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G = D12⋊(C3⋊S3)  order 432 = 24·33

3rd semidirect product of D12 and C3⋊S3 acting via C3⋊S3/C32=C2

metabelian, supersoluble, monomial

Aliases: C12.37S32, (C3×D12)⋊7S3, D12⋊3(C3⋊S3), (S3×C6).14D6, C33⋊7D4⋊7C2, (C3×C12).140D6, C33⋊13(C4○D4), C3⋊Dic3.35D6, C3⋊4(D12⋊S3), C32⋊4Q8⋊10S3, (C32×D12)⋊11C2, C3⋊1(C12.D6), (C32×C6).40C23, C32⋊13(D4⋊2S3), C32⋊14(Q8⋊3S3), (C32×C12).42C22, C33⋊5C4.13C22, C6.50(C2×S32), C4.19(S3×C3⋊S3), D6.2(C2×C3⋊S3), C12.22(C2×C3⋊S3), (S3×C3⋊Dic3)⋊4C2, C6.3(C22×C3⋊S3), (C4×C33⋊C2)⋊1C2, (S3×C3×C6).15C22, (C3×C32⋊4Q8)⋊9C2, (C3×C6).99(C22×S3), (C3×C3⋊Dic3).18C22, (C2×C33⋊C2).11C22, C2.7(C2×S3×C3⋊S3), SmallGroup(432,662)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C32×C6 — D12⋊(C3⋊S3)
Chief series
C1 — C3 — C32 — C33 — C32×C6 — S3×C3×C6 — S3×C3⋊Dic3 — D12⋊(C3⋊S3)
Lower central
C33 — C32×C6 — D12⋊(C3⋊S3)
Upper central
C1 — C2 — C4

Generators and relations for D12⋊(C3⋊S3)
 G = < a,b,c,d,e | a12=b2=c3=d3=e2=1, bab=a-1, ac=ca, ad=da, eae=a5, bc=cb, bd=db, ebe=a10b, cd=dc, ece=c-1, ede=d-1 >

Subgroups: 1744 in 304 conjugacy classes, 68 normal (20 characteristic)
C1, C2, C2, C3, C3, C3, C4, C4, C22, S3, C6, C6, C6, C2×C4, D4, Q8, C32, C32, C32, Dic3, C12, C12, C12, D6, D6, C2×C6, C4○D4, C3×S3, C3⋊S3, C3×C6, C3×C6, C3×C6, Dic6, C4×S3, D12, D12, C2×Dic3, C3⋊D4, C3×D4, C3×Q8, C33, C3×Dic3, C3⋊Dic3, C3⋊Dic3, C3×C12, C3×C12, C3×C12, S3×C6, C2×C3⋊S3, C62, D4⋊2S3, Q8⋊3S3, S3×C32, C33⋊C2, C32×C6, S3×Dic3, C3⋊D12, C3×Dic6, C3×D12, C32⋊4Q8, C4×C3⋊S3, C2×C3⋊Dic3, C32⋊7D4, D4×C32, C3×C3⋊Dic3, C33⋊5C4, C32×C12, S3×C3×C6, C2×C33⋊C2, D12⋊S3, C12.D6, S3×C3⋊Dic3, C33⋊7D4, C32×D12, C3×C32⋊4Q8, C4×C33⋊C2, D12⋊(C3⋊S3)
Quotients: C1, C2, C22, S3, C23, D6, C4○D4, C3⋊S3, C22×S3, S32, C2×C3⋊S3, D4⋊2S3, Q8⋊3S3, C2×S32, C22×C3⋊S3, S3×C3⋊S3, D12⋊S3, C12.D6, C2×S3×C3⋊S3, D12⋊(C3⋊S3)

Smallest permutation representation of D12⋊(C3⋊S3)
►On 72 points
Generators in S72
(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 17)(2 16)(3 15)(4 14)(5 13)(6 24)(7 23)(8 22)(9 21)(10 20)(11 19)(12 18)(25 58)(26 57)(27 56)(28 55)(29 54)(30 53)(31 52)(32 51)(33 50)(34 49)(35 60)(36 59)(37 66)(38 65)(39 64)(40 63)(41 62)(42 61)(43 72)(44 71)(45 70)(46 69)(47 68)(48 67)

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

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

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

51 conjugacy classes

class 1 2A2B2C2D3A···3E3F3G3H3I4A4B4C4D4E6A···6E6F6G6H6I6J···6Q12A···12M12N12O
order122223···33333444446···666666···612···121212
size1166542···244442181827272···2444412···124···43636

51 irreducible representations

dim11111122222244444
type++++++++++++-++
imageC1C2C2C2C2C2S3S3D6D6D6C4○D4S32D4⋊2S3Q8⋊3S3C2×S32D12⋊S3
kernelD12⋊(C3⋊S3)S3×C3⋊Dic3C33⋊7D4C32×D12C3×C32⋊4Q8C4×C33⋊C2C3×D12C32⋊4Q8C3⋊Dic3C3×C12S3×C6C33C12C32C32C6C3
# reps12211141258244148

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

80000000
25000000
00100000
00010000
00001000
00000100
00000001
0000001212
,
1011000000
43000000
00100000
00010000
000012000
000001200
00000001
00000010
,
10000000
01000000
001100000
001110000
00001000
00000100
00000010
00000001
,
10000000
01000000
001100000
001110000
000012100
000012000
00000010
00000001
,
10000000
1012000000
001230000
00010000
00000100
00001000
00000010
0000001212

G:=sub<GL(8,GF(13))| [8,2,0,0,0,0,0,0,0,5,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,1,12],[10,4,0,0,0,0,0,0,11,3,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,1,0,0,0,0,0,0,10,11,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,1,0,0,0,0,0,0,10,11,0,0,0,0,0,0,0,0,12,12,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[1,10,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,3,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,12,0,0,0,0,0,0,0,12] >;
 

D12⋊(C3⋊S3) in GAP, Magma, Sage, TeX 

D_{12}\rtimes (C_3\rtimes S_3)
 
% in TeX
 
G:=Group("D12:(C3:S3)");
 
// GroupNames label
 
G:=SmallGroup(432,662);
 
// by ID
 
G=gap.SmallGroup(432,662);
 
# by ID
 
G:=PCGroup([7,-2,-2,-2,-2,-3,-3,-3,64,254,135,58,571,2028,14118]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e|a^12=b^2=c^3=d^3=e^2=1,b*a*b=a^-1,a*c=c*a,a*d=d*a,e*a*e=a^5,b*c=c*b,b*d=d*b,e*b*e=a^10*b,c*d=d*c,e*c*e=c^-1,e*d*e=d^-1>;
 
// generators/relations
 

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