 59 44 53)(39 58 45 52)(40 57 46 51)(41 56 47 50)(42 55 48 49)

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

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

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

C9⋊Dic6 is a maximal subgroup of   D9×Dic6  Dic18⋊S3  S3×Dic18  D6.D18  D18.3D6  Dic3.D18  D18.4D6
C9⋊Dic6 is a maximal quotient of   Dic9⋊Dic3  C18.Dic6  Dic3⋊Dic9

33 conjugacy classes

class 1  2 3A3B3C4A4B4C6A6B6C9A9B9C9D9E9F12A12B12C12D18A18B18C18D18E18F36A···36F
order123334446669999991212121218181818181836···36
size11224618542242224446618182224446···6

33 irreducible representations

dim111122222222224444
type++++++-+++--+-+-+-
imageC1C2C2C2S3S3Q8D6D6D9Dic6Dic6D18Dic18S32C322Q8S3×D9C9⋊Dic6
kernelC9⋊Dic6C3×Dic9C9×Dic3C9⋊Dic3Dic9C3×Dic3C3×C9C18C3×C6Dic3C9C32C6C3C6C3C2C1
# reps111111111322361133

Matrix representation of C9⋊Dic6 in GL6(𝔽37)

100000
010000
00172600
0011600
000010
000001
,
010000
3600000
000100
001000
0000036
0000136
,
15120000
12220000
0036000
0003600
000001
000010

G:=sub<GL(6,GF(37))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,17,11,0,0,0,0,26,6,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[0,36,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,36,36],[15,12,0,0,0,0,12,22,0,0,0,0,0,0,36,0,0,0,0,0,0,36,0,0,0,0,0,0,0,1,0,0,0,0,1,0] >;

C9⋊Dic6 in GAP, Magma, Sage, TeX

C_9\rtimes {\rm Dic}_6
% in TeX

G:=Group("C9:Dic6");
// GroupNames label

G:=SmallGroup(216,26);
// by ID

G=gap.SmallGroup(216,26);
# by ID

G:=PCGroup([6,-2,-2,-2,-3,-3,-3,24,73,31,1065,453,1444,2603]);
// Polycyclic

G:=Group<a,b,c|a^9=b^12=1,c^2=b^6,b*a*b^-1=a^-1,a*c=c*a,c*b*c^-1=b^-1>;
// generators/relations

Export

Subgroup lattice of C9⋊Dic6 in TeX

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