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## G = C9×C4.Dic3order 432 = 24·33

### Direct product of C9 and C4.Dic3

Series: Derived Chief Lower central Upper central

 Derived series C1 — C6 — C9×C4.Dic3
 Chief series C1 — C3 — C6 — C3×C6 — C3×C12 — C3×C36 — C9×C3⋊C8 — C9×C4.Dic3
 Lower central C3 — C6 — C9×C4.Dic3
 Upper central C1 — C36 — C2×C36

Generators and relations for C9×C4.Dic3
G = < a,b,c,d | a9=b4=1, c6=b2, d2=b2c3, ab=ba, ac=ca, ad=da, bc=cb, dbd-1=b-1, dcd-1=c5 >

Subgroups: 116 in 76 conjugacy classes, 45 normal (39 characteristic)
C1, C2, C2, C3, C3, C4, C22, C6, C6, C8, C2×C4, C9, C9, C32, C12, C12, C2×C6, C2×C6, M4(2), C18, C18, C3×C6, C3×C6, C3⋊C8, C24, C2×C12, C2×C12, C3×C9, C36, C36, C2×C18, C2×C18, C3×C12, C62, C4.Dic3, C3×M4(2), C3×C18, C3×C18, C72, C2×C36, C2×C36, C3×C3⋊C8, C6×C12, C3×C36, C6×C18, C9×M4(2), C3×C4.Dic3, C9×C3⋊C8, C6×C36, C9×C4.Dic3
Quotients: C1, C2, C3, C4, C22, S3, C6, C2×C4, C9, Dic3, C12, D6, C2×C6, M4(2), C18, C3×S3, C2×Dic3, C2×C12, C36, C2×C18, C3×Dic3, S3×C6, C4.Dic3, C3×M4(2), S3×C9, C2×C36, C6×Dic3, C9×Dic3, S3×C18, C9×M4(2), C3×C4.Dic3, Dic3×C18, C9×C4.Dic3

Smallest permutation representation of C9×C4.Dic3
On 72 points
Generators in S72
(1 33 15 9 29 23 5 25 19)(2 34 16 10 30 24 6 26 20)(3 35 17 11 31 13 7 27 21)(4 36 18 12 32 14 8 28 22)(37 72 55 41 64 59 45 68 51)(38 61 56 42 65 60 46 69 52)(39 62 57 43 66 49 47 70 53)(40 63 58 44 67 50 48 71 54)
(1 10 7 4)(2 11 8 5)(3 12 9 6)(13 22 19 16)(14 23 20 17)(15 24 21 18)(25 34 31 28)(26 35 32 29)(27 36 33 30)(37 40 43 46)(38 41 44 47)(39 42 45 48)(49 52 55 58)(50 53 56 59)(51 54 57 60)(61 64 67 70)(62 65 68 71)(63 66 69 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 39 10 48 7 45 4 42)(2 44 11 41 8 38 5 47)(3 37 12 46 9 43 6 40)(13 59 22 56 19 53 16 50)(14 52 23 49 20 58 17 55)(15 57 24 54 21 51 18 60)(25 70 34 67 31 64 28 61)(26 63 35 72 32 69 29 66)(27 68 36 65 33 62 30 71)

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

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

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

162 conjugacy classes

 class 1 2A 2B 3A 3B 3C 3D 3E 4A 4B 4C 6A 6B 6C ··· 6M 8A 8B 8C 8D 9A ··· 9F 9G ··· 9L 12A 12B 12C 12D 12E ··· 12R 18A ··· 18F 18G ··· 18AD 24A ··· 24H 36A ··· 36L 36M ··· 36AP 72A ··· 72X order 1 2 2 3 3 3 3 3 4 4 4 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 2 1 1 2 2 2 1 1 2 1 1 2 ··· 2 6 6 6 6 1 ··· 1 2 ··· 2 1 1 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 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 type + + + + - + - image C1 C2 C2 C3 C4 C4 C6 C6 C9 C12 C12 C18 C18 C36 C36 S3 Dic3 D6 Dic3 M4(2) C3×S3 C3×Dic3 S3×C6 C3×Dic3 C4.Dic3 C3×M4(2) S3×C9 C9×Dic3 S3×C18 C9×Dic3 C9×M4(2) C3×C4.Dic3 C9×C4.Dic3 kernel C9×C4.Dic3 C9×C3⋊C8 C6×C36 C3×C4.Dic3 C3×C36 C6×C18 C3×C3⋊C8 C6×C12 C4.Dic3 C3×C12 C62 C3⋊C8 C2×C12 C12 C2×C6 C2×C36 C36 C36 C2×C18 C3×C9 C2×C12 C12 C12 C2×C6 C9 C32 C2×C4 C4 C4 C22 C3 C3 C1 # reps 1 2 1 2 2 2 4 2 6 4 4 12 6 12 12 1 1 1 1 2 2 2 2 2 4 4 6 6 6 6 12 8 24

Matrix representation of C9×C4.Dic3 in GL2(𝔽37) generated by

 9 0 0 9
,
 31 0 0 6
,
 8 0 0 23
,
 0 19 12 0
G:=sub<GL(2,GF(37))| [9,0,0,9],[31,0,0,6],[8,0,0,23],[0,12,19,0] >;

C9×C4.Dic3 in GAP, Magma, Sage, TeX

C_9\times C_4.{\rm Dic}_3
% in TeX

G:=Group("C9xC4.Dic3");
// GroupNames label

G:=SmallGroup(432,127);
// by ID

G=gap.SmallGroup(432,127);
# by ID

G:=PCGroup([7,-2,-2,-3,-2,-3,-2,-3,84,1037,142,192,14118]);
// Polycyclic

G:=Group<a,b,c,d|a^9=b^4=1,c^6=b^2,d^2=b^2*c^3,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,d*b*d^-1=b^-1,d*c*d^-1=c^5>;
// generators/relations

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