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## G = C3⋊S3⋊4Dic6order 432 = 24·33

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

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⋊S34Dic6
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, C322Q8, C3×Dic6, S3×C12, C324Q8, C4×C3⋊S3, C3×C3⋊Dic3, C3×C3⋊Dic3, C32×C12, C6×C3⋊S3, S3×Dic6, Dic3.D6, C339(C2×C4), C335Q8, C3×C324Q8, C12×C3⋊S3, C3⋊S34Dic6
Quotients: C1, C2, C22, S3, Q8, C23, D6, C2×Q8, Dic6, C22×S3, S32, C2×Dic6, S3×Q8, C2×S32, C324D6, S3×Dic6, Dic3.D6, C2×C324D6, C3⋊S34Dic6

Smallest permutation representation of C3⋊S34Dic6
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 2A 2B 2C 3A 3B 3C 3D ··· 3H 4A 4B ··· 4F 6A 6B 6C 6D ··· 6H 6I 6J 12A 12B 12C ··· 12N 12O 12P 12Q 12R 12S 12T order 1 2 2 2 3 3 3 3 ··· 3 4 4 ··· 4 6 6 6 6 ··· 6 6 6 12 12 12 ··· 12 12 12 12 12 12 12 size 1 1 9 9 2 2 2 4 ··· 4 2 18 ··· 18 2 2 2 4 ··· 4 18 18 2 2 4 ··· 4 18 18 36 36 36 36

48 irreducible representations

 dim 1 1 1 1 1 2 2 2 2 2 2 2 4 4 4 4 4 4 4 4 type + + + + + + + - + + + - + - + - image C1 C2 C2 C2 C2 S3 S3 Q8 D6 D6 D6 Dic6 S32 S3×Q8 C2×S32 C32⋊4D6 S3×Dic6 Dic3.D6 C2×C32⋊4D6 C3⋊S3⋊4Dic6 kernel C3⋊S3⋊4Dic6 C33⋊9(C2×C4) C33⋊5Q8 C3×C32⋊4Q8 C12×C3⋊S3 C32⋊4Q8 C4×C3⋊S3 C3×C3⋊S3 C3⋊Dic3 C3×C12 C2×C3⋊S3 C3⋊S3 C12 C32 C6 C4 C3 C3 C2 C1 # reps 1 2 2 2 1 2 1 2 5 3 1 4 3 2 3 2 4 2 2 4

Matrix representation of C3⋊S34Dic6 in GL8(𝔽13)

 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 12 0 0 0 0 0 0 1 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 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 0 0 0 0 0 0 0 1 1
,
 12 11 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 1 1 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 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1
,
 1 7 0 0 0 0 0 0 9 12 0 0 0 0 0 0 0 0 12 0 0 0 0 0 0 0 1 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

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⋊S34Dic6 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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