Copied to
clipboard

## G = S32×C8order 288 = 25·32

### Direct product of C8, S3 and S3

Series: Derived Chief Lower central Upper central

 Derived series C1 — C32 — S32×C8
 Chief series C1 — C3 — C32 — C3×C6 — C3×C12 — S3×C12 — C4×S32 — S32×C8
 Lower central C32 — S32×C8
 Upper central C1 — C8

Generators and relations for S32×C8
G = < a,b,c,d,e | a8=b3=c2=d3=e2=1, ab=ba, ac=ca, ad=da, ae=ea, cbc=b-1, bd=db, be=eb, cd=dc, ce=ec, ede=d-1 >

Subgroups: 498 in 163 conjugacy classes, 64 normal (18 characteristic)
C1, C2, C2 [×6], C3 [×2], C3, C4, C4 [×3], C22 [×7], S3 [×4], S3 [×6], C6 [×2], C6 [×5], C8, C8 [×3], C2×C4 [×6], C23, C32, Dic3 [×2], Dic3 [×3], C12 [×2], C12 [×3], D6 [×2], D6 [×11], C2×C6 [×2], C2×C8 [×6], C22×C4, C3×S3 [×4], C3⋊S3 [×2], C3×C6, C3⋊C8 [×2], C3⋊C8 [×3], C24 [×2], C24 [×3], C4×S3 [×2], C4×S3 [×7], C2×Dic3 [×2], C2×C12 [×2], C22×S3 [×2], C22×C8, C3×Dic3 [×2], C3⋊Dic3, C3×C12, S32 [×4], S3×C6 [×2], C2×C3⋊S3, S3×C8 [×2], S3×C8 [×7], C2×C3⋊C8 [×2], C2×C24 [×2], S3×C2×C4 [×2], C3×C3⋊C8 [×2], C324C8, C3×C24, S3×Dic3 [×2], C6.D6, S3×C12 [×2], C4×C3⋊S3, C2×S32, S3×C2×C8 [×2], S3×C3⋊C8 [×2], C12.29D6, S3×C24 [×2], C8×C3⋊S3, C4×S32, S32×C8
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], S3 [×2], C8 [×4], C2×C4 [×6], C23, D6 [×6], C2×C8 [×6], C22×C4, C4×S3 [×4], C22×S3 [×2], C22×C8, S32, S3×C8 [×4], S3×C2×C4 [×2], C2×S32, S3×C2×C8 [×2], C4×S32, S32×C8

Smallest permutation representation of S32×C8
On 48 points
Generators in S48
(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 35 28)(2 36 29)(3 37 30)(4 38 31)(5 39 32)(6 40 25)(7 33 26)(8 34 27)(9 46 17)(10 47 18)(11 48 19)(12 41 20)(13 42 21)(14 43 22)(15 44 23)(16 45 24)
(1 13)(2 14)(3 15)(4 16)(5 9)(6 10)(7 11)(8 12)(17 39)(18 40)(19 33)(20 34)(21 35)(22 36)(23 37)(24 38)(25 47)(26 48)(27 41)(28 42)(29 43)(30 44)(31 45)(32 46)
(1 28 35)(2 29 36)(3 30 37)(4 31 38)(5 32 39)(6 25 40)(7 26 33)(8 27 34)(9 46 17)(10 47 18)(11 48 19)(12 41 20)(13 42 21)(14 43 22)(15 44 23)(16 45 24)
(1 13)(2 14)(3 15)(4 16)(5 9)(6 10)(7 11)(8 12)(17 32)(18 25)(19 26)(20 27)(21 28)(22 29)(23 30)(24 31)(33 48)(34 41)(35 42)(36 43)(37 44)(38 45)(39 46)(40 47)

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

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

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

72 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 3A 3B 3C 4A 4B 4C 4D 4E 4F 4G 4H 6A 6B 6C 6D 6E 6F 6G 8A 8B 8C 8D 8E ··· 8L 8M 8N 8O 8P 12A 12B 12C 12D 12E 12F 12G 12H 12I 12J 24A ··· 24H 24I 24J 24K 24L 24M ··· 24T order 1 2 2 2 2 2 2 2 3 3 3 4 4 4 4 4 4 4 4 6 6 6 6 6 6 6 8 8 8 8 8 ··· 8 8 8 8 8 12 12 12 12 12 12 12 12 12 12 24 ··· 24 24 24 24 24 24 ··· 24 size 1 1 3 3 3 3 9 9 2 2 4 1 1 3 3 3 3 9 9 2 2 4 6 6 6 6 1 1 1 1 3 ··· 3 9 9 9 9 2 2 2 2 4 4 6 6 6 6 2 ··· 2 4 4 4 4 6 ··· 6

72 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 4 4 4 4 type + + + + + + + + + + + + image C1 C2 C2 C2 C2 C2 C4 C4 C4 C8 S3 D6 D6 D6 C4×S3 C4×S3 S3×C8 S32 C2×S32 C4×S32 S32×C8 kernel S32×C8 S3×C3⋊C8 C12.29D6 S3×C24 C8×C3⋊S3 C4×S32 S3×Dic3 C6.D6 C2×S32 S32 S3×C8 C3⋊C8 C24 C4×S3 Dic3 D6 S3 C8 C4 C2 C1 # reps 1 2 1 2 1 1 4 2 2 16 2 2 2 2 4 4 16 1 1 2 4

Matrix representation of S32×C8 in GL4(𝔽73) generated by

 10 0 0 0 0 10 0 0 0 0 1 0 0 0 0 1
,
 1 0 0 0 0 1 0 0 0 0 0 1 0 0 72 72
,
 1 0 0 0 0 1 0 0 0 0 1 0 0 0 72 72
,
 72 1 0 0 72 0 0 0 0 0 1 0 0 0 0 1
,
 0 1 0 0 1 0 0 0 0 0 1 0 0 0 0 1
G:=sub<GL(4,GF(73))| [10,0,0,0,0,10,0,0,0,0,1,0,0,0,0,1],[1,0,0,0,0,1,0,0,0,0,0,72,0,0,1,72],[1,0,0,0,0,1,0,0,0,0,1,72,0,0,0,72],[72,72,0,0,1,0,0,0,0,0,1,0,0,0,0,1],[0,1,0,0,1,0,0,0,0,0,1,0,0,0,0,1] >;

S32×C8 in GAP, Magma, Sage, TeX

S_3^2\times C_8
% in TeX

G:=Group("S3^2xC8");
// GroupNames label

G:=SmallGroup(288,437);
// by ID

G=gap.SmallGroup(288,437);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,58,80,1356,9414]);
// Polycyclic

G:=Group<a,b,c,d,e|a^8=b^3=c^2=d^3=e^2=1,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,c*b*c=b^-1,b*d=d*b,b*e=e*b,c*d=d*c,c*e=e*c,e*d*e=d^-1>;
// generators/relations

׿
×
𝔽