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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, (C3xD12):7S3, D12:3(C3:S3), (S3xC6).14D6, C33:7D4:7C2, (C3xC12).140D6, C33:13(C4oD4), C3:Dic3.35D6, C3:4(D12:S3), C32:4Q8:10S3, (C32xD12):11C2, C3:1(C12.D6), (C32xC6).40C23, C32:13(D4:2S3), C32:14(Q8:3S3), (C32xC12).42C22, C33:5C4.13C22, C6.50(C2xS32), C4.19(S3xC3:S3), D6.2(C2xC3:S3), C12.22(C2xC3:S3), (S3xC3:Dic3):4C2, C6.3(C22xC3:S3), (C4xC33:C2):1C2, (S3xC3xC6).15C22, (C3xC32:4Q8):9C2, (C3xC6).99(C22xS3), (C3xC3:Dic3).18C22, (C2xC33:C2).11C22, C2.7(C2xS3xC3:S3), SmallGroup(432,662)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C32xC6 — D12:(C3:S3)
Chief series
C1 — C3 — C32 — C33 — C32xC6 — S3xC3xC6 — S3xC3:Dic3 — D12:(C3:S3)
Lower central
C33 — C32xC6 — 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, C2xC4, D4, Q8, C32, C32, C32, Dic3, C12, C12, C12, D6, D6, C2xC6, C4oD4, C3xS3, C3:S3, C3xC6, C3xC6, C3xC6, Dic6, C4xS3, D12, D12, C2xDic3, C3:D4, C3xD4, C3xQ8, C33, C3xDic3, C3:Dic3, C3:Dic3, C3xC12, C3xC12, C3xC12, S3xC6, C2xC3:S3, C62, D4:2S3, Q8:3S3, S3xC32, C33:C2, C32xC6, S3xDic3, C3:D12, C3xDic6, C3xD12, C32:4Q8, C4xC3:S3, C2xC3:Dic3, C32:7D4, D4xC32, C3xC3:Dic3, C33:5C4, C32xC12, S3xC3xC6, C2xC33:C2, D12:S3, C12.D6, S3xC3:Dic3, C33:7D4, C32xD12, C3xC32:4Q8, C4xC33:C2, D12:(C3:S3)
Quotients: C1, C2, C22, S3, C23, D6, C4oD4, C3:S3, C22xS3, S32, C2xC3:S3, D4:2S3, Q8:3S3, C2xS32, C22xC3:S3, S3xC3:S3, D12:S3, C12.D6, C2xS3xC3: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++++++++++++-++
imageC1C2C2C2C2C2S3S3D6D6D6C4oD4S32D4:2S3Q8:3S3C2xS32D12:S3
kernelD12:(C3:S3)S3xC3:Dic3C33:7D4C32xD12C3xC32:4Q8C4xC33:C2C3xD12C32:4Q8C3:Dic3C3xC12S3xC6C33C12C32C32C6C3
# 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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