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G = C3×D6.6D6  order 432 = 24·33

Direct product of C3 and D6.6D6

direct product, metabelian, supersoluble, monomial

Aliases: C3×D6.6D6, C12.80S32, (S3×C12)⋊3S3, (S3×C12)⋊3C6, D6.6(S3×C6), C3⋊D12⋊4C6, C12.43(S3×C6), Dic6⋊5(C3×S3), (C3×Dic6)⋊7C6, (S3×C6).38D6, C6.D6⋊1C6, C12⋊S3⋊10C6, (C3×Dic6)⋊11S3, (C3×C12).177D6, C33⋊11(C4○D4), Dic3.9(S3×C6), (C3×Dic3).26D6, (C32×Dic6)⋊9C2, C32⋊19(C4○D12), (C32×C6).25C23, C32⋊11(Q8⋊3S3), (C32×C12).39C22, (C32×Dic3).11C22, C2.9(S32×C6), C4.7(C3×S32), C6.6(S3×C2×C6), (S3×C3×C12)⋊4C2, (C4×S3)⋊2(C3×S3), C6.109(C2×S32), C3⋊1(C3×C4○D12), C3⋊1(C3×Q8⋊3S3), C32⋊7(C3×C4○D4), (C3×C12⋊S3)⋊6C2, (S3×C6).13(C2×C6), (C3×C12).54(C2×C6), (C3×C6.D6)⋊4C2, (S3×C3×C6).20C22, (C3×C3⋊D12)⋊10C2, (C6×C3⋊S3).24C22, (C3×C6).16(C22×C6), (C3×Dic3).9(C2×C6), (C3×C6).130(C22×S3), (C2×C3⋊S3).7(C2×C6), SmallGroup(432,647)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C3×C6 — C3×D6.6D6
Chief series
C1 — C3 — C32 — C3×C6 — C32×C6 — S3×C3×C6 — C3×C3⋊D12 — C3×D6.6D6
Lower central
C32 — C3×C6 — C3×D6.6D6
Upper central
C1 — C6 — C12

Generators and relations for C3×D6.6D6
 G = < a,b,c,d,e | a3=b6=c2=1, d6=e2=b3, ab=ba, ac=ca, ad=da, ae=ea, cbc=b-1, bd=db, be=eb, cd=dc, ece-1=b3c, ede-1=b3d5 >

Subgroups: 744 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⋊S3, C3×C6, C3×C6, Dic6, C4×S3, C4×S3, D12, C3⋊D4, C2×C12, C3×D4, C3×Q8, C33, C3×Dic3, C3×Dic3, C3×Dic3, C3×C12, C3×C12, S3×C6, S3×C6, C2×C3⋊S3, C62, C4○D12, Q8⋊3S3, C3×C4○D4, S3×C32, C3×C3⋊S3, C32×C6, C6.D6, C3⋊D12, C3×Dic6, C3×Dic6, S3×C12, S3×C12, C3×D12, C3×C3⋊D4, C12⋊S3, C6×C12, Q8×C32, C32×Dic3, C32×Dic3, C32×C12, S3×C3×C6, C6×C3⋊S3, D6.6D6, C3×C4○D12, C3×Q8⋊3S3, C3×C6.D6, C3×C3⋊D12, C32×Dic6, S3×C3×C12, C3×C12⋊S3, C3×D6.6D6
Quotients: C1, C2, C3, C22, S3, C6, C23, D6, C2×C6, C4○D4, C3×S3, C22×S3, C22×C6, S32, S3×C6, C4○D12, Q8⋊3S3, C3×C4○D4, C2×S32, S3×C2×C6, C3×S32, D6.6D6, C3×C4○D12, C3×Q8⋊3S3, S32×C6, C3×D6.6D6

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

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

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

81 conjugacy classes

class 1 2A2B2C2D3A3B3C···3H3I3J3K4A4B4C4D4E6A6B6C···6H6I6J6K6L···6S6T6U6V6W12A···12H12I12J12K12L12M···12U12V···12AE12AF···12AK
order12222333···333344444666···66666···6666612···121212121212···1212···1212···12
size1161818112···244423366112···24446···6181818182···233334···46···612···12

81 irreducible representations

dim1111111111112222222222222244444444
type+++++++++++++++
imageC1C2C2C2C2C2C3C6C6C6C6C6S3S3D6D6D6C4○D4C3×S3C3×S3S3×C6S3×C6S3×C6C4○D12C3×C4○D4C3×C4○D12S32Q8⋊3S3C2×S32C3×S32D6.6D6C3×Q8⋊3S3S32×C6C3×D6.6D6
kernelC3×D6.6D6C3×C6.D6C3×C3⋊D12C32×Dic6S3×C3×C12C3×C12⋊S3D6.6D6C6.D6C3⋊D12C3×Dic6S3×C12C12⋊S3C3×Dic6S3×C12C3×Dic3C3×C12S3×C6C33Dic6C4×S3Dic3C12D6C32C32C3C12C32C6C4C3C3C2C1
# reps1221112442221132122264244811122224

Matrix representation of C3×D6.6D6 ►in GL6(𝔽13)

300000
030000
009000
000900
000010
000001
,
1200000
0120000
000100
00121200
000010
000001
,
130000
0120000
00121200
000100
000010
000001
,
8110000
050000
0012000
0001200
0000012
0000112
,
1080000
230000
001000
000100
0000112
0000012

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],[12,0,0,0,0,0,0,12,0,0,0,0,0,0,0,12,0,0,0,0,1,12,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,3,12,0,0,0,0,0,0,12,0,0,0,0,0,12,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[8,0,0,0,0,0,11,5,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,0,1,0,0,0,0,12,12],[10,2,0,0,0,0,8,3,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,12,12] >;
 

C3×D6.6D6 in GAP, Magma, Sage, TeX 

C_3\times D_6._6D_6
 
% in TeX
 
G:=Group("C3xD6.6D6");
 
// GroupNames label
 
G:=SmallGroup(432,647);
 
// by ID
 
G=gap.SmallGroup(432,647);
 
# by ID
 
G:=PCGroup([7,-2,-2,-2,-3,-2,-3,-3,365,176,303,142,2028,14118]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e|a^3=b^6=c^2=1,d^6=e^2=b^3,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^-1=b^3*c,e*d*e^-1=b^3*d^5>;
 
// generators/relations
 

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