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G = C23×C32⋊C4order 288 = 25·32

Direct product of C23 and C32⋊C4

direct product, metabelian, soluble, monomial, A-group

Aliases: C23×C32⋊C4, (C2×C62)⋊6C4, C623(C2×C4), C3⋊S3.3C24, C322(C23×C4), C3⋊S33(C22×C4), (C3×C6)⋊2(C22×C4), (C22×C3⋊S3)⋊10C4, (C23×C3⋊S3).7C2, (C2×C3⋊S3).58C23, (C22×C3⋊S3).110C22, (C2×C3⋊S3)⋊20(C2×C4), SmallGroup(288,1039)

Series: Derived Chief Lower central Upper central

C1C32 — C23×C32⋊C4
C1C32C3⋊S3C32⋊C4C2×C32⋊C4C22×C32⋊C4 — C23×C32⋊C4
C32 — C23×C32⋊C4
C1C23

Generators and relations for C23×C32⋊C4
 G = < a,b,c,d,e,f | a2=b2=c2=d3=e3=f4=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, fef-1=de=ed, fdf-1=d-1e >

Subgroups: 1728 in 370 conjugacy classes, 134 normal (7 characteristic)
C1, C2 [×7], C2 [×8], C3 [×2], C4 [×8], C22 [×7], C22 [×28], S3 [×16], C6 [×14], C2×C4 [×28], C23, C23 [×14], C32, D6 [×56], C2×C6 [×14], C22×C4 [×14], C24, C3⋊S3, C3⋊S3 [×7], C3×C6 [×7], C22×S3 [×28], C22×C6 [×2], C23×C4, C32⋊C4 [×8], C2×C3⋊S3 [×28], C62 [×7], S3×C23 [×2], C2×C32⋊C4 [×28], C22×C3⋊S3 [×14], C2×C62, C22×C32⋊C4 [×14], C23×C3⋊S3, C23×C32⋊C4
Quotients: C1, C2 [×15], C4 [×8], C22 [×35], C2×C4 [×28], C23 [×15], C22×C4 [×14], C24, C23×C4, C32⋊C4, C2×C32⋊C4 [×7], C22×C32⋊C4 [×7], C23×C32⋊C4

Smallest permutation representation of C23×C32⋊C4
On 48 points
Generators in S48
(1 6)(2 5)(3 12)(4 11)(7 16)(8 15)(9 13)(10 14)(17 31)(18 32)(19 29)(20 30)(21 44)(22 41)(23 42)(24 43)(25 38)(26 39)(27 40)(28 37)(33 45)(34 46)(35 47)(36 48)
(1 10)(2 9)(3 7)(4 8)(5 13)(6 14)(11 15)(12 16)(17 42)(18 43)(19 44)(20 41)(21 29)(22 30)(23 31)(24 32)(25 33)(26 34)(27 35)(28 36)(37 48)(38 45)(39 46)(40 47)
(1 16)(2 15)(3 14)(4 13)(5 8)(6 7)(9 11)(10 12)(17 37)(18 38)(19 39)(20 40)(21 34)(22 35)(23 36)(24 33)(25 32)(26 29)(27 30)(28 31)(41 47)(42 48)(43 45)(44 46)
(2 48 46)(4 31 29)(5 36 34)(8 23 21)(9 37 39)(11 17 19)(13 28 26)(15 42 44)
(1 47 45)(2 48 46)(3 30 32)(4 31 29)(5 36 34)(6 35 33)(7 22 24)(8 23 21)(9 37 39)(10 40 38)(11 17 19)(12 20 18)(13 28 26)(14 27 25)(15 42 44)(16 41 43)
(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)

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

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

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

48 conjugacy classes

class 1 2A···2G2H···2O3A3B4A···4P6A···6N
order12···22···2334···46···6
size11···19···9449···94···4

48 irreducible representations

dim1111144
type+++++
imageC1C2C2C4C4C32⋊C4C2×C32⋊C4
kernelC23×C32⋊C4C22×C32⋊C4C23×C3⋊S3C22×C3⋊S3C2×C62C23C22
# reps1141142214

Matrix representation of C23×C32⋊C4 in GL6(𝔽13)

1200000
010000
0012000
0001200
0000120
0000012
,
100000
0120000
0012000
0001200
0000120
0000012
,
1200000
0120000
0012000
0001200
0000120
0000012
,
100000
010000
001000
000100
0000121
0000120
,
100000
010000
0001200
0011200
0000121
0000120
,
500000
080000
000010
000001
000100
001000

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

C23×C32⋊C4 in GAP, Magma, Sage, TeX

C_2^3\times C_3^2\rtimes C_4
% in TeX

G:=Group("C2^3xC3^2:C4");
// GroupNames label

G:=SmallGroup(288,1039);
// by ID

G=gap.SmallGroup(288,1039);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,3,112,9413,201,12550,622]);
// Polycyclic

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

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