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G = S3xC4:1D4order 192 = 26·3

Direct product of S3 and C4:1D4

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

Aliases: S3xC4:1D4, C42:36D6, C4:1(S3xD4), (C4xS3):7D4, C12:2(C2xD4), (C2xD4):25D6, Dic3:1(C2xD4), D6.61(C2xD4), C12:3D4:25C2, C4:D12:16C2, (S3xC42):12C2, (C4xC12):25C22, (C6xD4):17C22, C6.92(C22xD4), (C2xD12):30C22, (C2xC6).258C24, (C2xC12).507C23, (C4xDic3):65C22, C23.74(C22xS3), (C22xC6).72C23, (S3xC23).71C22, C22.279(S3xC23), (C22xS3).259C23, (C2xDic3).268C23, (C2xS3xD4):18C2, C3:2(C2xC4:1D4), C2.65(C2xS3xD4), (C3xC4:1D4):5C2, (C2xC3:D4):25C22, (S3xC2xC4).250C22, (C2xC4).596(C22xS3), SmallGroup(192,1273)

Series: Derived Chief Lower central Upper central

C1C2xC6 — S3xC4:1D4
C1C3C6C2xC6C22xS3S3xC2xC4C2xS3xD4 — S3xC4:1D4
C3C2xC6 — S3xC4:1D4
C1C22C4:1D4

Generators and relations for S3xC4:1D4
 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, C2, C3, C4, C4, C22, C22, S3, S3, C6, C6, C2xC4, C2xC4, D4, C23, C23, Dic3, C12, D6, D6, C2xC6, C2xC6, C42, C42, C22xC4, C2xD4, C2xD4, C24, C4xS3, D12, C2xDic3, C3:D4, C2xC12, C3xD4, C22xS3, C22xS3, C22xS3, C22xC6, C2xC42, C4:1D4, C4:1D4, C22xD4, C4xDic3, C4xC12, S3xC2xC4, C2xD12, S3xD4, C2xC3:D4, C6xD4, S3xC23, C2xC4:1D4, S3xC42, C4:D12, C12:3D4, C3xC4:1D4, C2xS3xD4, S3xC4:1D4
Quotients: C1, C2, C22, S3, D4, C23, D6, C2xD4, C24, C22xS3, C4:1D4, C22xD4, S3xD4, S3xC23, C2xC4:1D4, C2xS3xD4, S3xC4:1D4

Smallest permutation representation of S3xC4:1D4
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+++++++++++
imageC1C2C2C2C2C2S3D4D6D6S3xD4
kernelS3xC4:1D4S3xC42C4:D12C12:3D4C3xC4:1D4C2xS3xD4C4:1D4C4xS3C42C2xD4C4
# reps111616112166

Matrix representation of S3xC4:1D4 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] >;

S3xC4:1D4 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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