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G = (S3×C12)⋊S3  order 432 = 24·33

1st semidirect product of S3×C12 and S3 acting via S3/C3=C2

metabelian, supersoluble, monomial

Aliases: (S3×C12)⋊1S3, C12.73(S32), (S3×C6).40D6, (C3×C12).170D6, C33⋊16(C4○D4), C33⋊8D4⋊11C2, C33⋊7D4⋊11C2, C33⋊6D4⋊11C2, C3⋊Dic3.45D6, C33⋊4Q8⋊11C2, C3⋊4(D6.D6), (C3×Dic3).35D6, C3⋊2(C12.59D6), C32⋊10(C4○D12), (C32×C6).45C23, (C32×C12).73C22, C33⋊5C4.16C22, (C32×Dic3).22C22, (S3×C3×C12)⋊1C2, C6.55(C2×S32), (C12×C3⋊S3)⋊1C2, (C4×C3⋊S3)⋊11S3, C4.28(S3×C3⋊S3), D6.5(C2×C3⋊S3), (C4×S3)⋊4(C3⋊S3), C12.43(C2×C3⋊S3), (C2×C3⋊S3).44D6, C6.8(C22×C3⋊S3), (C4×C33⋊C2)⋊7C2, (S3×C3×C6).24C22, Dic3.8(C2×C3⋊S3), (C6×C3⋊S3).53C22, (C3×C6).103(C22×S3), (C3×C3⋊Dic3).43C22, (C2×C33⋊C2).14C22, C2.12(C2×S3×C3⋊S3), SmallGroup(432,667)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C32×C6 — (S3×C12)⋊S3
Chief series
C1 — C3 — C32 — C33 — C32×C6 — S3×C3×C6 — C33⋊6D4 — (S3×C12)⋊S3
Lower central
C33 — C32×C6 — (S3×C12)⋊S3
Upper central
C1 — C4

Generators and relations for (S3×C12)⋊S3
 G = < a,b,c,d,e | a12=b3=c2=d3=e2=1, ab=ba, ac=ca, ad=da, eae=a5, cbc=b-1, bd=db, be=eb, cd=dc, ece=a6c, ede=d-1 >

Subgroups: 1784 in 304 conjugacy classes, 68 normal (32 characteristic)
C1, C2, C2 [×3], C3, C3 [×4], C3 [×4], C4, C4 [×3], C22 [×3], S3 [×18], C6, C6 [×4], C6 [×9], C2×C4 [×3], D4 [×3], Q8, C32, C32 [×4], C32 [×4], Dic3, Dic3 [×13], C12, C12 [×4], C12 [×9], D6, D6 [×13], C2×C6 [×5], C4○D4, C3×S3 [×8], C3⋊S3 [×14], C3×C6, C3×C6 [×4], C3×C6 [×5], Dic6 [×5], C4×S3, C4×S3 [×13], D12 [×5], C3⋊D4 [×10], C2×C12 [×5], C33, C3×Dic3 [×4], C3×Dic3 [×4], C3⋊Dic3, C3⋊Dic3 [×9], C3×C12, C3×C12 [×4], C3×C12 [×5], S3×C6 [×4], S3×C6 [×4], C2×C3⋊S3, C2×C3⋊S3 [×9], C62, C4○D12 [×5], S3×C32, C3×C3⋊S3, C33⋊C2, C32×C6, D6⋊S3 [×4], C3⋊D12 [×8], C32⋊2Q8 [×4], S3×C12 [×4], S3×C12 [×4], C32⋊4Q8, C4×C3⋊S3, C4×C3⋊S3 [×9], C12⋊S3, C32⋊7D4 [×2], C6×C12, C32×Dic3, C3×C3⋊Dic3, C33⋊5C4, C32×C12, S3×C3×C6, C6×C3⋊S3, C2×C33⋊C2, D6.D6 [×4], C12.59D6, C33⋊6D4, C33⋊7D4, C33⋊8D4, C33⋊4Q8, S3×C3×C12, C12×C3⋊S3, C4×C33⋊C2, (S3×C12)⋊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], C4○D12 [×5], C2×S32 [×4], C22×C3⋊S3, S3×C3⋊S3, D6.D6 [×4], C12.59D6, C2×S3×C3⋊S3, (S3×C12)⋊S3

Smallest permutation representation of (S3×C12)⋊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 58 35)(2 59 36)(3 60 25)(4 49 26)(5 50 27)(6 51 28)(7 52 29)(8 53 30)(9 54 31)(10 55 32)(11 56 33)(12 57 34)(13 61 42)(14 62 43)(15 63 44)(16 64 45)(17 65 46)(18 66 47)(19 67 48)(20 68 37)(21 69 38)(22 70 39)(23 71 40)(24 72 41)

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

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

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

66 conjugacy classes

class 1 2A2B2C2D3A···3E3F3G3H3I4A4B4C4D4E6A···6E6F6G6H6I6J···6Q6R6S12A···12J12K···12R12S···12Z12AA12AB
order122223···33333444446···666666···66612···1212···1212···121212
size11618542···2444411618542···244446···618182···24···46···61818

66 irreducible representations

dim11111111222222222444
type+++++++++++++++++
imageC1C2C2C2C2C2C2C2S3S3D6D6D6D6D6C4○D4C4○D12S32C2×S32D6.D6
kernel(S3×C12)⋊S3C33⋊6D4C33⋊7D4C33⋊8D4C33⋊4Q8S3×C3×C12C12×C3⋊S3C4×C33⋊C2S3×C12C4×C3⋊S3C3×Dic3C3⋊Dic3C3×C12S3×C6C2×C3⋊S3C33C32C12C6C3
# reps111111114141541220448

Matrix representation of (S3×C12)⋊S3 ►in GL8(𝔽13)

50000000
05000000
001120000
00100000
000012000
000001200
00000010
00000001
,
10000000
01000000
00100000
00010000
00001000
00000100
000000012
000000112
,
72000000
26000000
001200000
000120000
000012000
000001200
00000001
00000010
,
10000000
01000000
001210000
001200000
000012100
000012000
00000010
00000001
,
43000000
89000000
00010000
00100000
00000100
00001000
00000010
00000001

G:=sub<GL(8,GF(13))| [5,0,0,0,0,0,0,0,0,5,0,0,0,0,0,0,0,0,1,1,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,12,12],[7,2,0,0,0,0,0,0,2,6,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,12,12,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,12,12,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[4,8,0,0,0,0,0,0,3,9,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1] >;
 

(S3×C12)⋊S3 in GAP, Magma, Sage, TeX 

(S_3\times C_{12})\rtimes S_3
 
% in TeX
 
G:=Group("(S3xC12):S3");
 
// GroupNames label
 
G:=SmallGroup(432,667);
 
// by ID
 
G=gap.SmallGroup(432,667);
 
# by ID
 
G:=PCGroup([7,-2,-2,-2,-2,-3,-3,-3,141,58,571,2028,14118]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e|a^12=b^3=c^2=d^3=e^2=1,a*b=b*a,a*c=c*a,a*d=d*a,e*a*e=a^5,c*b*c=b^-1,b*d=d*b,b*e=e*b,c*d=d*c,e*c*e=a^6*c,e*d*e=d^-1>;
 
// generators/relations
 

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