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G = C3×D12⋊5S3  order 432 = 24·33

Direct product of C3 and D12⋊5S3

direct product, metabelian, supersoluble, monomial

Aliases: C3×D12⋊5S3, C12.79S32, (S3×C12)⋊2C6, (S3×C12)⋊2S3, (C3×D12)⋊4C6, D12⋊5(C3×S3), D6.1(S3×C6), D6⋊S3⋊2C6, (C3×D12)⋊11S3, C12.36(S3×C6), (S3×Dic3)⋊4C6, (S3×C6).36D6, C33⋊8(C4○D4), (C3×C12).135D6, (C32×D12)⋊6C2, Dic3.8(S3×C6), C32⋊4Q8⋊11C6, (C3×Dic3).49D6, C32⋊22(C4○D12), (C32×C6).21C23, C32⋊21(D4⋊2S3), (C32×C12).36C22, (C32×Dic3).26C22, C2.5(S32×C6), C4.6(C3×S32), C6.2(S3×C2×C6), (S3×C3×C12)⋊2C2, (C4×S3)⋊1(C3×S3), C6.105(C2×S32), C3⋊3(C3×C4○D12), C3⋊2(C3×D4⋊2S3), (S3×C6).1(C2×C6), (C3×S3×Dic3)⋊11C2, C32⋊4(C3×C4○D4), (C3×D6⋊S3)⋊9C2, (C3×C12).51(C2×C6), (S3×C3×C6).10C22, C3⋊Dic3.9(C2×C6), (C3×C32⋊4Q8)⋊7C2, (C3×C6).12(C22×C6), (C3×Dic3).8(C2×C6), (C3×C6).126(C22×S3), (C3×C3⋊Dic3).35C22, SmallGroup(432,643)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C3×C6 — C3×D12⋊5S3
Chief series
C1 — C3 — C32 — C3×C6 — C32×C6 — S3×C3×C6 — C3×S3×Dic3 — C3×D12⋊5S3
Lower central
C32 — C3×C6 — C3×D12⋊5S3
Upper central
C1 — C6 — C12

Generators and relations for C3×D12⋊5S3
 G = < a,b,c,d,e | a3=b12=c2=d3=e2=1, ab=ba, ac=ca, ad=da, ae=ea, cbc=b-1, bd=db, be=eb, cd=dc, ece=b6c, ede=d-1 >

Subgroups: 664 in 210 conjugacy classes, 64 normal (44 characteristic)
C1, C2, C2, C3, C3, C4, C4, C22, S3, C6, C6, C2×C4, D4, Q8, C32, C32, Dic3, Dic3, C12, C12, D6, D6, C2×C6, C4○D4, C3×S3, C3×C6, C3×C6, Dic6, C4×S3, C4×S3, D12, C2×Dic3, C3⋊D4, C2×C12, C3×D4, C3×Q8, C33, C3×Dic3, C3×Dic3, C3⋊Dic3, C3×C12, C3×C12, S3×C6, S3×C6, S3×C6, C62, C4○D12, D4⋊2S3, C3×C4○D4, S3×C32, C32×C6, S3×Dic3, D6⋊S3, C3×Dic6, S3×C12, S3×C12, C3×D12, C3×D12, C6×Dic3, C3×C3⋊D4, C32⋊4Q8, C6×C12, D4×C32, C32×Dic3, C3×C3⋊Dic3, C32×C12, S3×C3×C6, S3×C3×C6, D12⋊5S3, C3×C4○D12, C3×D4⋊2S3, C3×S3×Dic3, C3×D6⋊S3, S3×C3×C12, C32×D12, C3×C32⋊4Q8, C3×D12⋊5S3
Quotients: C1, C2, C3, C22, S3, C6, C23, D6, C2×C6, C4○D4, C3×S3, C22×S3, C22×C6, S32, S3×C6, C4○D12, D4⋊2S3, C3×C4○D4, C2×S32, S3×C2×C6, C3×S32, D12⋊5S3, C3×C4○D12, C3×D4⋊2S3, S32×C6, C3×D12⋊5S3

Smallest permutation representation of C3×D12⋊5S3
►On 48 points
Generators in S48
(1 5 9)(2 6 10)(3 7 11)(4 8 12)(13 17 21)(14 18 22)(15 19 23)(16 20 24)(25 33 29)(26 34 30)(27 35 31)(28 36 32)(37 45 41)(38 46 42)(39 47 43)(40 48 44)

(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)

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

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

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

81 conjugacy classes

class 1 2A2B2C2D3A3B3C···3H3I3J3K4A4B4C4D4E6A6B6C···6H6I6J6K6L···6W6X···6AC12A···12H12I12J12K12L12M···12U12V···12AA12AB12AC12AD12AE
order12222333···333344444666···66666···66···612···121212121212···1212···1212121212
size11666112···24442331818112···24446···612···122···233334···46···618181818

81 irreducible representations

dim1111111111112222222222222244444444
type++++++++++++-+-
imageC1C2C2C2C2C2C3C6C6C6C6C6S3S3D6D6D6C4○D4C3×S3C3×S3S3×C6S3×C6S3×C6C4○D12C3×C4○D4C3×C4○D12S32D4⋊2S3C2×S32C3×S32D12⋊5S3C3×D4⋊2S3S32×C6C3×D12⋊5S3
kernelC3×D12⋊5S3C3×S3×Dic3C3×D6⋊S3S3×C3×C12C32×D12C3×C32⋊4Q8D12⋊5S3S3×Dic3D6⋊S3S3×C12C3×D12C32⋊4Q8S3×C12C3×D12C3×Dic3C3×C12S3×C6C33C4×S3D12Dic3C12D6C32C32C3C12C32C6C4C3C3C2C1
# reps1221112442221112322224644811122224

Matrix representation of C3×D12⋊5S3 ►in GL6(𝔽13)

300000
030000
009000
000900
000010
000001
,
800000
050000
0011200
001000
000010
000001
,
050000
800000
000100
001000
000010
000001
,
100000
010000
001000
000100
000001
00001212
,
1200000
010000
001000
000100
000010
00001212

G:=sub<GL(6,GF(13))| [3,0,0,0,0,0,0,3,0,0,0,0,0,0,9,0,0,0,0,0,0,9,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[8,0,0,0,0,0,0,5,0,0,0,0,0,0,1,1,0,0,0,0,12,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[0,8,0,0,0,0,5,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,12,0,0,0,0,1,12],[12,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,12,0,0,0,0,0,12] >;
 

C3×D12⋊5S3 in GAP, Magma, Sage, TeX 

C_3\times D_{12}\rtimes_5S_3
 
% in TeX
 
G:=Group("C3xD12:5S3");
 
// GroupNames label
 
G:=SmallGroup(432,643);
 
// by ID
 
G=gap.SmallGroup(432,643);
 
# by ID
 
G:=PCGroup([7,-2,-2,-2,-3,-2,-3,-3,176,590,142,2028,14118]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e|a^3=b^12=c^2=d^3=e^2=1,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,c*b*c=b^-1,b*d=d*b,b*e=e*b,c*d=d*c,e*c*e=b^6*c,e*d*e=d^-1>;
 
// generators/relations
 

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