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G = S3×C41D4order 192 = 26·3

Direct product of S3 and C41D4

direct product, metabelian, supersoluble, monomial, rational, 2-hyperelementary

Aliases: S3×C41D4, C4236D6, C41(S3×D4), (C4×S3)⋊7D4, C122(C2×D4), (C2×D4)⋊25D6, Dic31(C2×D4), D6.61(C2×D4), C123D425C2, C4⋊D1216C2, (S3×C42)⋊12C2, (C4×C12)⋊25C22, (C6×D4)⋊17C22, C6.92(C22×D4), (C2×D12)⋊30C22, (C2×C6).258C24, (C2×C12).507C23, (C4×Dic3)⋊65C22, C23.74(C22×S3), (C22×C6).72C23, (S3×C23).71C22, C22.279(S3×C23), (C22×S3).259C23, (C2×Dic3).268C23, (C2×S3×D4)⋊18C2, C32(C2×C41D4), C2.65(C2×S3×D4), (C3×C41D4)⋊5C2, (C2×C3⋊D4)⋊25C22, (S3×C2×C4).250C22, (C2×C4).596(C22×S3), SmallGroup(192,1273)

Series: Derived Chief Lower central Upper central

C1C2×C6 — S3×C41D4
C1C3C6C2×C6C22×S3S3×C2×C4C2×S3×D4 — S3×C41D4
C3C2×C6 — S3×C41D4
C1C22C41D4

Generators and relations for S3×C41D4
 G = < a,b,c,d,e | a3=b2=c4=d4=e2=1, bab=a-1, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, cd=dc, ece=c-1, ede=d-1 >

Subgroups: 1584 in 498 conjugacy classes, 131 normal (12 characteristic)
C1, C2 [×3], C2 [×12], C3, C4 [×6], C4 [×6], C22, C22 [×46], S3 [×4], S3 [×4], C6 [×3], C6 [×4], C2×C4 [×3], C2×C4 [×15], D4 [×48], C23 [×4], C23 [×29], Dic3 [×6], C12 [×6], D6 [×6], D6 [×28], C2×C6, C2×C6 [×12], C42, C42 [×3], C22×C4 [×3], C2×D4 [×6], C2×D4 [×42], C24 [×4], C4×S3 [×12], D12 [×12], C2×Dic3 [×3], C3⋊D4 [×24], C2×C12 [×3], C3×D4 [×12], C22×S3, C22×S3 [×4], C22×S3 [×24], C22×C6 [×4], C2×C42, C41D4, C41D4 [×7], C22×D4 [×6], C4×Dic3 [×3], C4×C12, S3×C2×C4 [×3], C2×D12 [×6], S3×D4 [×24], C2×C3⋊D4 [×12], C6×D4 [×6], S3×C23 [×4], C2×C41D4, S3×C42, C4⋊D12, C123D4 [×6], C3×C41D4, C2×S3×D4 [×6], S3×C41D4
Quotients: C1, C2 [×15], C22 [×35], S3, D4 [×12], C23 [×15], D6 [×7], C2×D4 [×18], C24, C22×S3 [×7], C41D4 [×4], C22×D4 [×3], S3×D4 [×6], S3×C23, C2×C41D4, C2×S3×D4 [×3], S3×C41D4

Smallest permutation representation of S3×C41D4
On 48 points
Generators in S48
(1 23 34)(2 24 35)(3 21 36)(4 22 33)(5 30 42)(6 31 43)(7 32 44)(8 29 41)(9 38 25)(10 39 26)(11 40 27)(12 37 28)(13 18 46)(14 19 47)(15 20 48)(16 17 45)
(1 3)(2 4)(5 32)(6 29)(7 30)(8 31)(9 40)(10 37)(11 38)(12 39)(13 15)(14 16)(17 47)(18 48)(19 45)(20 46)(21 34)(22 35)(23 36)(24 33)(25 27)(26 28)(41 43)(42 44)
(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)
(1 41 16 26)(2 42 13 27)(3 43 14 28)(4 44 15 25)(5 18 11 24)(6 19 12 21)(7 20 9 22)(8 17 10 23)(29 45 39 34)(30 46 40 35)(31 47 37 36)(32 48 38 33)
(1 2)(3 4)(5 10)(6 9)(7 12)(8 11)(13 16)(14 15)(17 18)(19 20)(21 22)(23 24)(25 43)(26 42)(27 41)(28 44)(29 40)(30 39)(31 38)(32 37)(33 36)(34 35)(45 46)(47 48)

G:=sub<Sym(48)| (1,23,34)(2,24,35)(3,21,36)(4,22,33)(5,30,42)(6,31,43)(7,32,44)(8,29,41)(9,38,25)(10,39,26)(11,40,27)(12,37,28)(13,18,46)(14,19,47)(15,20,48)(16,17,45), (1,3)(2,4)(5,32)(6,29)(7,30)(8,31)(9,40)(10,37)(11,38)(12,39)(13,15)(14,16)(17,47)(18,48)(19,45)(20,46)(21,34)(22,35)(23,36)(24,33)(25,27)(26,28)(41,43)(42,44), (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), (1,41,16,26)(2,42,13,27)(3,43,14,28)(4,44,15,25)(5,18,11,24)(6,19,12,21)(7,20,9,22)(8,17,10,23)(29,45,39,34)(30,46,40,35)(31,47,37,36)(32,48,38,33), (1,2)(3,4)(5,10)(6,9)(7,12)(8,11)(13,16)(14,15)(17,18)(19,20)(21,22)(23,24)(25,43)(26,42)(27,41)(28,44)(29,40)(30,39)(31,38)(32,37)(33,36)(34,35)(45,46)(47,48)>;

G:=Group( (1,23,34)(2,24,35)(3,21,36)(4,22,33)(5,30,42)(6,31,43)(7,32,44)(8,29,41)(9,38,25)(10,39,26)(11,40,27)(12,37,28)(13,18,46)(14,19,47)(15,20,48)(16,17,45), (1,3)(2,4)(5,32)(6,29)(7,30)(8,31)(9,40)(10,37)(11,38)(12,39)(13,15)(14,16)(17,47)(18,48)(19,45)(20,46)(21,34)(22,35)(23,36)(24,33)(25,27)(26,28)(41,43)(42,44), (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), (1,41,16,26)(2,42,13,27)(3,43,14,28)(4,44,15,25)(5,18,11,24)(6,19,12,21)(7,20,9,22)(8,17,10,23)(29,45,39,34)(30,46,40,35)(31,47,37,36)(32,48,38,33), (1,2)(3,4)(5,10)(6,9)(7,12)(8,11)(13,16)(14,15)(17,18)(19,20)(21,22)(23,24)(25,43)(26,42)(27,41)(28,44)(29,40)(30,39)(31,38)(32,37)(33,36)(34,35)(45,46)(47,48) );

G=PermutationGroup([(1,23,34),(2,24,35),(3,21,36),(4,22,33),(5,30,42),(6,31,43),(7,32,44),(8,29,41),(9,38,25),(10,39,26),(11,40,27),(12,37,28),(13,18,46),(14,19,47),(15,20,48),(16,17,45)], [(1,3),(2,4),(5,32),(6,29),(7,30),(8,31),(9,40),(10,37),(11,38),(12,39),(13,15),(14,16),(17,47),(18,48),(19,45),(20,46),(21,34),(22,35),(23,36),(24,33),(25,27),(26,28),(41,43),(42,44)], [(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)], [(1,41,16,26),(2,42,13,27),(3,43,14,28),(4,44,15,25),(5,18,11,24),(6,19,12,21),(7,20,9,22),(8,17,10,23),(29,45,39,34),(30,46,40,35),(31,47,37,36),(32,48,38,33)], [(1,2),(3,4),(5,10),(6,9),(7,12),(8,11),(13,16),(14,15),(17,18),(19,20),(21,22),(23,24),(25,43),(26,42),(27,41),(28,44),(29,40),(30,39),(31,38),(32,37),(33,36),(34,35),(45,46),(47,48)])

42 conjugacy classes

class 1 2A2B2C2D2E2F2G2H2I2J2K2L2M2N2O 3 4A···4F4G···4L6A6B6C6D6E6F6G12A···12F
order122222222222222234···44···4666666612···12
size1111333344441212121222···26···622288884···4

42 irreducible representations

dim11111122224
type+++++++++++
imageC1C2C2C2C2C2S3D4D6D6S3×D4
kernelS3×C41D4S3×C42C4⋊D12C123D4C3×C41D4C2×S3×D4C41D4C4×S3C42C2×D4C4
# reps111616112166

Matrix representation of S3×C41D4 in GL6(𝔽13)

100000
010000
001000
000100
0000012
0000112
,
1200000
0120000
001000
000100
000001
000010
,
1110000
1120000
001000
000100
0000120
0000012
,
1110000
1120000
0012500
0010100
0000120
0000012
,
1220000
010000
001000
0031200
0000120
0000012

G:=sub<GL(6,GF(13))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,12,12],[12,0,0,0,0,0,0,12,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,1,0],[1,1,0,0,0,0,11,12,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,12,0,0,0,0,0,0,12],[1,1,0,0,0,0,11,12,0,0,0,0,0,0,12,10,0,0,0,0,5,1,0,0,0,0,0,0,12,0,0,0,0,0,0,12],[12,0,0,0,0,0,2,1,0,0,0,0,0,0,1,3,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,12] >;

S3×C41D4 in GAP, Magma, Sage, TeX

S_3\times C_4\rtimes_1D_4
% in TeX

G:=Group("S3xC4:1D4");
// GroupNames label

G:=SmallGroup(192,1273);
// by ID

G=gap.SmallGroup(192,1273);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,232,387,570,185,6278]);
// Polycyclic

G:=Group<a,b,c,d,e|a^3=b^2=c^4=d^4=e^2=1,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=c^-1,e*d*e=d^-1>;
// generators/relations

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