Copied to
clipboard

## G = C23×C32⋊C4order 288 = 25·32

### Direct product of C23 and C32⋊C4

Series: Derived Chief Lower central Upper central

 Derived series C1 — C32 — C23×C32⋊C4
 Chief series C1 — C32 — C3⋊S3 — C32⋊C4 — C2×C32⋊C4 — C22×C32⋊C4 — C23×C32⋊C4
 Lower central C32 — C23×C32⋊C4
 Upper central C1 — C23

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 ··· 2G 2H ··· 2O 3A 3B 4A ··· 4P 6A ··· 6N order 1 2 ··· 2 2 ··· 2 3 3 4 ··· 4 6 ··· 6 size 1 1 ··· 1 9 ··· 9 4 4 9 ··· 9 4 ··· 4

48 irreducible representations

 dim 1 1 1 1 1 4 4 type + + + + + image C1 C2 C2 C4 C4 C32⋊C4 C2×C32⋊C4 kernel C23×C32⋊C4 C22×C32⋊C4 C23×C3⋊S3 C22×C3⋊S3 C2×C62 C23 C22 # reps 1 14 1 14 2 2 14

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

 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 1 0 0 0 0 12 0
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 12 0 0 0 0 1 12 0 0 0 0 0 0 12 1 0 0 0 0 12 0
,
 5 0 0 0 0 0 0 8 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 1 0 0 0 0 1 0 0 0

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

׿
×
𝔽