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G = C3×D6⋊3D4  order 288 = 25·32

Direct product of C3 and D6⋊3D4

direct product, metabelian, supersoluble, monomial

Aliases: C3×D6⋊3D4, C62.203C23, D6⋊3(C3×D4), (C6×D4)⋊3C6, C12⋊2(C3×D4), (C6×D4)⋊11S3, (S3×C6)⋊13D4, (C3×C12)⋊10D4, C6.50(C6×D4), C4⋊Dic3⋊14C6, C6.199(S3×D4), C12⋊11(C3⋊D4), (C2×C12).327D6, C23.13(S3×C6), (C22×C6).32D6, C6.D4⋊11C6, C32⋊22(C4⋊D4), (C6×C12).122C22, (C2×C62).57C22, C6.125(D4⋊2S3), (C6×Dic3).101C22, (S3×C2×C4)⋊2C6, (D4×C3×C6)⋊3C2, (S3×C2×C12)⋊6C2, C4⋊2(C3×C3⋊D4), C2.26(C3×S3×D4), (C2×D4)⋊4(C3×S3), (C2×C3⋊D4)⋊5C6, C3⋊4(C3×C4⋊D4), (C6×C3⋊D4)⋊19C2, (C2×C4).51(S3×C6), C6.31(C3×C4○D4), C2.14(C6×C3⋊D4), C22.60(S3×C2×C6), (C2×C12).33(C2×C6), (C3×C4⋊Dic3)⋊23C2, (C3×C6).260(C2×D4), C6.151(C2×C3⋊D4), C2.17(C3×D4⋊2S3), (S3×C2×C6).100C22, (C2×C6).58(C22×C6), (C22×C6).31(C2×C6), (C3×C6).139(C4○D4), (C3×C6.D4)⋊27C2, (C22×S3).27(C2×C6), (C2×C6).336(C22×S3), (C2×Dic3).12(C2×C6), SmallGroup(288,709)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C2×C6 — C3×D6⋊3D4
Chief series
C1 — C3 — C6 — C2×C6 — C62 — S3×C2×C6 — S3×C2×C12 — C3×D6⋊3D4
Lower central
C3 — C2×C6 — C3×D6⋊3D4
Upper central
C1 — C2×C6 — C6×D4

Generators and relations for C3×D6⋊3D4
 G = < a,b,c,d,e | a3=b6=c2=d4=e2=1, ab=ba, ac=ca, ad=da, ae=ea, cbc=b-1, bd=db, be=eb, cd=dc, ece=b3c, ede=d-1 >

Subgroups: 522 in 215 conjugacy classes, 70 normal (38 characteristic)
C1, C2, C2, C3, C3, C4, C4, C22, C22, S3, C6, C6, C2×C4, C2×C4, D4, C23, C23, C32, Dic3, C12, C12, D6, D6, C2×C6, C2×C6, C22⋊C4, C4⋊C4, C22×C4, C2×D4, C2×D4, C3×S3, C3×C6, C3×C6, C4×S3, C2×Dic3, C2×Dic3, C3⋊D4, C2×C12, C2×C12, C3×D4, C22×S3, C22×C6, C22×C6, C4⋊D4, C3×Dic3, C3×C12, S3×C6, S3×C6, C62, C62, C4⋊Dic3, C6.D4, C3×C22⋊C4, C3×C4⋊C4, S3×C2×C4, C2×C3⋊D4, C22×C12, C6×D4, C6×D4, S3×C12, C6×Dic3, C6×Dic3, C3×C3⋊D4, C6×C12, D4×C32, S3×C2×C6, C2×C62, D6⋊3D4, C3×C4⋊D4, C3×C4⋊Dic3, C3×C6.D4, S3×C2×C12, C6×C3⋊D4, D4×C3×C6, C3×D6⋊3D4
Quotients: C1, C2, C3, C22, S3, C6, D4, C23, D6, C2×C6, C2×D4, C4○D4, C3×S3, C3⋊D4, C3×D4, C22×S3, C22×C6, C4⋊D4, S3×C6, S3×D4, D4⋊2S3, C2×C3⋊D4, C6×D4, C3×C4○D4, C3×C3⋊D4, S3×C2×C6, D6⋊3D4, C3×C4⋊D4, C3×S3×D4, C3×D4⋊2S3, C6×C3⋊D4, C3×D6⋊3D4

Smallest permutation representation of C3×D6⋊3D4
►On 48 points
Generators in S48
(1 3 5)(2 4 6)(7 11 9)(8 12 10)(13 15 17)(14 16 18)(19 21 23)(20 22 24)(25 27 29)(26 28 30)(31 35 33)(32 36 34)(37 41 39)(38 42 40)(43 47 45)(44 48 46)

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

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

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

72 conjugacy classes

class 1 2A2B2C2D2E2F2G3A3B3C3D3E4A4B4C4D4E4F6A···6F6G···6O6P···6AE6AF6AG6AH6AI12A12B12C12D12E···12J12K12L12M12N12O12P12Q12R
order12222222333334444446···66···66···666661212121212···121212121212121212
size1111446611222226612121···12···24···4666622224···4666612121212

72 irreducible representations

dim111111111111222222222222224444
type++++++++++++-
imageC1C2C2C2C2C2C3C6C6C6C6C6S3D4D4D6D6C4○D4C3×S3C3⋊D4C3×D4C3×D4S3×C6S3×C6C3×C4○D4C3×C3⋊D4S3×D4D4⋊2S3C3×S3×D4C3×D4⋊2S3
kernelC3×D6⋊3D4C3×C4⋊Dic3C3×C6.D4S3×C2×C12C6×C3⋊D4D4×C3×C6D6⋊3D4C4⋊Dic3C6.D4S3×C2×C4C2×C3⋊D4C6×D4C6×D4C3×C12S3×C6C2×C12C22×C6C3×C6C2×D4C12C12D6C2×C4C23C6C4C6C6C2C2
# reps112121224242122122244424481122

Matrix representation of C3×D6⋊3D4 ►in GL6(𝔽13)

100000
010000
001000
000100
000090
000009
,
100000
010000
0012000
0001200
000030
0000109
,
100000
010000
000800
005000
000045
0000109
,
0120000
100000
0001200
001000
0000120
0000012
,
100000
0120000
001000
0001200
000010
000001

G:=sub<GL(6,GF(13))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,9,0,0,0,0,0,0,9],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,3,10,0,0,0,0,0,9],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,5,0,0,0,0,8,0,0,0,0,0,0,0,4,10,0,0,0,0,5,9],[0,1,0,0,0,0,12,0,0,0,0,0,0,0,0,1,0,0,0,0,12,0,0,0,0,0,0,0,12,0,0,0,0,0,0,12],[1,0,0,0,0,0,0,12,0,0,0,0,0,0,1,0,0,0,0,0,0,12,0,0,0,0,0,0,1,0,0,0,0,0,0,1] >;
 

C3×D6⋊3D4 in GAP, Magma, Sage, TeX 

C_3\times D_6\rtimes_3D_4
 
% in TeX
 
G:=Group("C3xD6:3D4");
 
// GroupNames label
 
G:=SmallGroup(288,709);
 
// by ID
 
G=gap.SmallGroup(288,709);
 
# by ID
 
G:=PCGroup([7,-2,-2,-2,-3,-2,-2,-3,344,590,555,9414]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e|a^3=b^6=c^2=d^4=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^3*c,e*d*e=d^-1>;
 
// generators/relations
 

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