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G = C3⋊D12⋊7S3  order 432 = 24·33

The semidirect product of C3⋊D12 and S3 acting through Inn(C3⋊D12)

metabelian, supersoluble, monomial

Aliases: D6.2(S32), C3⋊D12⋊7S3, (S3×Dic3)⋊5S3, (S3×C6).20D6, C6.D6⋊2S3, C33⋊2(C4○D4), C33⋊9D4⋊3C2, C33⋊6D4⋊3C2, C33⋊4Q8⋊3C2, Dic3.9(S32), C3⋊Dic3.18D6, C3⋊1(D12⋊5S3), C3⋊2(D6.3D6), C3⋊2(D6.D6), C32⋊7(C4○D12), (C3×Dic3).10D6, C32⋊8(D4⋊2S3), (C32×C6).14C23, C33⋊5C4.3C22, (C32×Dic3).20C22, C2.14(S33), C6.14(C2×S32), (C3×S3×Dic3)⋊2C2, (Dic3×C3⋊S3)⋊8C2, (C2×C3⋊S3).32D6, (S3×C3×C6).5C22, (C3×C3⋊D12)⋊5C2, (C3×C6.D6)⋊2C2, (C6×C3⋊S3).19C22, (C3×C6).63(C22×S3), (C3×C3⋊Dic3).6C22, SmallGroup(432,607)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C32×C6 — C3⋊D12⋊7S3
Chief series
C1 — C3 — C32 — C33 — C32×C6 — S3×C3×C6 — C3×S3×Dic3 — C3⋊D12⋊7S3
Lower central
C33 — C32×C6 — C3⋊D12⋊7S3
Upper central
C1 — C2

Generators and relations for C3⋊D12⋊7S3
 G = < a,b,c,d,e | a3=b12=c2=d3=e2=1, bab-1=cac=eae=a-1, ad=da, cbc=b-1, bd=db, be=eb, cd=dc, ece=b6c, ede=d-1 >

Subgroups: 1156 in 210 conjugacy classes, 46 normal (all characteristic)
C1, C2, C2 [×3], C3 [×3], C3 [×4], C4 [×4], C22 [×3], S3 [×8], C6 [×3], C6 [×9], C2×C4 [×3], D4 [×3], Q8, C32 [×3], C32 [×4], Dic3 [×2], Dic3 [×10], C12 [×7], D6, D6 [×6], C2×C6 [×5], C4○D4, C3×S3 [×10], C3⋊S3 [×2], C3×C6 [×3], C3×C6 [×5], Dic6 [×4], C4×S3 [×6], D12 [×2], C2×Dic3 [×2], C3⋊D4 [×7], C2×C12 [×2], C3×D4, C33, C3×Dic3 [×4], C3×Dic3 [×5], C3⋊Dic3, C3⋊Dic3 [×7], C3×C12 [×2], S3×C6 [×2], S3×C6 [×7], C2×C3⋊S3 [×2], C62, C4○D12 [×2], D4⋊2S3, S3×C32, C3×C3⋊S3 [×2], C32×C6, S3×Dic3, S3×Dic3 [×3], C6.D6, D6⋊S3 [×4], C3⋊D12, C3⋊D12 [×2], C32⋊2Q8 [×3], S3×C12 [×3], C3×D12, C6×Dic3, C3×C3⋊D4, C32⋊4Q8, C4×C3⋊S3, C32⋊7D4, C32×Dic3 [×2], C3×C3⋊Dic3, C33⋊5C4, S3×C3×C6, C6×C3⋊S3 [×2], D12⋊5S3, D6.D6, D6.3D6, C3×S3×Dic3, C3×C6.D6, C3×C3⋊D12, Dic3×C3⋊S3, C33⋊6D4, C33⋊4Q8, C33⋊9D4, C3⋊D12⋊7S3
Quotients: C1, C2 [×7], C22 [×7], S3 [×3], C23, D6 [×9], C4○D4, C22×S3 [×3], S32 [×3], C4○D12 [×2], D4⋊2S3, C2×S32 [×3], D12⋊5S3, D6.D6, D6.3D6, S33, C3⋊D12⋊7S3

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

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

(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 33 29)(26 34 30)(27 35 31)(28 36 32)(37 41 45)(38 42 46)(39 43 47)(40 44 48)

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

45 conjugacy classes

class 1 2A2B2C2D3A3B3C3D3E3F3G4A4B4C4D4E6A6B6C6D6E6F6G6H6I6J6K6L6M6N6O12A···12F12G···12K12L12M
order1222233333334444466666666666666612···1212···121212
size1161818222444833618542224446681212121818366···612···121818

45 irreducible representations

dim11111111222222222444444488
type+++++++++++++++++-+-+-
imageC1C2C2C2C2C2C2C2S3S3S3D6D6D6D6C4○D4C4○D12S32S32D4⋊2S3C2×S32D12⋊5S3D6.D6D6.3D6S33C3⋊D12⋊7S3
kernelC3⋊D12⋊7S3C3×S3×Dic3C3×C6.D6C3×C3⋊D12Dic3×C3⋊S3C33⋊6D4C33⋊4Q8C33⋊9D4S3×Dic3C6.D6C3⋊D12C3×Dic3C3⋊Dic3S3×C6C2×C3⋊S3C33C32Dic3D6C32C6C3C3C3C2C1
# reps11111111111412228211322211

Matrix representation of C3⋊D12⋊7S3 ►in GL8(𝔽13)

10000000
01000000
00100000
00010000
00001000
00000100
00000001
0000001212
,
80000000
05000000
00110000
001200000
000012000
000001200
00000010
0000001212
,
05000000
80000000
00110000
000120000
00001000
00000100
000000120
00000011
,
10000000
01000000
00100000
00010000
000012100
000012000
00000010
00000001
,
120000000
01000000
001200000
000120000
00000100
00001000
00000010
0000001212

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

C3⋊D12⋊7S3 in GAP, Magma, Sage, TeX 

C_3\rtimes D_{12}\rtimes_7S_3
 
% in TeX
 
G:=Group("C3:D12:7S3");
 
// GroupNames label
 
G:=SmallGroup(432,607);
 
// by ID
 
G=gap.SmallGroup(432,607);
 
# by ID
 
G:=PCGroup([7,-2,-2,-2,-2,-3,-3,-3,64,254,58,298,2028,14118]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e|a^3=b^12=c^2=d^3=e^2=1,b*a*b^-1=c*a*c=e*a*e=a^-1,a*d=d*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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