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G = C2×C4×C9⋊C6  order 432 = 24·33

Direct product of C2×C4 and C9⋊C6

direct product, metabelian, supersoluble, monomial

Aliases: C2×C4×C9⋊C6, D18⋊2C12, C62.38D6, D9⋊(C2×C12), C36⋊3(C2×C6), (C2×C36)⋊3C6, (C4×D9)⋊5C6, C18⋊1(C2×C12), C12.99(S3×C6), C6.22(S3×C12), C9⋊C12⋊3C22, (C6×C12).22S3, C9⋊1(C22×C12), (C2×Dic9)⋊5C6, Dic9⋊3(C2×C6), D18.4(C2×C6), (C3×C12).68D6, C18.2(C22×C6), (C22×D9).2C6, 3- 1+2⋊1(C22×C4), (C4×3- 1+2)⋊3C22, (C2×3- 1+2).2C23, (C22×3- 1+2).8C22, (C2×C4×D9)⋊C3, C32.(S3×C2×C4), C6.28(S3×C2×C6), C3.3(S3×C2×C12), (C2×C9⋊C12)⋊5C2, (C3×C6).25(C4×S3), (C2×C6).58(S3×C6), (C2×C18).8(C2×C6), C22.9(C2×C9⋊C6), C2.1(C22×C9⋊C6), (C2×C12).37(C3×S3), (C22×C9⋊C6).2C2, (C2×C9⋊C6).4C22, (C3×C6).24(C22×S3), (C2×C4×3- 1+2)⋊3C2, (C2×3- 1+2)⋊1(C2×C4), SmallGroup(432,353)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C9 — C2×C4×C9⋊C6
Chief series
C1 — C3 — C9 — C18 — C2×3- 1+2 — C2×C9⋊C6 — C22×C9⋊C6 — C2×C4×C9⋊C6
Lower central
C9 — C2×C4×C9⋊C6
Upper central
C1 — C2×C4

Generators and relations for C2×C4×C9⋊C6
 G = < a,b,c,d | a2=b4=c9=d6=1, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd-1=c2 >

Subgroups: 574 in 170 conjugacy classes, 78 normal (26 characteristic)
C1, C2, C2, C2, C3, C3, C4, C4, C22, C22, S3, C6, C6, C6, C2×C4, C2×C4, C23, C9, C9, C32, Dic3, C12, C12, D6, C2×C6, C2×C6, C22×C4, D9, C18, C18, C18, C3×S3, C3×C6, C3×C6, C4×S3, C2×Dic3, C2×C12, C2×C12, C22×S3, C22×C6, 3- 1+2, Dic9, C36, C36, D18, C2×C18, C2×C18, C3×Dic3, C3×C12, S3×C6, C62, S3×C2×C4, C22×C12, C9⋊C6, C2×3- 1+2, C2×3- 1+2, C4×D9, C2×Dic9, C2×C36, C2×C36, C22×D9, S3×C12, C6×Dic3, C6×C12, S3×C2×C6, C9⋊C12, C4×3- 1+2, C2×C9⋊C6, C22×3- 1+2, C2×C4×D9, S3×C2×C12, C4×C9⋊C6, C2×C9⋊C12, C2×C4×3- 1+2, C22×C9⋊C6, C2×C4×C9⋊C6
Quotients: C1, C2, C3, C4, C22, S3, C6, C2×C4, C23, C12, D6, C2×C6, C22×C4, C3×S3, C4×S3, C2×C12, C22×S3, C22×C6, S3×C6, S3×C2×C4, C22×C12, C9⋊C6, S3×C12, S3×C2×C6, C2×C9⋊C6, S3×C2×C12, C4×C9⋊C6, C22×C9⋊C6, C2×C4×C9⋊C6

Smallest permutation representation of C2×C4×C9⋊C6
►On 72 points
Generators in S72
(1 38)(2 39)(3 40)(4 41)(5 42)(6 43)(7 44)(8 45)(9 37)(10 46)(11 47)(12 48)(13 49)(14 50)(15 51)(16 52)(17 53)(18 54)(19 55)(20 56)(21 57)(22 58)(23 59)(24 60)(25 61)(26 62)(27 63)(28 64)(29 65)(30 66)(31 67)(32 68)(33 69)(34 70)(35 71)(36 72)

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

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

80 conjugacy classes

class 1 2A2B2C2D2E2F2G3A3B3C4A4B4C4D4E4F4G4H6A6B6C6D···6I6J···6Q9A9B9C12A12B12C12D12E···12L12M···12T18A···18I36A···36L
order12222222333444444446666···66···69991212121212···1212···1218···1836···36
size11119999233111199992223···39···966622223···39···96···66···6

80 irreducible representations

dim111111111111222222226666
type+++++++++++
imageC1C2C2C2C2C3C4C6C6C6C6C12S3D6D6C3×S3C4×S3S3×C6S3×C6S3×C12C9⋊C6C2×C9⋊C6C2×C9⋊C6C4×C9⋊C6
kernelC2×C4×C9⋊C6C4×C9⋊C6C2×C9⋊C12C2×C4×3- 1+2C22×C9⋊C6C2×C4×D9C2×C9⋊C6C4×D9C2×Dic9C2×C36C22×D9D18C6×C12C3×C12C62C2×C12C3×C6C12C2×C6C6C2×C4C4C22C2
# reps1411128822216121244281214

Matrix representation of C2×C4×C9⋊C6 ►in GL8(𝔽37)

360000000
036000000
003600000
000360000
000036000
000003600
000000360
000000036
,
10000000
01000000
003100000
000310000
000031000
000003100
000000310
000000031
,
01000000
3636000000
000036100
000036000
00363636363635
000000136
00001001
00101001
,
110000000
2626000000
000360000
003600000
00111121
000000136
0000036360
0000360360

G:=sub<GL(8,GF(37))| [36,0,0,0,0,0,0,0,0,36,0,0,0,0,0,0,0,0,36,0,0,0,0,0,0,0,0,36,0,0,0,0,0,0,0,0,36,0,0,0,0,0,0,0,0,36,0,0,0,0,0,0,0,0,36,0,0,0,0,0,0,0,0,36],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,31,0,0,0,0,0,0,0,0,31,0,0,0,0,0,0,0,0,31,0,0,0,0,0,0,0,0,31,0,0,0,0,0,0,0,0,31,0,0,0,0,0,0,0,0,31],[0,36,0,0,0,0,0,0,1,36,0,0,0,0,0,0,0,0,0,0,36,0,0,1,0,0,0,0,36,0,0,0,0,0,36,36,36,0,1,1,0,0,1,0,36,0,0,0,0,0,0,0,36,1,0,0,0,0,0,0,35,36,1,1],[11,26,0,0,0,0,0,0,0,26,0,0,0,0,0,0,0,0,0,36,1,0,0,0,0,0,36,0,1,0,0,0,0,0,0,0,1,0,0,36,0,0,0,0,1,0,36,0,0,0,0,0,2,1,36,36,0,0,0,0,1,36,0,0] >;
 

C2×C4×C9⋊C6 in GAP, Magma, Sage, TeX 

C_2\times C_4\times C_9\rtimes C_6
 
% in TeX
 
G:=Group("C2xC4xC9:C6");
 
// GroupNames label
 
G:=SmallGroup(432,353);
 
// by ID
 
G=gap.SmallGroup(432,353);
 
# by ID
 
G:=PCGroup([7,-2,-2,-2,-3,-2,-3,-3,142,10085,1034,292,14118]);
 
// Polycyclic
 
G:=Group<a,b,c,d|a^2=b^4=c^9=d^6=1,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d^-1=c^2>;
 
// generators/relations
 

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