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## G = C9×C6.D4order 432 = 24·33

### Direct product of C9 and C6.D4

Series: Derived Chief Lower central Upper central

 Derived series C1 — C6 — C9×C6.D4
 Chief series C1 — C3 — C32 — C3×C6 — C62 — C6×C18 — Dic3×C18 — C9×C6.D4
 Lower central C3 — C6 — C9×C6.D4
 Upper central C1 — C2×C18 — C22×C18

Generators and relations for C9×C6.D4
G = < a,b,c,d | a9=b6=c4=1, d2=b3, ab=ba, ac=ca, ad=da, cbc-1=dbd-1=b-1, dcd-1=b3c-1 >

Subgroups: 244 in 134 conjugacy classes, 57 normal (27 characteristic)
C1, C2, C2 [×2], C2 [×2], C3 [×2], C3, C4 [×2], C22, C22 [×2], C22 [×2], C6 [×2], C6 [×4], C6 [×11], C2×C4 [×2], C23, C9, C9, C32, Dic3 [×2], C12 [×2], C2×C6 [×2], C2×C6 [×4], C2×C6 [×11], C22⋊C4, C18, C18 [×2], C18 [×9], C3×C6, C3×C6 [×2], C3×C6 [×2], C2×Dic3 [×2], C2×C12 [×2], C22×C6 [×2], C22×C6, C3×C9, C36 [×2], C2×C18, C2×C18 [×2], C2×C18 [×9], C3×Dic3 [×2], C62, C62 [×2], C62 [×2], C6.D4, C3×C22⋊C4, C3×C18, C3×C18 [×2], C3×C18 [×2], C2×C36 [×2], C22×C18, C22×C18, C6×Dic3 [×2], C2×C62, C9×Dic3 [×2], C6×C18, C6×C18 [×2], C6×C18 [×2], C9×C22⋊C4, C3×C6.D4, Dic3×C18 [×2], C2×C6×C18, C9×C6.D4
Quotients: C1, C2 [×3], C3, C4 [×2], C22, S3, C6 [×3], C2×C4, D4 [×2], C9, Dic3 [×2], C12 [×2], D6, C2×C6, C22⋊C4, C18 [×3], C3×S3, C2×Dic3, C3⋊D4 [×2], C2×C12, C3×D4 [×2], C36 [×2], C2×C18, C3×Dic3 [×2], S3×C6, C6.D4, C3×C22⋊C4, S3×C9, C2×C36, D4×C9 [×2], C6×Dic3, C3×C3⋊D4 [×2], C9×Dic3 [×2], S3×C18, C9×C22⋊C4, C3×C6.D4, Dic3×C18, C9×C3⋊D4 [×2], C9×C6.D4

Smallest permutation representation of C9×C6.D4
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 50 7 47 4 53)(2 51 8 48 5 54)(3 52 9 49 6 46)(10 19 13 22 16 25)(11 20 14 23 17 26)(12 21 15 24 18 27)(28 41 34 38 31 44)(29 42 35 39 32 45)(30 43 36 40 33 37)(55 65 58 68 61 71)(56 66 59 69 62 72)(57 67 60 70 63 64)
(1 57 44 14)(2 58 45 15)(3 59 37 16)(4 60 38 17)(5 61 39 18)(6 62 40 10)(7 63 41 11)(8 55 42 12)(9 56 43 13)(19 49 72 36)(20 50 64 28)(21 51 65 29)(22 52 66 30)(23 53 67 31)(24 54 68 32)(25 46 69 33)(26 47 70 34)(27 48 71 35)
(1 26 47 14)(2 27 48 15)(3 19 49 16)(4 20 50 17)(5 21 51 18)(6 22 52 10)(7 23 53 11)(8 24 54 12)(9 25 46 13)(28 60 38 64)(29 61 39 65)(30 62 40 66)(31 63 41 67)(32 55 42 68)(33 56 43 69)(34 57 44 70)(35 58 45 71)(36 59 37 72)

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

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

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

162 conjugacy classes

 class 1 2A 2B 2C 2D 2E 3A 3B 3C 3D 3E 4A 4B 4C 4D 6A ··· 6F 6G ··· 6AE 9A ··· 9F 9G ··· 9L 12A ··· 12H 18A ··· 18R 18S ··· 18BT 36A ··· 36X order 1 2 2 2 2 2 3 3 3 3 3 4 4 4 4 6 ··· 6 6 ··· 6 9 ··· 9 9 ··· 9 12 ··· 12 18 ··· 18 18 ··· 18 36 ··· 36 size 1 1 1 1 2 2 1 1 2 2 2 6 6 6 6 1 ··· 1 2 ··· 2 1 ··· 1 2 ··· 2 6 ··· 6 1 ··· 1 2 ··· 2 6 ··· 6

162 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 type + + + + + - + image C1 C2 C2 C3 C4 C6 C6 C9 C12 C18 C18 C36 S3 D4 Dic3 D6 C3×S3 C3⋊D4 C3×D4 C3×Dic3 S3×C6 S3×C9 D4×C9 C3×C3⋊D4 C9×Dic3 S3×C18 C9×C3⋊D4 kernel C9×C6.D4 Dic3×C18 C2×C6×C18 C3×C6.D4 C6×C18 C6×Dic3 C2×C62 C6.D4 C62 C2×Dic3 C22×C6 C2×C6 C22×C18 C3×C18 C2×C18 C2×C18 C22×C6 C18 C3×C6 C2×C6 C2×C6 C23 C6 C6 C22 C22 C2 # reps 1 2 1 2 4 4 2 6 8 12 6 24 1 2 2 1 2 4 4 4 2 6 12 8 12 6 24

Matrix representation of C9×C6.D4 in GL4(𝔽37) generated by

 34 0 0 0 0 34 0 0 0 0 9 0 0 0 0 9
,
 27 0 0 0 0 11 0 0 0 0 11 0 0 0 0 27
,
 0 1 0 0 36 0 0 0 0 0 0 1 0 0 1 0
,
 0 1 0 0 36 0 0 0 0 0 0 1 0 0 36 0
G:=sub<GL(4,GF(37))| [34,0,0,0,0,34,0,0,0,0,9,0,0,0,0,9],[27,0,0,0,0,11,0,0,0,0,11,0,0,0,0,27],[0,36,0,0,1,0,0,0,0,0,0,1,0,0,1,0],[0,36,0,0,1,0,0,0,0,0,0,36,0,0,1,0] >;

C9×C6.D4 in GAP, Magma, Sage, TeX

C_9\times C_6.D_4
% in TeX

G:=Group("C9xC6.D4");
// GroupNames label

G:=SmallGroup(432,165);
// by ID

G=gap.SmallGroup(432,165);
# by ID

G:=PCGroup([7,-2,-2,-3,-2,-2,-3,-3,84,365,268,14118]);
// Polycyclic

G:=Group<a,b,c,d|a^9=b^6=c^4=1,d^2=b^3,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c^-1=d*b*d^-1=b^-1,d*c*d^-1=b^3*c^-1>;
// generators/relations

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