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G = S3×C3⋊Dic3order 216 = 23·33

Direct product of S3 and C3⋊Dic3

direct product, metabelian, supersoluble, monomial, A-group

Aliases: S3×C3⋊Dic3, C6.9S32, D6.(C3⋊S3), (C3×S3)⋊Dic3, C336(C2×C4), (S3×C6).5S3, C33(S3×Dic3), (C3×C6).43D6, (S3×C32)⋊3C4, C3213(C4×S3), C335C41C2, C325(C2×Dic3), (C32×C6).6C22, C6.1(C2×C3⋊S3), C2.1(S3×C3⋊S3), (S3×C3×C6).2C2, C31(C2×C3⋊Dic3), (C3×C3⋊Dic3)⋊4C2, SmallGroup(216,124)

Series: Derived Chief Lower central Upper central

C1C33 — S3×C3⋊Dic3
C1C3C32C33C32×C6S3×C3×C6 — S3×C3⋊Dic3
C33 — S3×C3⋊Dic3
C1C2

Generators and relations for S3×C3⋊Dic3
 G = < a,b,c,d,e | a3=b2=c3=d6=1, e2=d3, bab=a-1, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, cd=dc, ece-1=c-1, ede-1=d-1 >

Subgroups: 444 in 120 conjugacy classes, 46 normal (14 characteristic)
C1, C2, C2, C3, C3, C3, C4, C22, S3, C6, C6, C6, C2×C4, C32, C32, C32, Dic3, C12, D6, C2×C6, C3×S3, C3×C6, C3×C6, C3×C6, C4×S3, C2×Dic3, C33, C3×Dic3, C3⋊Dic3, C3⋊Dic3, S3×C6, C62, S3×C32, C32×C6, S3×Dic3, C2×C3⋊Dic3, C3×C3⋊Dic3, C335C4, S3×C3×C6, S3×C3⋊Dic3
Quotients: C1, C2, C4, C22, S3, C2×C4, Dic3, D6, C3⋊S3, C4×S3, C2×Dic3, C3⋊Dic3, S32, C2×C3⋊S3, S3×Dic3, C2×C3⋊Dic3, S3×C3⋊S3, S3×C3⋊Dic3

Smallest permutation representation of S3×C3⋊Dic3
On 72 points
Generators in S72
(1 24 25)(2 19 26)(3 20 27)(4 21 28)(5 22 29)(6 23 30)(7 71 14)(8 72 15)(9 67 16)(10 68 17)(11 69 18)(12 70 13)(31 45 39)(32 46 40)(33 47 41)(34 48 42)(35 43 37)(36 44 38)(49 59 66)(50 60 61)(51 55 62)(52 56 63)(53 57 64)(54 58 65)
(1 60)(2 55)(3 56)(4 57)(5 58)(6 59)(7 43)(8 44)(9 45)(10 46)(11 47)(12 48)(13 42)(14 37)(15 38)(16 39)(17 40)(18 41)(19 51)(20 52)(21 53)(22 54)(23 49)(24 50)(25 61)(26 62)(27 63)(28 64)(29 65)(30 66)(31 67)(32 68)(33 69)(34 70)(35 71)(36 72)
(1 20 29)(2 21 30)(3 22 25)(4 23 26)(5 24 27)(6 19 28)(7 67 18)(8 68 13)(9 69 14)(10 70 15)(11 71 16)(12 72 17)(31 41 43)(32 42 44)(33 37 45)(34 38 46)(35 39 47)(36 40 48)(49 62 57)(50 63 58)(51 64 59)(52 65 60)(53 66 55)(54 61 56)
(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 39 4 42)(2 38 5 41)(3 37 6 40)(7 66 10 63)(8 65 11 62)(9 64 12 61)(13 60 16 57)(14 59 17 56)(15 58 18 55)(19 36 22 33)(20 35 23 32)(21 34 24 31)(25 45 28 48)(26 44 29 47)(27 43 30 46)(49 68 52 71)(50 67 53 70)(51 72 54 69)

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

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

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

S3×C3⋊Dic3 is a maximal subgroup of
C33⋊M4(2)  S32×Dic3  D6.4S32  D6⋊S3⋊S3  D6.6S32  (C3×D12)⋊S3  D12⋊(C3⋊S3)  C4×S3×C3⋊S3  C62.90D6  C62.91D6
S3×C3⋊Dic3 is a maximal quotient of
C337M4(2)  C62.77D6  C62.80D6

36 conjugacy classes

class 1 2A2B2C3A···3E3F3G3H3I4A4B4C4D6A···6E6F6G6H6I6J···6Q12A12B
order12223···3333344446···666666···61212
size11332···244449927272···244446···61818

36 irreducible representations

dim111112222244
type++++++-++-
imageC1C2C2C2C4S3S3Dic3D6C4×S3S32S3×Dic3
kernelS3×C3⋊Dic3C3×C3⋊Dic3C335C4S3×C3×C6S3×C32C3⋊Dic3S3×C6C3×S3C3×C6C32C6C3
# reps111141485244

Matrix representation of S3×C3⋊Dic3 in GL6(𝔽13)

100000
010000
001000
000100
0000121
0000120
,
100000
010000
0012000
0001200
0000012
0000120
,
1210000
1200000
001000
000100
000010
000001
,
0120000
1120000
00121200
001000
0000120
0000012
,
0120000
1200000
001000
00121200
000050
000005

G:=sub<GL(6,GF(13))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,12,12,0,0,0,0,1,0],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,0,12,0,0,0,0,12,0],[12,12,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[0,1,0,0,0,0,12,12,0,0,0,0,0,0,12,1,0,0,0,0,12,0,0,0,0,0,0,0,12,0,0,0,0,0,0,12],[0,12,0,0,0,0,12,0,0,0,0,0,0,0,1,12,0,0,0,0,0,12,0,0,0,0,0,0,5,0,0,0,0,0,0,5] >;

S3×C3⋊Dic3 in GAP, Magma, Sage, TeX

S_3\times C_3\rtimes {\rm Dic}_3
% in TeX

G:=Group("S3xC3:Dic3");
// GroupNames label

G:=SmallGroup(216,124);
// by ID

G=gap.SmallGroup(216,124);
# by ID

G:=PCGroup([6,-2,-2,-2,-3,-3,-3,31,201,730,5189]);
// Polycyclic

G:=Group<a,b,c,d,e|a^3=b^2=c^3=d^6=1,e^2=d^3,b*a*b=a^-1,a*c=c*a,a*d=d*a,a*e=e*a,b*c=c*b,b*d=d*b,b*e=e*b,c*d=d*c,e*c*e^-1=c^-1,e*d*e^-1=d^-1>;
// generators/relations

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