direct product, metabelian, supersoluble, monomial, A-group
Aliases: S32×C8, C24⋊19D6, C3⋊C8⋊29D6, D6.8(C4×S3), (S3×C24)⋊11C2, (C4×S3).43D6, C32⋊2(C22×C8), (C3×C24)⋊19C22, (S3×Dic3).3C4, C6.D6.5C4, Dic3.11(C4×S3), C12.29D6⋊12C2, (S3×C12).55C22, (C3×C12).135C23, C12.134(C22×S3), C32⋊4C8⋊25C22, C3⋊1(S3×C2×C8), C2.1(C4×S32), C6.1(S3×C2×C4), (S3×C3⋊C8)⋊14C2, C3⋊S3⋊2(C2×C8), (C2×S32).5C4, (C4×S32).6C2, C4.81(C2×S32), (C8×C3⋊S3)⋊10C2, (C3×S3)⋊1(C2×C8), (C3×C3⋊C8)⋊33C22, (S3×C6).8(C2×C4), (C3×C6).1(C22×C4), (C4×C3⋊S3).84C22, C3⋊Dic3.29(C2×C4), (C3×Dic3).15(C2×C4), (C2×C3⋊S3).25(C2×C4), SmallGroup(288,437)
Series: Derived ►Chief ►Lower central ►Upper central
| C32 — S32×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], C32⋊4C8, 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
►On 48 points
(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