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G = (C3×Dic6)⋊S3  order 432 = 24·33

7th semidirect product of C3×Dic6 and S3 acting via S3/C3=C2

metabelian, supersoluble, monomial

Aliases: C12.39(S32), (C3×Dic6)⋊7S3, C12⋊S3⋊10S3, C33⋊8D4⋊7C2, Dic6⋊3(C3⋊S3), (C3×C12).142D6, C33⋊14(C4○D4), C3⋊3(D12⋊S3), (C3×Dic3).14D6, C3⋊2(C12.26D6), C32⋊6(Q8⋊3S3), (C32×Dic6)⋊11C2, (C32×C6).42C23, C32⋊20(D4⋊2S3), (C32×C12).44C22, C33⋊5C4.14C22, (C32×Dic3).14C22, C6.52(C2×S32), C4.20(S3×C3⋊S3), C12.23(C2×C3⋊S3), (Dic3×C3⋊S3)⋊3C2, (C2×C3⋊S3).34D6, (C3×C12⋊S3)⋊8C2, C6.5(C22×C3⋊S3), (C4×C33⋊C2)⋊2C2, Dic3.2(C2×C3⋊S3), (C6×C3⋊S3).26C22, (C3×C6).100(C22×S3), (C2×C33⋊C2).12C22, C2.9(C2×S3×C3⋊S3), SmallGroup(432,664)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C32×C6 — (C3×Dic6)⋊S3
Chief series
C1 — C3 — C32 — C33 — C32×C6 — C32×Dic3 — Dic3×C3⋊S3 — (C3×Dic6)⋊S3
Lower central
C33 — C32×C6 — (C3×Dic6)⋊S3
Upper central
C1 — C2 — C4

Generators and relations for (C3×Dic6)⋊S3
 G = < a,b,c,d,e | a3=b12=d3=e2=1, c2=b6, ab=ba, ac=ca, ad=da, eae=a-1, cbc-1=b-1, bd=db, ebe=b7, cd=dc, ce=ec, ede=d-1 >

Subgroups: 1824 in 304 conjugacy classes, 68 normal (20 characteristic)
C1, C2, C2 [×3], C3, C3 [×4], C3 [×4], C4, C4 [×3], C22 [×3], S3 [×21], C6, C6 [×4], C6 [×6], C2×C4 [×3], D4 [×3], Q8, C32, C32 [×4], C32 [×4], Dic3 [×2], Dic3 [×9], C12, C12 [×4], C12 [×12], D6 [×17], C2×C6 [×2], C4○D4, C3×S3 [×8], C3⋊S3 [×15], C3×C6, C3×C6 [×4], C3×C6 [×4], Dic6, C4×S3 [×17], D12 [×12], C2×Dic3 [×2], C3⋊D4 [×2], C3×D4, C3×Q8 [×4], C33, C3×Dic3 [×8], C3⋊Dic3 [×9], C3×C12, C3×C12 [×4], C3×C12 [×6], S3×C6 [×8], C2×C3⋊S3 [×2], C2×C3⋊S3 [×9], D4⋊2S3, Q8⋊3S3 [×4], C3×C3⋊S3 [×2], C33⋊C2, C32×C6, S3×Dic3 [×8], C3⋊D12 [×8], C3×Dic6 [×4], C3×D12 [×4], C4×C3⋊S3 [×11], C12⋊S3, C12⋊S3 [×2], Q8×C32, C32×Dic3 [×2], C33⋊5C4, C32×C12, C6×C3⋊S3 [×2], C2×C33⋊C2, D12⋊S3 [×4], C12.26D6, Dic3×C3⋊S3 [×2], C33⋊8D4 [×2], C32×Dic6, C3×C12⋊S3, C4×C33⋊C2, (C3×Dic6)⋊S3
Quotients: C1, C2 [×7], C22 [×7], S3 [×5], C23, D6 [×15], C4○D4, C3⋊S3, C22×S3 [×5], S32 [×4], C2×C3⋊S3 [×3], D4⋊2S3, Q8⋊3S3 [×4], C2×S32 [×4], C22×C3⋊S3, S3×C3⋊S3, D12⋊S3 [×4], C12.26D6, C2×S3×C3⋊S3, (C3×Dic6)⋊S3

Smallest permutation representation of (C3×Dic6)⋊S3
►On 72 points
Generators in S72
(1 25 45)(2 26 46)(3 27 47)(4 28 48)(5 29 37)(6 30 38)(7 31 39)(8 32 40)(9 33 41)(10 34 42)(11 35 43)(12 36 44)(13 54 67)(14 55 68)(15 56 69)(16 57 70)(17 58 71)(18 59 72)(19 60 61)(20 49 62)(21 50 63)(22 51 64)(23 52 65)(24 53 66)

(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)(49 50 51 52 53 54 55 56 57 58 59 60)(61 62 63 64 65 66 67 68 69 70 71 72)

(1 59 7 53)(2 58 8 52)(3 57 9 51)(4 56 10 50)(5 55 11 49)(6 54 12 60)(13 44 19 38)(14 43 20 37)(15 42 21 48)(16 41 22 47)(17 40 23 46)(18 39 24 45)(25 72 31 66)(26 71 32 65)(27 70 33 64)(28 69 34 63)(29 68 35 62)(30 67 36 61)

(1 41 29)(2 42 30)(3 43 31)(4 44 32)(5 45 33)(6 46 34)(7 47 35)(8 48 36)(9 37 25)(10 38 26)(11 39 27)(12 40 28)(13 71 50)(14 72 51)(15 61 52)(16 62 53)(17 63 54)(18 64 55)(19 65 56)(20 66 57)(21 67 58)(22 68 59)(23 69 60)(24 70 49)

(1 50)(2 57)(3 52)(4 59)(5 54)(6 49)(7 56)(8 51)(9 58)(10 53)(11 60)(12 55)(13 29)(14 36)(15 31)(16 26)(17 33)(18 28)(19 35)(20 30)(21 25)(22 32)(23 27)(24 34)(37 67)(38 62)(39 69)(40 64)(41 71)(42 66)(43 61)(44 68)(45 63)(46 70)(47 65)(48 72)
 
G:=sub<Sym(72)| (1,25,45)(2,26,46)(3,27,47)(4,28,48)(5,29,37)(6,30,38)(7,31,39)(8,32,40)(9,33,41)(10,34,42)(11,35,43)(12,36,44)(13,54,67)(14,55,68)(15,56,69)(16,57,70)(17,58,71)(18,59,72)(19,60,61)(20,49,62)(21,50,63)(22,51,64)(23,52,65)(24,53,66), (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)(49,50,51,52,53,54,55,56,57,58,59,60)(61,62,63,64,65,66,67,68,69,70,71,72), (1,59,7,53)(2,58,8,52)(3,57,9,51)(4,56,10,50)(5,55,11,49)(6,54,12,60)(13,44,19,38)(14,43,20,37)(15,42,21,48)(16,41,22,47)(17,40,23,46)(18,39,24,45)(25,72,31,66)(26,71,32,65)(27,70,33,64)(28,69,34,63)(29,68,35,62)(30,67,36,61), (1,41,29)(2,42,30)(3,43,31)(4,44,32)(5,45,33)(6,46,34)(7,47,35)(8,48,36)(9,37,25)(10,38,26)(11,39,27)(12,40,28)(13,71,50)(14,72,51)(15,61,52)(16,62,53)(17,63,54)(18,64,55)(19,65,56)(20,66,57)(21,67,58)(22,68,59)(23,69,60)(24,70,49), (1,50)(2,57)(3,52)(4,59)(5,54)(6,49)(7,56)(8,51)(9,58)(10,53)(11,60)(12,55)(13,29)(14,36)(15,31)(16,26)(17,33)(18,28)(19,35)(20,30)(21,25)(22,32)(23,27)(24,34)(37,67)(38,62)(39,69)(40,64)(41,71)(42,66)(43,61)(44,68)(45,63)(46,70)(47,65)(48,72)>;
 
G:=Group( (1,25,45)(2,26,46)(3,27,47)(4,28,48)(5,29,37)(6,30,38)(7,31,39)(8,32,40)(9,33,41)(10,34,42)(11,35,43)(12,36,44)(13,54,67)(14,55,68)(15,56,69)(16,57,70)(17,58,71)(18,59,72)(19,60,61)(20,49,62)(21,50,63)(22,51,64)(23,52,65)(24,53,66), (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)(49,50,51,52,53,54,55,56,57,58,59,60)(61,62,63,64,65,66,67,68,69,70,71,72), (1,59,7,53)(2,58,8,52)(3,57,9,51)(4,56,10,50)(5,55,11,49)(6,54,12,60)(13,44,19,38)(14,43,20,37)(15,42,21,48)(16,41,22,47)(17,40,23,46)(18,39,24,45)(25,72,31,66)(26,71,32,65)(27,70,33,64)(28,69,34,63)(29,68,35,62)(30,67,36,61), (1,41,29)(2,42,30)(3,43,31)(4,44,32)(5,45,33)(6,46,34)(7,47,35)(8,48,36)(9,37,25)(10,38,26)(11,39,27)(12,40,28)(13,71,50)(14,72,51)(15,61,52)(16,62,53)(17,63,54)(18,64,55)(19,65,56)(20,66,57)(21,67,58)(22,68,59)(23,69,60)(24,70,49), (1,50)(2,57)(3,52)(4,59)(5,54)(6,49)(7,56)(8,51)(9,58)(10,53)(11,60)(12,55)(13,29)(14,36)(15,31)(16,26)(17,33)(18,28)(19,35)(20,30)(21,25)(22,32)(23,27)(24,34)(37,67)(38,62)(39,69)(40,64)(41,71)(42,66)(43,61)(44,68)(45,63)(46,70)(47,65)(48,72) );
 
G=PermutationGroup([(1,25,45),(2,26,46),(3,27,47),(4,28,48),(5,29,37),(6,30,38),(7,31,39),(8,32,40),(9,33,41),(10,34,42),(11,35,43),(12,36,44),(13,54,67),(14,55,68),(15,56,69),(16,57,70),(17,58,71),(18,59,72),(19,60,61),(20,49,62),(21,50,63),(22,51,64),(23,52,65),(24,53,66)], [(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),(49,50,51,52,53,54,55,56,57,58,59,60),(61,62,63,64,65,66,67,68,69,70,71,72)], [(1,59,7,53),(2,58,8,52),(3,57,9,51),(4,56,10,50),(5,55,11,49),(6,54,12,60),(13,44,19,38),(14,43,20,37),(15,42,21,48),(16,41,22,47),(17,40,23,46),(18,39,24,45),(25,72,31,66),(26,71,32,65),(27,70,33,64),(28,69,34,63),(29,68,35,62),(30,67,36,61)], [(1,41,29),(2,42,30),(3,43,31),(4,44,32),(5,45,33),(6,46,34),(7,47,35),(8,48,36),(9,37,25),(10,38,26),(11,39,27),(12,40,28),(13,71,50),(14,72,51),(15,61,52),(16,62,53),(17,63,54),(18,64,55),(19,65,56),(20,66,57),(21,67,58),(22,68,59),(23,69,60),(24,70,49)], [(1,50),(2,57),(3,52),(4,59),(5,54),(6,49),(7,56),(8,51),(9,58),(10,53),(11,60),(12,55),(13,29),(14,36),(15,31),(16,26),(17,33),(18,28),(19,35),(20,30),(21,25),(22,32),(23,27),(24,34),(37,67),(38,62),(39,69),(40,64),(41,71),(42,66),(43,61),(44,68),(45,63),(46,70),(47,65),(48,72)])
 

51 conjugacy classes

class 1 2A2B2C2D3A···3E3F3G3H3I4A4B4C4D4E6A···6E6F6G6H6I6J6K12A···12M12N···12U
order122223···33333444446···666666612···1212···12
size111818542···2444426627272···2444436364···412···12

51 irreducible representations

dim11111122222244444
type++++++++++++-++
imageC1C2C2C2C2C2S3S3D6D6D6C4○D4S32D4⋊2S3Q8⋊3S3C2×S32D12⋊S3
kernel(C3×Dic6)⋊S3Dic3×C3⋊S3C33⋊8D4C32×Dic6C3×C12⋊S3C4×C33⋊C2C3×Dic6C12⋊S3C3×Dic3C3×C12C2×C3⋊S3C33C12C32C32C6C3
# reps12211141852241448

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

10000000
01000000
001210000
001200000
00001000
00000100
00000010
00000001
,
80000000
05000000
00100000
00010000
000012000
000001200
000000121
000000120
,
05000000
50000000
00100000
00010000
000012000
000001200
00000001
00000010
,
10000000
01000000
000120000
001120000
000012100
000012000
00000010
00000001
,
01000000
10000000
00010000
00100000
00000100
00001000
00000010
00000001

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

(C3×Dic6)⋊S3 in GAP, Magma, Sage, TeX 

(C_3\times {\rm Dic}_6)\rtimes S_3
 
% in TeX
 
G:=Group("(C3xDic6):S3");
 
// GroupNames label
 
G:=SmallGroup(432,664);
 
// by ID
 
G=gap.SmallGroup(432,664);
 
# by ID
 
G:=PCGroup([7,-2,-2,-2,-2,-3,-3,-3,56,254,135,58,571,2028,14118]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e|a^3=b^12=d^3=e^2=1,c^2=b^6,a*b=b*a,a*c=c*a,a*d=d*a,e*a*e=a^-1,c*b*c^-1=b^-1,b*d=d*b,e*b*e=b^7,c*d=d*c,c*e=e*c,e*d*e=d^-1>;
 
// generators/relations
 

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