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## G = S3×C24order 96 = 25·3

### Direct product of C24 and S3

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

 Derived series C1 — C3 — S3×C24
 Chief series C1 — C3 — S3 — D6 — C22×S3 — S3×C23 — S3×C24
 Lower central C3 — S3×C24
 Upper central C1 — C24

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, C2, C3, C22, C22, S3, C6, C23, C23, D6, C2×C6, C24, C24, C22×S3, C22×C6, C25, S3×C23, C23×C6, S3×C24
Quotients: C1, C2, C22, S3, C23, D6, C24, C22×S3, C25, S3×C23, 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 ··· 2O 2P ··· 2AE 3 6A ··· 6O order 1 2 ··· 2 2 ··· 2 3 6 ··· 6 size 1 1 ··· 1 3 ··· 3 2 2 ··· 2

48 irreducible representations

 dim 1 1 1 2 2 type + + + + + image C1 C2 C2 S3 D6 kernel S3×C24 S3×C23 C23×C6 C24 C23 # reps 1 30 1 1 15

Matrix representation of S3×C24 in GL5(ℤ)

 -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 0 0 0 0 1 1

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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