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G = C2×C33⋊9(C2×C4)  order 432 = 24·33

Direct product of C2 and C33⋊9(C2×C4)

direct product, metabelian, supersoluble, monomial, A-group

Aliases: C2×C33⋊9(C2×C4), C62.95D6, C6⋊2(S3×Dic3), C3⋊Dic3⋊21D6, C6⋊2(C6.D6), C33⋊15(C22×C4), (C3×C62).33C22, (C32×C6).70C23, C32⋊7(C22×Dic3), C22.5(C32⋊4D6), (C6×C3⋊S3)⋊9C4, C6.99(C2×S32), (C3×C6)⋊7(C4×S3), (C2×C6).63S32, C3⋊3(C2×S3×Dic3), C32⋊14(S3×C2×C4), (C2×C3⋊S3)⋊6Dic3, C3⋊S3⋊3(C2×Dic3), (C2×C3⋊S3).49D6, (C32×C6)⋊9(C2×C4), (C3×C6)⋊6(C2×Dic3), C3⋊3(C2×C6.D6), (C6×C3⋊Dic3)⋊14C2, (C2×C3⋊Dic3)⋊12S3, (C22×C3⋊S3).7S3, (C6×C3⋊S3).60C22, C2.2(C2×C32⋊4D6), (C3×C6).120(C22×S3), (C3×C3⋊Dic3)⋊21C22, (C2×C6×C3⋊S3).9C2, (C3×C3⋊S3)⋊11(C2×C4), SmallGroup(432,692)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C33 — C2×C33⋊9(C2×C4)
Chief series
C1 — C3 — C32 — C33 — C32×C6 — C6×C3⋊S3 — C33⋊9(C2×C4) — C2×C33⋊9(C2×C4)
Lower central
C33 — C2×C33⋊9(C2×C4)
Upper central
C1 — C22

Generators and relations for C2×C33⋊9(C2×C4)
 G = < a,b,c,d,e,f | a2=b3=c3=d3=e2=f4=1, ab=ba, ac=ca, ad=da, ae=ea, af=fa, bc=cb, bd=db, ebe=fbf-1=b-1, cd=dc, ce=ec, fcf-1=c-1, ede=d-1, df=fd, ef=fe >

Subgroups: 1208 in 290 conjugacy classes, 79 normal (15 characteristic)
C1, C2, C2, C2, C3, C3, C3, C4, C22, C22, S3, C6, C6, C6, C2×C4, C23, C32, C32, C32, Dic3, C12, D6, C2×C6, C2×C6, C2×C6, C22×C4, C3×S3, C3⋊S3, C3×C6, C3×C6, C3×C6, C4×S3, C2×Dic3, C2×C12, C22×S3, C22×C6, C33, C3×Dic3, C3⋊Dic3, S3×C6, C2×C3⋊S3, C62, C62, C62, S3×C2×C4, C22×Dic3, C3×C3⋊S3, C32×C6, C32×C6, S3×Dic3, C6.D6, C6×Dic3, C2×C3⋊Dic3, S3×C2×C6, C22×C3⋊S3, C3×C3⋊Dic3, C6×C3⋊S3, C3×C62, C2×S3×Dic3, C2×C6.D6, C33⋊9(C2×C4), C6×C3⋊Dic3, C2×C6×C3⋊S3, C2×C33⋊9(C2×C4)
Quotients: C1, C2, C4, C22, S3, C2×C4, C23, Dic3, D6, C22×C4, C4×S3, C2×Dic3, C22×S3, S32, S3×C2×C4, C22×Dic3, S3×Dic3, C6.D6, C2×S32, C32⋊4D6, C2×S3×Dic3, C2×C6.D6, C33⋊9(C2×C4), C2×C32⋊4D6, C2×C33⋊9(C2×C4)

Smallest permutation representation of C2×C33⋊9(C2×C4)
►On 48 points
Generators in S48
(1 11)(2 12)(3 9)(4 10)(5 24)(6 21)(7 22)(8 23)(13 31)(14 32)(15 29)(16 30)(17 25)(18 26)(19 27)(20 28)(33 43)(34 44)(35 41)(36 42)(37 45)(38 46)(39 47)(40 48)

(1 19 6)(2 7 20)(3 17 8)(4 5 18)(9 25 23)(10 24 26)(11 27 21)(12 22 28)(13 43 47)(14 48 44)(15 41 45)(16 46 42)(29 35 37)(30 38 36)(31 33 39)(32 40 34)

(1 19 6)(2 7 20)(3 17 8)(4 5 18)(9 25 23)(10 24 26)(11 27 21)(12 22 28)(13 47 43)(14 44 48)(15 45 41)(16 42 46)(29 37 35)(30 36 38)(31 39 33)(32 34 40)

(1 6 19)(2 7 20)(3 8 17)(4 5 18)(9 23 25)(10 24 26)(11 21 27)(12 22 28)(13 47 43)(14 48 44)(15 45 41)(16 46 42)(29 37 35)(30 38 36)(31 39 33)(32 40 34)

(1 35)(2 36)(3 33)(4 34)(5 40)(6 37)(7 38)(8 39)(9 43)(10 44)(11 41)(12 42)(13 25)(14 26)(15 27)(16 28)(17 31)(18 32)(19 29)(20 30)(21 45)(22 46)(23 47)(24 48)

(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)
 
G:=sub<Sym(48)| (1,11)(2,12)(3,9)(4,10)(5,24)(6,21)(7,22)(8,23)(13,31)(14,32)(15,29)(16,30)(17,25)(18,26)(19,27)(20,28)(33,43)(34,44)(35,41)(36,42)(37,45)(38,46)(39,47)(40,48), (1,19,6)(2,7,20)(3,17,8)(4,5,18)(9,25,23)(10,24,26)(11,27,21)(12,22,28)(13,43,47)(14,48,44)(15,41,45)(16,46,42)(29,35,37)(30,38,36)(31,33,39)(32,40,34), (1,19,6)(2,7,20)(3,17,8)(4,5,18)(9,25,23)(10,24,26)(11,27,21)(12,22,28)(13,47,43)(14,44,48)(15,45,41)(16,42,46)(29,37,35)(30,36,38)(31,39,33)(32,34,40), (1,6,19)(2,7,20)(3,8,17)(4,5,18)(9,23,25)(10,24,26)(11,21,27)(12,22,28)(13,47,43)(14,48,44)(15,45,41)(16,46,42)(29,37,35)(30,38,36)(31,39,33)(32,40,34), (1,35)(2,36)(3,33)(4,34)(5,40)(6,37)(7,38)(8,39)(9,43)(10,44)(11,41)(12,42)(13,25)(14,26)(15,27)(16,28)(17,31)(18,32)(19,29)(20,30)(21,45)(22,46)(23,47)(24,48), (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)>;
 
G:=Group( (1,11)(2,12)(3,9)(4,10)(5,24)(6,21)(7,22)(8,23)(13,31)(14,32)(15,29)(16,30)(17,25)(18,26)(19,27)(20,28)(33,43)(34,44)(35,41)(36,42)(37,45)(38,46)(39,47)(40,48), (1,19,6)(2,7,20)(3,17,8)(4,5,18)(9,25,23)(10,24,26)(11,27,21)(12,22,28)(13,43,47)(14,48,44)(15,41,45)(16,46,42)(29,35,37)(30,38,36)(31,33,39)(32,40,34), (1,19,6)(2,7,20)(3,17,8)(4,5,18)(9,25,23)(10,24,26)(11,27,21)(12,22,28)(13,47,43)(14,44,48)(15,45,41)(16,42,46)(29,37,35)(30,36,38)(31,39,33)(32,34,40), (1,6,19)(2,7,20)(3,8,17)(4,5,18)(9,23,25)(10,24,26)(11,21,27)(12,22,28)(13,47,43)(14,48,44)(15,45,41)(16,46,42)(29,37,35)(30,38,36)(31,39,33)(32,40,34), (1,35)(2,36)(3,33)(4,34)(5,40)(6,37)(7,38)(8,39)(9,43)(10,44)(11,41)(12,42)(13,25)(14,26)(15,27)(16,28)(17,31)(18,32)(19,29)(20,30)(21,45)(22,46)(23,47)(24,48), (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) );
 
G=PermutationGroup([[(1,11),(2,12),(3,9),(4,10),(5,24),(6,21),(7,22),(8,23),(13,31),(14,32),(15,29),(16,30),(17,25),(18,26),(19,27),(20,28),(33,43),(34,44),(35,41),(36,42),(37,45),(38,46),(39,47),(40,48)], [(1,19,6),(2,7,20),(3,17,8),(4,5,18),(9,25,23),(10,24,26),(11,27,21),(12,22,28),(13,43,47),(14,48,44),(15,41,45),(16,46,42),(29,35,37),(30,38,36),(31,33,39),(32,40,34)], [(1,19,6),(2,7,20),(3,17,8),(4,5,18),(9,25,23),(10,24,26),(11,27,21),(12,22,28),(13,47,43),(14,44,48),(15,45,41),(16,42,46),(29,37,35),(30,36,38),(31,39,33),(32,34,40)], [(1,6,19),(2,7,20),(3,8,17),(4,5,18),(9,23,25),(10,24,26),(11,21,27),(12,22,28),(13,47,43),(14,48,44),(15,45,41),(16,46,42),(29,37,35),(30,38,36),(31,39,33),(32,40,34)], [(1,35),(2,36),(3,33),(4,34),(5,40),(6,37),(7,38),(8,39),(9,43),(10,44),(11,41),(12,42),(13,25),(14,26),(15,27),(16,28),(17,31),(18,32),(19,29),(20,30),(21,45),(22,46),(23,47),(24,48)], [(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)]])
 

60 conjugacy classes

class 1 2A2B2C2D2E2F2G3A3B3C3D···3H4A···4H6A···6I6J···6X6Y6Z6AA6AB12A···12H
order122222223333···34···46···66···6666612···12
size111199992224···49···92···24···41818181818···18

60 irreducible representations

dim1111122222224444444
type+++++++-+++-++
imageC1C2C2C2C4S3S3D6Dic3D6D6C4×S3S32S3×Dic3C6.D6C2×S32C32⋊4D6C33⋊9(C2×C4)C2×C32⋊4D6
kernelC2×C33⋊9(C2×C4)C33⋊9(C2×C4)C6×C3⋊Dic3C2×C6×C3⋊S3C6×C3⋊S3C2×C3⋊Dic3C22×C3⋊S3C3⋊Dic3C2×C3⋊S3C2×C3⋊S3C62C3×C6C2×C6C6C6C6C22C2C2
# reps1421821442383423242

Matrix representation of C2×C33⋊9(C2×C4) ►in GL6(𝔽13)

1200000
0120000
001000
000100
000010
000001
,
100000
010000
001000
000100
0000121
0000120
,
010000
12120000
001000
000100
000010
000001
,
100000
010000
0012100
0012000
000010
000001
,
100000
010000
000100
001000
0000112
0000012
,
800000
550000
0012000
0001200
000058
000008

G:=sub<GL(6,GF(13))| [12,0,0,0,0,0,0,12,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,12,12,0,0,0,0,1,0],[0,12,0,0,0,0,1,12,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,12,12,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,12,12],[8,5,0,0,0,0,0,5,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,5,0,0,0,0,0,8,8] >;
 

C2×C33⋊9(C2×C4) in GAP, Magma, Sage, TeX 

C_2\times C_3^3\rtimes_9(C_2\times C_4)
 
% in TeX
 
G:=Group("C2xC3^3:9(C2xC4)");
 
// GroupNames label
 
G:=SmallGroup(432,692);
 
// by ID
 
G=gap.SmallGroup(432,692);
 
# by ID
 
G:=PCGroup([7,-2,-2,-2,-2,-3,-3,-3,64,1124,571,2028,14118]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e,f|a^2=b^3=c^3=d^3=e^2=f^4=1,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,a*f=f*a,b*c=c*b,b*d=d*b,e*b*e=f*b*f^-1=b^-1,c*d=d*c,c*e=e*c,f*c*f^-1=c^-1,e*d*e=d^-1,d*f=f*d,e*f=f*e>;
 
// generators/relations
 

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