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## G = C9×C4.D4order 288 = 25·32

### Direct product of C9 and C4.D4

direct product, metabelian, nilpotent (class 3), monomial, 2-elementary

Series: Derived Chief Lower central Upper central

 Derived series C1 — C22 — C9×C4.D4
 Chief series C1 — C2 — C6 — C12 — C2×C12 — C2×C36 — C9×M4(2) — C9×C4.D4
 Lower central C1 — C2 — C22 — C9×C4.D4
 Upper central C1 — C18 — C2×C36 — C9×C4.D4

Generators and relations for C9×C4.D4
G = < a,b,c,d | a9=b4=1, c4=b2, d2=b, ab=ba, ac=ca, ad=da, cbc-1=b-1, bd=db, dcd-1=b-1c3 >

Subgroups: 126 in 69 conjugacy classes, 36 normal (18 characteristic)
C1, C2, C2, C3, C4, C22, C22, C6, C6, C8, C2×C4, D4, C23, C9, C12, C2×C6, C2×C6, M4(2), C2×D4, C18, C18, C24, C2×C12, C3×D4, C22×C6, C4.D4, C36, C2×C18, C2×C18, C3×M4(2), C6×D4, C72, C2×C36, D4×C9, C22×C18, C3×C4.D4, C9×M4(2), D4×C18, C9×C4.D4
Quotients: C1, C2, C3, C4, C22, C6, C2×C4, D4, C9, C12, C2×C6, C22⋊C4, C18, C2×C12, C3×D4, C4.D4, C36, C2×C18, C3×C22⋊C4, C2×C36, D4×C9, C3×C4.D4, C9×C22⋊C4, C9×C4.D4

Smallest permutation representation of C9×C4.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 59 42 53)(2 60 43 54)(3 61 44 46)(4 62 45 47)(5 63 37 48)(6 55 38 49)(7 56 39 50)(8 57 40 51)(9 58 41 52)(10 34 20 72)(11 35 21 64)(12 36 22 65)(13 28 23 66)(14 29 24 67)(15 30 25 68)(16 31 26 69)(17 32 27 70)(18 33 19 71)
(1 66 59 23 42 28 53 13)(2 67 60 24 43 29 54 14)(3 68 61 25 44 30 46 15)(4 69 62 26 45 31 47 16)(5 70 63 27 37 32 48 17)(6 71 55 19 38 33 49 18)(7 72 56 20 39 34 50 10)(8 64 57 21 40 35 51 11)(9 65 58 22 41 36 52 12)
(1 28 59 23 42 66 53 13)(2 29 60 24 43 67 54 14)(3 30 61 25 44 68 46 15)(4 31 62 26 45 69 47 16)(5 32 63 27 37 70 48 17)(6 33 55 19 38 71 49 18)(7 34 56 20 39 72 50 10)(8 35 57 21 40 64 51 11)(9 36 58 22 41 65 52 12)

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

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

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

99 conjugacy classes

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

99 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 4 4 4 type + + + + + image C1 C2 C2 C3 C4 C6 C6 C9 C12 C18 C18 C36 D4 C3×D4 D4×C9 C4.D4 C3×C4.D4 C9×C4.D4 kernel C9×C4.D4 C9×M4(2) D4×C18 C3×C4.D4 C22×C18 C3×M4(2) C6×D4 C4.D4 C22×C6 M4(2) C2×D4 C23 C36 C12 C4 C9 C3 C1 # reps 1 2 1 2 4 4 2 6 8 12 6 24 2 4 12 1 2 6

Matrix representation of C9×C4.D4 in GL6(𝔽73)

 4 0 0 0 0 0 0 4 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 72 71 0 0 0 0 1 1 0 0 0 0 0 46 0 1 0 0 27 27 72 0
,
 0 1 0 0 0 0 72 0 0 0 0 0 0 0 27 0 71 0 0 0 46 0 1 72 0 0 0 1 46 0 0 0 0 0 27 0
,
 0 1 0 0 0 0 1 0 0 0 0 0 0 0 27 0 71 0 0 0 0 0 1 1 0 0 0 1 46 0 0 0 1 0 27 0

G:=sub<GL(6,GF(73))| [4,0,0,0,0,0,0,4,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,72,1,0,27,0,0,71,1,46,27,0,0,0,0,0,72,0,0,0,0,1,0],[0,72,0,0,0,0,1,0,0,0,0,0,0,0,27,46,0,0,0,0,0,0,1,0,0,0,71,1,46,27,0,0,0,72,0,0],[0,1,0,0,0,0,1,0,0,0,0,0,0,0,27,0,0,1,0,0,0,0,1,0,0,0,71,1,46,27,0,0,0,1,0,0] >;

C9×C4.D4 in GAP, Magma, Sage, TeX

C_9\times C_4.D_4
% in TeX

G:=Group("C9xC4.D4");
// GroupNames label

G:=SmallGroup(288,50);
// by ID

G=gap.SmallGroup(288,50);
# by ID

G:=PCGroup([7,-2,-2,-3,-2,-2,-3,-2,168,197,268,4371,2951,242]);
// Polycyclic

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

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