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G = C3⋊S3⋊4Dic6  order 432 = 24·33

The semidirect product of C3⋊S3 and Dic6 acting via Dic6/C12=C2

metabelian, supersoluble, monomial

Aliases: C12.42S32, C3⋊S3⋊4Dic6, C3⋊5(S3×Dic6), C33⋊12(C2×Q8), C32⋊10(S3×Q8), C33⋊5Q8⋊7C2, (C3×C12).147D6, C3⋊Dic3.20D6, C32⋊4Q8⋊12S3, C32⋊9(C2×Dic6), C3⋊4(Dic3.D6), C4.4(C32⋊4D6), (C32×C6).65C23, (C32×C12).49C22, C6.94(C2×S32), (C3×C3⋊S3)⋊6Q8, (C4×C3⋊S3).5S3, (C12×C3⋊S3).7C2, (C2×C3⋊S3).45D6, C33⋊9(C2×C4).2C2, (C6×C3⋊S3).57C22, (C3×C32⋊4Q8)⋊12C2, C2.3(C2×C32⋊4D6), (C3×C6).115(C22×S3), (C3×C3⋊Dic3).23C22, SmallGroup(432,687)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C32×C6 — C3⋊S3⋊4Dic6
Chief series
C1 — C3 — C32 — C33 — C32×C6 — C6×C3⋊S3 — C33⋊9(C2×C4) — C3⋊S3⋊4Dic6
Lower central
C33 — C32×C6 — C3⋊S3⋊4Dic6
Upper central
C1 — C2 — C4

Generators and relations for C3⋊S3⋊4Dic6
 G = < a,b,c,d,e | a3=b3=c2=d12=1, e2=d6, ab=ba, cac=a-1, ad=da, ae=ea, cbc=ebe-1=b-1, bd=db, cd=dc, ce=ec, ede-1=d-1 >

Subgroups: 888 in 198 conjugacy classes, 51 normal (19 characteristic)
C1, C2, C2, C3, C3, C3, C4, C4, C22, S3, C6, C6, C6, C2×C4, Q8, C32, C32, C32, Dic3, C12, C12, C12, D6, C2×C6, C2×Q8, C3×S3, C3⋊S3, C3×C6, C3×C6, C3×C6, Dic6, C4×S3, C2×Dic3, C2×C12, C3×Q8, C33, C3×Dic3, C3⋊Dic3, C3⋊Dic3, C3×C12, C3×C12, C3×C12, S3×C6, C2×C3⋊S3, C2×Dic6, S3×Q8, C3×C3⋊S3, C32×C6, S3×Dic3, C6.D6, C32⋊2Q8, C3×Dic6, S3×C12, C32⋊4Q8, C4×C3⋊S3, C3×C3⋊Dic3, C3×C3⋊Dic3, C32×C12, C6×C3⋊S3, S3×Dic6, Dic3.D6, C33⋊9(C2×C4), C33⋊5Q8, C3×C32⋊4Q8, C12×C3⋊S3, C3⋊S3⋊4Dic6
Quotients: C1, C2, C22, S3, Q8, C23, D6, C2×Q8, Dic6, C22×S3, S32, C2×Dic6, S3×Q8, C2×S32, C32⋊4D6, S3×Dic6, Dic3.D6, C2×C32⋊4D6, C3⋊S3⋊4Dic6

Smallest permutation representation of C3⋊S3⋊4Dic6
►On 48 points
Generators in S48
(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 9 5)(2 10 6)(3 11 7)(4 12 8)(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 41 45)(38 42 46)(39 43 47)(40 44 48)

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

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

48 conjugacy classes

class 1 2A2B2C3A3B3C3D···3H4A4B···4F6A6B6C6D···6H6I6J12A12B12C···12N12O12P12Q12R12S12T
order12223333···344···46666···666121212···12121212121212
size11992224···4218···182224···41818224···4181836363636

48 irreducible representations

dim11111222222244444444
type+++++++-+++-+-+-
imageC1C2C2C2C2S3S3Q8D6D6D6Dic6S32S3×Q8C2×S32C32⋊4D6S3×Dic6Dic3.D6C2×C32⋊4D6C3⋊S3⋊4Dic6
kernelC3⋊S3⋊4Dic6C33⋊9(C2×C4)C33⋊5Q8C3×C32⋊4Q8C12×C3⋊S3C32⋊4Q8C4×C3⋊S3C3×C3⋊S3C3⋊Dic3C3×C12C2×C3⋊S3C3⋊S3C12C32C6C4C3C3C2C1
# reps12221212531432324224

Matrix representation of C3⋊S3⋊4Dic6 ►in GL8(𝔽13)

10000000
01000000
00100000
00010000
00001000
00000100
0000001212
00000010
,
10000000
01000000
00100000
00010000
000012100
000012000
00000010
00000001
,
120000000
012000000
00100000
00010000
00000100
00001000
000000120
00000011
,
1211000000
11000000
00110000
001200000
000012000
000001200
00000010
00000001
,
17000000
912000000
001200000
00110000
000001200
000012000
000000120
000000012

G:=sub<GL(8,GF(13))| [1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,12,1,0,0,0,0,0,0,12,0],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,12,12,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[12,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,12,1,0,0,0,0,0,0,0,1],[12,1,0,0,0,0,0,0,11,1,0,0,0,0,0,0,0,0,1,12,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[1,9,0,0,0,0,0,0,7,12,0,0,0,0,0,0,0,0,12,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,12] >;
 

C3⋊S3⋊4Dic6 in GAP, Magma, Sage, TeX 

C_3\rtimes S_3\rtimes_4{\rm Dic}_6
 
% in TeX
 
G:=Group("C3:S3:4Dic6");
 
// GroupNames label
 
G:=SmallGroup(432,687);
 
// by ID
 
G=gap.SmallGroup(432,687);
 
# by ID
 
G:=PCGroup([7,-2,-2,-2,-2,-3,-3,-3,64,135,58,1124,571,2028,14118]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e|a^3=b^3=c^2=d^12=1,e^2=d^6,a*b=b*a,c*a*c=a^-1,a*d=d*a,a*e=e*a,c*b*c=e*b*e^-1=b^-1,b*d=d*b,c*d=d*c,c*e=e*c,e*d*e^-1=d^-1>;
 
// generators/relations
 

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