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## G = C9×C3⋊C8order 216 = 23·33

### Direct product of C9 and C3⋊C8

Series: Derived Chief Lower central Upper central

 Derived series C1 — C3 — C9×C3⋊C8
 Chief series C1 — C3 — C6 — C3×C6 — C3×C12 — C3×C36 — C9×C3⋊C8
 Lower central C3 — C9×C3⋊C8
 Upper central C1 — C36

Generators and relations for C9×C3⋊C8
G = < a,b,c | a9=b3=c8=1, ab=ba, ac=ca, cbc-1=b-1 >

Smallest permutation representation of C9×C3⋊C8
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 4 7)(2 5 8)(3 6 9)(10 16 13)(11 17 14)(12 18 15)(19 25 22)(20 26 23)(21 27 24)(28 31 34)(29 32 35)(30 33 36)(37 40 43)(38 41 44)(39 42 45)(46 49 52)(47 50 53)(48 51 54)(55 61 58)(56 62 59)(57 63 60)(64 70 67)(65 71 68)(66 72 69)
(1 22 52 67 31 63 42 18)(2 23 53 68 32 55 43 10)(3 24 54 69 33 56 44 11)(4 25 46 70 34 57 45 12)(5 26 47 71 35 58 37 13)(6 27 48 72 36 59 38 14)(7 19 49 64 28 60 39 15)(8 20 50 65 29 61 40 16)(9 21 51 66 30 62 41 17)

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

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

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

C9×C3⋊C8 is a maximal subgroup of
C36.38D6  C36.39D6  C36.40D6  D36.S3  C6.D36  C3⋊D72  C3⋊Dic36  S3×C72

108 conjugacy classes

 class 1 2 3A 3B 3C 3D 3E 4A 4B 6A 6B 6C 6D 6E 8A 8B 8C 8D 9A ··· 9F 9G ··· 9L 12A 12B 12C 12D 12E ··· 12J 18A ··· 18F 18G ··· 18L 24A ··· 24H 36A ··· 36L 36M ··· 36X 72A ··· 72X order 1 2 3 3 3 3 3 4 4 6 6 6 6 6 8 8 8 8 9 ··· 9 9 ··· 9 12 12 12 12 12 ··· 12 18 ··· 18 18 ··· 18 24 ··· 24 36 ··· 36 36 ··· 36 72 ··· 72 size 1 1 1 1 2 2 2 1 1 1 1 2 2 2 3 3 3 3 1 ··· 1 2 ··· 2 1 1 1 1 2 ··· 2 1 ··· 1 2 ··· 2 3 ··· 3 1 ··· 1 2 ··· 2 3 ··· 3

108 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 type + + + - image C1 C2 C3 C4 C6 C8 C9 C12 C18 C24 C36 C72 S3 Dic3 C3×S3 C3⋊C8 C3×Dic3 S3×C9 C3×C3⋊C8 C9×Dic3 C9×C3⋊C8 kernel C9×C3⋊C8 C3×C36 C3×C3⋊C8 C3×C18 C3×C12 C3×C9 C3⋊C8 C3×C6 C12 C32 C6 C3 C36 C18 C12 C9 C6 C4 C3 C2 C1 # reps 1 1 2 2 2 4 6 4 6 8 12 24 1 1 2 2 2 6 4 6 12

Matrix representation of C9×C3⋊C8 in GL2(𝔽37) generated by

 7 0 0 7
,
 26 0 0 10
,
 0 18 12 0
G:=sub<GL(2,GF(37))| [7,0,0,7],[26,0,0,10],[0,12,18,0] >;

C9×C3⋊C8 in GAP, Magma, Sage, TeX

C_9\times C_3\rtimes C_8
% in TeX

G:=Group("C9xC3:C8");
// GroupNames label

G:=SmallGroup(216,13);
// by ID

G=gap.SmallGroup(216,13);
# by ID

G:=PCGroup([6,-2,-3,-2,-3,-2,-3,36,79,122,5189]);
// Polycyclic

G:=Group<a,b,c|a^9=b^3=c^8=1,a*b=b*a,a*c=c*a,c*b*c^-1=b^-1>;
// generators/relations

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