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G = C9⋊Dic6order 216 = 23·33

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

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

 Derived series C1 — C3×C18 — C9⋊Dic6
 Chief series C1 — C3 — C32 — C3×C9 — C3×C18 — C9×Dic3 — C9⋊Dic6
 Lower central C3×C9 — C3×C18 — C9⋊Dic6
 Upper central C1 — C2

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 >

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

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

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

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

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 3A 3B 3C 4A 4B 4C 6A 6B 6C 9A 9B 9C 9D 9E 9F 12A 12B 12C 12D 18A 18B 18C 18D 18E 18F 36A ··· 36F order 1 2 3 3 3 4 4 4 6 6 6 9 9 9 9 9 9 12 12 12 12 18 18 18 18 18 18 36 ··· 36 size 1 1 2 2 4 6 18 54 2 2 4 2 2 2 4 4 4 6 6 18 18 2 2 2 4 4 4 6 ··· 6

33 irreducible representations

 dim 1 1 1 1 2 2 2 2 2 2 2 2 2 2 4 4 4 4 type + + + + + + - + + + - - + - + - + - image C1 C2 C2 C2 S3 S3 Q8 D6 D6 D9 Dic6 Dic6 D18 Dic18 S32 C32⋊2Q8 S3×D9 C9⋊Dic6 kernel C9⋊Dic6 C3×Dic9 C9×Dic3 C9⋊Dic3 Dic9 C3×Dic3 C3×C9 C18 C3×C6 Dic3 C9 C32 C6 C3 C6 C3 C2 C1 # reps 1 1 1 1 1 1 1 1 1 3 2 2 3 6 1 1 3 3

Matrix representation of C9⋊Dic6 in GL6(𝔽37)

 1 0 0 0 0 0 0 1 0 0 0 0 0 0 17 26 0 0 0 0 11 6 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 0 1 0 0 0 0 36 0 0 0 0 0 0 0 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 36 0 0 0 0 1 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

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

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