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G = S3xC32:2Q8order 432 = 24·33

Direct product of S3 and C32:2Q8

direct product, metabelian, supersoluble, monomial

Aliases: S3xC32:2Q8, D6.9S32, C33:4(C2xQ8), C3:1(S3xDic6), Dic3.1S32, (S3xDic3).S3, (C3xS3):1Dic6, (S3xC6).19D6, (S3xC32):2Q8, C32:11(S3xQ8), C33:4Q8:1C2, C33:5Q8:3C2, C3:Dic3.27D6, C32:6(C2xDic6), (C3xDic3).19D6, (C32xC6).10C23, C33:5C4.1C22, (C32xDic3).1C22, C2.10S33, C6.10(C2xS32), C3:1(C2xC32:2Q8), (S3xC3xC6).3C22, (C3xS3xDic3).1C2, (C3xC32:2Q8):2C2, (S3xC3:Dic3).1C2, (C3xC6).59(C22xS3), (C3xC3:Dic3).4C22, SmallGroup(432,603)

Series: Derived Chief Lower central Upper central

C1C32xC6 — S3xC32:2Q8
C1C3C32C33C32xC6S3xC3xC6C3xS3xDic3 — S3xC32:2Q8
C33C32xC6 — S3xC32:2Q8
C1C2

Generators and relations for S3xC32:2Q8
 G = < a,b,c,d,e,f | a3=b2=c3=d3=e4=1, f2=e2, bab=a-1, ac=ca, ad=da, ae=ea, af=fa, bc=cb, bd=db, be=eb, bf=fb, cd=dc, ece-1=c-1, cf=fc, de=ed, fdf-1=d-1, fef-1=e-1 >

Subgroups: 980 in 198 conjugacy classes, 54 normal (18 characteristic)
C1, C2, C2, C3, C3, C3, C4, C22, S3, C6, C6, C6, C2xC4, Q8, C32, C32, C32, Dic3, Dic3, C12, D6, C2xC6, C2xQ8, C3xS3, C3xS3, C3xC6, C3xC6, C3xC6, Dic6, C4xS3, C2xDic3, C2xC12, C3xQ8, C33, C3xDic3, C3xDic3, C3:Dic3, C3:Dic3, C3:Dic3, C3xC12, S3xC6, S3xC6, C62, C2xDic6, S3xQ8, S3xC32, C32xC6, S3xDic3, S3xDic3, C32:2Q8, C32:2Q8, C3xDic6, S3xC12, C6xDic3, C32:4Q8, C2xC3:Dic3, C32xDic3, C3xC3:Dic3, C3xC3:Dic3, C33:5C4, S3xC3xC6, S3xDic6, C2xC32:2Q8, C3xS3xDic3, C3xC32:2Q8, S3xC3:Dic3, C33:4Q8, C33:5Q8, S3xC32:2Q8
Quotients: C1, C2, C22, S3, Q8, C23, D6, C2xQ8, Dic6, C22xS3, S32, C2xDic6, S3xQ8, C32:2Q8, C2xS32, S3xDic6, C2xC32:2Q8, S33, S3xC32:2Q8

Smallest permutation representation of S3xC32:2Q8
On 48 points
Generators in S48
(1 19 14)(2 20 15)(3 17 16)(4 18 13)(5 45 10)(6 46 11)(7 47 12)(8 48 9)(21 25 30)(22 26 31)(23 27 32)(24 28 29)(33 37 42)(34 38 43)(35 39 44)(36 40 41)
(1 34)(2 35)(3 36)(4 33)(5 25)(6 26)(7 27)(8 28)(9 29)(10 30)(11 31)(12 32)(13 37)(14 38)(15 39)(16 40)(17 41)(18 42)(19 43)(20 44)(21 45)(22 46)(23 47)(24 48)
(1 14 19)(2 20 15)(3 16 17)(4 18 13)(5 10 45)(6 46 11)(7 12 47)(8 48 9)(21 25 30)(22 31 26)(23 27 32)(24 29 28)(33 42 37)(34 38 43)(35 44 39)(36 40 41)
(1 14 19)(2 15 20)(3 16 17)(4 13 18)(5 10 45)(6 11 46)(7 12 47)(8 9 48)(21 25 30)(22 26 31)(23 27 32)(24 28 29)(33 37 42)(34 38 43)(35 39 44)(36 40 41)
(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 24 3 22)(2 23 4 21)(5 44 7 42)(6 43 8 41)(9 40 11 38)(10 39 12 37)(13 30 15 32)(14 29 16 31)(17 26 19 28)(18 25 20 27)(33 45 35 47)(34 48 36 46)

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

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

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

45 conjugacy classes

class 1 2A2B2C3A3B3C3D3E3F3G4A4B4C4D4E4F6A6B6C6D6E6F6G6H6I6J6K6L6M12A12B12C12D12E···12J12K12L12M12N12O
order1222333333344444466666666666661212121212···121212121212
size113322244486618181854222444666681212666612···121818181836

45 irreducible representations

dim111111222222244444488
type++++++++-+++-++--+-+-
imageC1C2C2C2C2C2S3S3Q8D6D6D6Dic6S32S32S3xQ8C32:2Q8C2xS32S3xDic6S33S3xC32:2Q8
kernelS3xC32:2Q8C3xS3xDic3C3xC32:2Q8S3xC3:Dic3C33:4Q8C33:5Q8S3xDic3C32:2Q8S3xC32C3xDic3C3:Dic3S3xC6C3xS3Dic3D6C32S3C6C3C2C1
# reps121121212432821123411

Matrix representation of S3xC32:2Q8 in GL6(F13)

1210000
1200000
001000
000100
000010
000001
,
0120000
1200000
0012000
0001200
000010
000001
,
100000
010000
001000
000100
0000121
0000120
,
100000
010000
009000
000300
000010
000001
,
100000
010000
008000
000500
000001
000010
,
1200000
0120000
000100
0012000
000010
000001

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

S3xC32:2Q8 in GAP, Magma, Sage, TeX

S_3\times C_3^2\rtimes_2Q_8
% in TeX

G:=Group("S3xC3^2:2Q8");
// GroupNames label

G:=SmallGroup(432,603);
// by ID

G=gap.SmallGroup(432,603);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-3,-3,-3,64,135,58,298,2028,14118]);
// Polycyclic

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

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