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G = S3×C24order 96 = 25·3

Direct product of C24 and S3

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

Aliases: S3×C24, C3⋊C25, C6⋊C24, (C2×C6)⋊4C23, (C23×C6)⋊5C2, (C22×C6)⋊8C22, SmallGroup(96,230)

Series: Derived Chief Lower central Upper central

C1C3 — S3×C24
C1C3S3D6C22×S3S3×C23 — S3×C24
C3 — S3×C24
C1C24

Generators and relations for S3×C24
 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 [×15], C2 [×16], C3, C22 [×35], C22 [×120], S3 [×16], C6 [×15], C23 [×15], C23 [×140], D6 [×120], C2×C6 [×35], C24, C24 [×30], C22×S3 [×140], C22×C6 [×15], C25, S3×C23 [×30], C23×C6, S3×C24
Quotients: C1, C2 [×31], C22 [×155], S3, C23 [×155], D6 [×15], C24 [×31], C22×S3 [×35], C25, S3×C23 [×15], S3×C24

Smallest permutation representation of S3×C24
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)])

S3×C24 is a maximal subgroup of   C24.59D6
S3×C24 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
kernelS3×C24S3×C23C23×C6C24C23
# reps1301115

Matrix representation of S3×C24 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] >;

S3×C24 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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