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G = S3xC24order 96 = 25·3

Direct product of C24 and S3

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

Aliases: S3xC24, C3:C25, C6:C24, (C2xC6):4C23, (C23xC6):5C2, (C22xC6):8C22, SmallGroup(96,230)

Series: Derived Chief Lower central Upper central

C1C3 — S3xC24
C1C3S3D6C22xS3S3xC23 — S3xC24
C3 — S3xC24
C1C24

Generators and relations for S3xC24
 G = < a,b,c,d,e,f | a2=b2=c2=d2=e3=f2=1, ab=ba, ac=ca, ad=da, ae=ea, af=fa, bc=cb, bd=db, be=eb, bf=fb, cd=dc, ce=ec, cf=fc, de=ed, df=fd, fef=e-1 >

Subgroups: 1362 in 748 conjugacy classes, 441 normal (5 characteristic)
C1, C2, C2, C3, C22, C22, S3, C6, C23, C23, D6, C2xC6, C24, C24, C22xS3, C22xC6, C25, S3xC23, C23xC6, S3xC24
Quotients: C1, C2, C22, S3, C23, D6, C24, C22xS3, C25, S3xC23, S3xC24

Smallest permutation representation of S3xC24
On 48 points
Generators in S48
(1 46)(2 47)(3 48)(4 43)(5 44)(6 45)(7 40)(8 41)(9 42)(10 37)(11 38)(12 39)(13 34)(14 35)(15 36)(16 31)(17 32)(18 33)(19 28)(20 29)(21 30)(22 25)(23 26)(24 27)
(1 22)(2 23)(3 24)(4 19)(5 20)(6 21)(7 16)(8 17)(9 18)(10 13)(11 14)(12 15)(25 46)(26 47)(27 48)(28 43)(29 44)(30 45)(31 40)(32 41)(33 42)(34 37)(35 38)(36 39)
(1 7)(2 8)(3 9)(4 10)(5 11)(6 12)(13 19)(14 20)(15 21)(16 22)(17 23)(18 24)(25 31)(26 32)(27 33)(28 34)(29 35)(30 36)(37 43)(38 44)(39 45)(40 46)(41 47)(42 48)
(1 4)(2 5)(3 6)(7 10)(8 11)(9 12)(13 16)(14 17)(15 18)(19 22)(20 23)(21 24)(25 28)(26 29)(27 30)(31 34)(32 35)(33 36)(37 40)(38 41)(39 42)(43 46)(44 47)(45 48)
(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 34)(2 36)(3 35)(4 31)(5 33)(6 32)(7 28)(8 30)(9 29)(10 25)(11 27)(12 26)(13 46)(14 48)(15 47)(16 43)(17 45)(18 44)(19 40)(20 42)(21 41)(22 37)(23 39)(24 38)

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

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

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

S3xC24 is a maximal subgroup of   C24.59D6
S3xC24 is a maximal quotient of   C6.C25  D6.C24  D12.39C23

48 conjugacy classes

class 1 2A···2O2P···2AE 3 6A···6O
order12···22···236···6
size11···13···322···2

48 irreducible representations

dim11122
type+++++
imageC1C2C2S3D6
kernelS3xC24S3xC23C23xC6C24C23
# reps1301115

Matrix representation of S3xC24 in GL5(Z)

-10000
0-1000
00-100
00010
00001
,
-10000
0-1000
00100
000-10
0000-1
,
10000
01000
00100
000-10
0000-1
,
-10000
01000
00-100
000-10
0000-1
,
10000
01000
00100
000-1-1
00010
,
10000
0-1000
00-100
000-10
00011

G:=sub<GL(5,Integers())| [-1,0,0,0,0,0,-1,0,0,0,0,0,-1,0,0,0,0,0,1,0,0,0,0,0,1],[-1,0,0,0,0,0,-1,0,0,0,0,0,1,0,0,0,0,0,-1,0,0,0,0,0,-1],[1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,-1,0,0,0,0,0,-1],[-1,0,0,0,0,0,1,0,0,0,0,0,-1,0,0,0,0,0,-1,0,0,0,0,0,-1],[1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,-1,1,0,0,0,-1,0],[1,0,0,0,0,0,-1,0,0,0,0,0,-1,0,0,0,0,0,-1,1,0,0,0,0,1] >;

S3xC24 in GAP, Magma, Sage, TeX

S_3\times C_2^4
% in TeX

G:=Group("S3xC2^4");
// GroupNames label

G:=SmallGroup(96,230);
// by ID

G=gap.SmallGroup(96,230);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-2,-3,2309]);
// Polycyclic

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

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