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, C3, C3, C4, C4, C22, S3, S3, C6, C6, C8, C8, C2×C4, C23, C32, Dic3, Dic3, C12, C12, D6, D6, C2×C6, C2×C8, C22×C4, C3×S3, C3⋊S3, C3×C6, C3⋊C8, C3⋊C8, C24, C24, C4×S3, C4×S3, C2×Dic3, C2×C12, C22×S3, C22×C8, C3×Dic3, C3⋊Dic3, C3×C12, S32, S3×C6, C2×C3⋊S3, S3×C8, S3×C8, C2×C3⋊C8, C2×C24, S3×C2×C4, C3×C3⋊C8, C32⋊4C8, C3×C24, S3×Dic3, C6.D6, S3×C12, C4×C3⋊S3, C2×S32, S3×C2×C8, S3×C3⋊C8, C12.29D6, S3×C24, C8×C3⋊S3, C4×S32, S32×C8
Quotients: C1, C2, C4, C22, S3, C8, C2×C4, C23, D6, C2×C8, C22×C4, C4×S3, C22×S3, C22×C8, S32, S3×C8, S3×C2×C4, C2×S32, S3×C2×C8, 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 25)(2 36 26)(3 37 27)(4 38 28)(5 39 29)(6 40 30)(7 33 31)(8 34 32)(9 44 23)(10 45 24)(11 46 17)(12 47 18)(13 48 19)(14 41 20)(15 42 21)(16 43 22)
(1 15)(2 16)(3 9)(4 10)(5 11)(6 12)(7 13)(8 14)(17 39)(18 40)(19 33)(20 34)(21 35)(22 36)(23 37)(24 38)(25 42)(26 43)(27 44)(28 45)(29 46)(30 47)(31 48)(32 41)
(1 25 35)(2 26 36)(3 27 37)(4 28 38)(5 29 39)(6 30 40)(7 31 33)(8 32 34)(9 44 23)(10 45 24)(11 46 17)(12 47 18)(13 48 19)(14 41 20)(15 42 21)(16 43 22)
(1 15)(2 16)(3 9)(4 10)(5 11)(6 12)(7 13)(8 14)(17 29)(18 30)(19 31)(20 32)(21 25)(22 26)(23 27)(24 28)(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), (1,35,25)(2,36,26)(3,37,27)(4,38,28)(5,39,29)(6,40,30)(7,33,31)(8,34,32)(9,44,23)(10,45,24)(11,46,17)(12,47,18)(13,48,19)(14,41,20)(15,42,21)(16,43,22), (1,15)(2,16)(3,9)(4,10)(5,11)(6,12)(7,13)(8,14)(17,39)(18,40)(19,33)(20,34)(21,35)(22,36)(23,37)(24,38)(25,42)(26,43)(27,44)(28,45)(29,46)(30,47)(31,48)(32,41), (1,25,35)(2,26,36)(3,27,37)(4,28,38)(5,29,39)(6,30,40)(7,31,33)(8,32,34)(9,44,23)(10,45,24)(11,46,17)(12,47,18)(13,48,19)(14,41,20)(15,42,21)(16,43,22), (1,15)(2,16)(3,9)(4,10)(5,11)(6,12)(7,13)(8,14)(17,29)(18,30)(19,31)(20,32)(21,25)(22,26)(23,27)(24,28)(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), (1,35,25)(2,36,26)(3,37,27)(4,38,28)(5,39,29)(6,40,30)(7,33,31)(8,34,32)(9,44,23)(10,45,24)(11,46,17)(12,47,18)(13,48,19)(14,41,20)(15,42,21)(16,43,22), (1,15)(2,16)(3,9)(4,10)(5,11)(6,12)(7,13)(8,14)(17,39)(18,40)(19,33)(20,34)(21,35)(22,36)(23,37)(24,38)(25,42)(26,43)(27,44)(28,45)(29,46)(30,47)(31,48)(32,41), (1,25,35)(2,26,36)(3,27,37)(4,28,38)(5,29,39)(6,30,40)(7,31,33)(8,32,34)(9,44,23)(10,45,24)(11,46,17)(12,47,18)(13,48,19)(14,41,20)(15,42,21)(16,43,22), (1,15)(2,16)(3,9)(4,10)(5,11)(6,12)(7,13)(8,14)(17,29)(18,30)(19,31)(20,32)(21,25)(22,26)(23,27)(24,28)(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)], [(1,35,25),(2,36,26),(3,37,27),(4,38,28),(5,39,29),(6,40,30),(7,33,31),(8,34,32),(9,44,23),(10,45,24),(11,46,17),(12,47,18),(13,48,19),(14,41,20),(15,42,21),(16,43,22)], [(1,15),(2,16),(3,9),(4,10),(5,11),(6,12),(7,13),(8,14),(17,39),(18,40),(19,33),(20,34),(21,35),(22,36),(23,37),(24,38),(25,42),(26,43),(27,44),(28,45),(29,46),(30,47),(31,48),(32,41)], [(1,25,35),(2,26,36),(3,27,37),(4,28,38),(5,29,39),(6,30,40),(7,31,33),(8,32,34),(9,44,23),(10,45,24),(11,46,17),(12,47,18),(13,48,19),(14,41,20),(15,42,21),(16,43,22)], [(1,15),(2,16),(3,9),(4,10),(5,11),(6,12),(7,13),(8,14),(17,29),(18,30),(19,31),(20,32),(21,25),(22,26),(23,27),(24,28),(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