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G = S32×C23order 288 = 25·32

Direct product of C23, S3 and S3

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

Aliases: S32×C23, C32⋊C25, C626C23, C3⋊S3⋊C24, (C3×C6)⋊C24, (C3×S3)⋊C24, C31(S3×C24), C61(S3×C23), (C22×C6)⋊14D6, (S3×C6)⋊11C23, (C2×C62)⋊13C22, (C2×C3⋊S3)⋊9C23, (C23×C3⋊S3)⋊9C2, (S3×C22×C6)⋊11C2, (S3×C2×C6)⋊22C22, (C2×C6)⋊9(C22×S3), (C22×C3⋊S3)⋊18C22, SmallGroup(288,1040)

Series: Derived Chief Lower central Upper central

C1C32 — S32×C23
C1C3C32C3×S3S32C2×S32C22×S32 — S32×C23
C32 — S32×C23
C1C23

Generators and relations for S32×C23
 G = < a,b,c,d,e,f,g | a2=b2=c2=d3=e2=f3=g2=1, ab=ba, ac=ca, ad=da, ae=ea, af=fa, ag=ga, bc=cb, bd=db, be=eb, bf=fb, bg=gb, cd=dc, ce=ec, cf=fc, cg=gc, ede=d-1, df=fd, dg=gd, ef=fe, eg=ge, gfg=f-1 >

Subgroups: 5314 in 1563 conjugacy classes, 524 normal (6 characteristic)
C1, C2 [×7], C2 [×24], C3 [×2], C3, C22 [×7], C22 [×148], S3 [×16], S3 [×24], C6 [×14], C6 [×23], C23, C23 [×154], C32, D6 [×56], D6 [×212], C2×C6 [×14], C2×C6 [×63], C24 [×31], C3×S3 [×16], C3⋊S3 [×8], C3×C6 [×7], C22×S3 [×28], C22×S3 [×266], C22×C6 [×2], C22×C6 [×29], C25, S32 [×64], S3×C6 [×56], C2×C3⋊S3 [×28], C62 [×7], S3×C23 [×2], S3×C23 [×59], C23×C6 [×2], C2×S32 [×112], S3×C2×C6 [×28], C22×C3⋊S3 [×14], C2×C62, S3×C24 [×2], C22×S32 [×28], S3×C22×C6 [×2], C23×C3⋊S3, S32×C23
Quotients: C1, C2 [×31], C22 [×155], S3 [×2], C23 [×155], D6 [×30], C24 [×31], C22×S3 [×70], C25, S32, S3×C23 [×30], C2×S32 [×7], S3×C24 [×2], C22×S32 [×7], S32×C23

Smallest permutation representation of S32×C23
On 48 points
Generators in S48
(1 20)(2 21)(3 19)(4 22)(5 23)(6 24)(7 13)(8 14)(9 15)(10 16)(11 17)(12 18)(25 43)(26 44)(27 45)(28 46)(29 47)(30 48)(31 37)(32 38)(33 39)(34 40)(35 41)(36 42)
(1 8)(2 9)(3 7)(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 5)(2 6)(3 4)(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 37)(2 39)(3 38)(4 41)(5 40)(6 42)(7 44)(8 43)(9 45)(10 47)(11 46)(12 48)(13 26)(14 25)(15 27)(16 29)(17 28)(18 30)(19 32)(20 31)(21 33)(22 35)(23 34)(24 36)
(1 3 2)(4 6 5)(7 9 8)(10 12 11)(13 15 14)(16 18 17)(19 21 20)(22 24 23)(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 43)(2 44)(3 45)(4 48)(5 46)(6 47)(7 39)(8 37)(9 38)(10 42)(11 40)(12 41)(13 33)(14 31)(15 32)(16 36)(17 34)(18 35)(19 27)(20 25)(21 26)(22 30)(23 28)(24 29)

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

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

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

72 conjugacy classes

class 1 2A···2G2H···2W2X···2AE3A3B3C6A···6N6O···6U6V···6AK
order12···22···22···23336···66···66···6
size11···13···39···92242···24···46···6

72 irreducible representations

dim111122244
type+++++++++
imageC1C2C2C2S3D6D6S32C2×S32
kernelS32×C23C22×S32S3×C22×C6C23×C3⋊S3S3×C23C22×S3C22×C6C23C22
# reps12821228217

Matrix representation of S32×C23 in GL8(ℤ)

10000000
01000000
00-100000
000-10000
00001000
00000100
00000010
00000001
,
-10000000
0-1000000
00100000
00010000
0000-1000
00000-100
00000010
00000001
,
-10000000
0-1000000
00-100000
000-10000
00001000
00000100
00000010
00000001
,
10000000
01000000
00100000
00010000
00001000
00000100
000000-11
000000-10
,
-10000000
0-1000000
00100000
00010000
0000-1000
00000-100
00000001
00000010
,
01000000
-1-1000000
00010000
00-1-10000
00000100
0000-1-100
00000010
00000001
,
10000000
-1-1000000
00100000
00-1-10000
00001000
0000-1-100
00000010
00000001

G:=sub<GL(8,Integers())| [1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[-1,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[-1,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,-1,-1,0,0,0,0,0,0,1,0],[-1,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0],[0,-1,0,0,0,0,0,0,1,-1,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,1,-1,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,1,-1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[1,-1,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,1,-1,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,1,-1,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1] >;

S32×C23 in GAP, Magma, Sage, TeX

S_3^2\times C_2^3
% in TeX

G:=Group("S3^2xC2^3");
// GroupNames label

G:=SmallGroup(288,1040);
// by ID

G=gap.SmallGroup(288,1040);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,1356,9414]);
// Polycyclic

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

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