direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C2×D8⋊S3, D8⋊9D6, C24⋊5C23, C12.2C24, Dic6⋊1C23, D12.1C23, (C2×C8)⋊8D6, C3⋊C8⋊1C23, (C2×D4)⋊28D6, (C2×D8)⋊11S3, (C6×D8)⋊11C2, C8⋊3(C22×S3), C4.40(S3×D4), C6⋊2(C8⋊C22), D4⋊S3⋊9C22, (C4×S3).14D4, D6.49(C2×D4), C12.77(C2×D4), (C3×D4)⋊2C23, D4⋊2(C22×S3), (S3×D4)⋊5C22, C4.2(S3×C23), (C2×C24)⋊16C22, (C3×D8)⋊14C22, (C6×D4)⋊19C22, (C4×S3).1C23, D4.S3⋊7C22, C8⋊S3⋊12C22, C24⋊C2⋊13C22, D4⋊2S3⋊5C22, Dic3.54(C2×D4), (C22×S3).97D4, C6.103(C22×D4), C22.136(S3×D4), (C2×C12).519C23, (C2×Dic3).191D4, (C2×Dic6)⋊36C22, (C2×D12).176C22, (C2×S3×D4)⋊22C2, C3⋊2(C2×C8⋊C22), C2.76(C2×S3×D4), (C2×C8⋊S3)⋊8C2, (C2×D4⋊S3)⋊26C2, (C2×C3⋊C8)⋊14C22, (C2×C24⋊C2)⋊24C2, (C2×D4.S3)⋊25C2, (C2×C6).392(C2×D4), (C2×D4⋊2S3)⋊23C2, (S3×C2×C4).155C22, (C2×C4).609(C22×S3), SmallGroup(192,1314)
Series: Derived ►Chief ►Lower central ►Upper central
Subgroups: 920 in 298 conjugacy classes, 103 normal (33 characteristic)
C1, C2, C2 [×2], C2 [×8], C3, C4 [×2], C4 [×4], C22, C22 [×24], S3 [×4], C6, C6 [×2], C6 [×4], C8 [×2], C8 [×2], C2×C4, C2×C4 [×10], D4 [×4], D4 [×13], Q8 [×3], C23 [×12], Dic3 [×2], Dic3 [×2], C12 [×2], D6 [×2], D6 [×14], C2×C6, C2×C6 [×8], C2×C8, C2×C8, M4(2) [×4], D8 [×4], D8 [×4], SD16 [×8], C22×C4 [×2], C2×D4 [×2], C2×D4 [×9], C2×Q8, C4○D4 [×6], C24, C3⋊C8 [×2], C24 [×2], Dic6 [×2], Dic6, C4×S3 [×4], D12 [×2], D12, C2×Dic3, C2×Dic3 [×5], C3⋊D4 [×8], C2×C12, C3×D4 [×4], C3×D4 [×2], C22×S3, C22×S3 [×9], C22×C6 [×2], C2×M4(2), C2×D8, C2×D8, C2×SD16 [×2], C8⋊C22 [×8], C22×D4, C2×C4○D4, C8⋊S3 [×4], C24⋊C2 [×4], C2×C3⋊C8, D4⋊S3 [×4], D4.S3 [×4], C2×C24, C3×D8 [×4], C2×Dic6, S3×C2×C4, C2×D12, S3×D4 [×4], S3×D4 [×2], D4⋊2S3 [×4], D4⋊2S3 [×2], C22×Dic3, C2×C3⋊D4 [×2], C6×D4 [×2], S3×C23, C2×C8⋊C22, C2×C8⋊S3, C2×C24⋊C2, D8⋊S3 [×8], C2×D4⋊S3, C2×D4.S3, C6×D8, C2×S3×D4, C2×D4⋊2S3, C2×D8⋊S3
Quotients:
C1, C2 [×15], C22 [×35], S3, D4 [×4], C23 [×15], D6 [×7], C2×D4 [×6], C24, C22×S3 [×7], C8⋊C22 [×2], C22×D4, S3×D4 [×2], S3×C23, C2×C8⋊C22, D8⋊S3 [×2], C2×S3×D4, C2×D8⋊S3
Generators and relations
G = < a,b,c,d,e | a2=b8=c2=d3=e2=1, ab=ba, ac=ca, ad=da, ae=ea, cbc=b-1, bd=db, ebe=b5, cd=dc, ce=ec, ede=d-1 >
►On 48 points
(1 22)(2 23)(3 24)(4 17)(5 18)(6 19)(7 20)(8 21)(9 48)(10 41)(11 42)(12 43)(13 44)(14 45)(15 46)(16 47)(25 40)(26 33)(27 34)(28 35)(29 36)(30 37)(31 38)(32 39)
(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 7)(2 6)(3 5)(9 15)(10 14)(11 13)(18 24)(19 23)(20 22)(25 27)(28 32)(29 31)(34 40)(35 39)(36 38)(41 45)(42 44)(46 48)
(1 34 13)(2 35 14)(3 36 15)(4 37 16)(5 38 9)(6 39 10)(7 40 11)(8 33 12)(17 30 47)(18 31 48)(19 32 41)(20 25 42)(21 26 43)(22 27 44)(23 28 45)(24 29 46)
(1 18)(2 23)(3 20)(4 17)(5 22)(6 19)(7 24)(8 21)(9 27)(10 32)(11 29)(12 26)(13 31)(14 28)(15 25)(16 30)(33 43)(34 48)(35 45)(36 42)(37 47)(38 44)(39 41)(40 46)
G:=sub<Sym(48)| (1,22)(2,23)(3,24)(4,17)(5,18)(6,19)(7,20)(8,21)(9,48)(10,41)(11,42)(12,43)(13,44)(14,45)(15,46)(16,47)(25,40)(26,33)(27,34)(28,35)(29,36)(30,37)(31,38)(32,39), (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), (1,7)(2,6)(3,5)(9,15)(10,14)(11,13)(18,24)(19,23)(20,22)(25,27)(28,32)(29,31)(34,40)(35,39)(36,38)(41,45)(42,44)(46,48), (1,34,13)(2,35,14)(3,36,15)(4,37,16)(5,38,9)(6,39,10)(7,40,11)(8,33,12)(17,30,47)(18,31,48)(19,32,41)(20,25,42)(21,26,43)(22,27,44)(23,28,45)(24,29,46), (1,18)(2,23)(3,20)(4,17)(5,22)(6,19)(7,24)(8,21)(9,27)(10,32)(11,29)(12,26)(13,31)(14,28)(15,25)(16,30)(33,43)(34,48)(35,45)(36,42)(37,47)(38,44)(39,41)(40,46)>;
G:=Group( (1,22)(2,23)(3,24)(4,17)(5,18)(6,19)(7,20)(8,21)(9,48)(10,41)(11,42)(12,43)(13,44)(14,45)(15,46)(16,47)(25,40)(26,33)(27,34)(28,35)(29,36)(30,37)(31,38)(32,39), (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), (1,7)(2,6)(3,5)(9,15)(10,14)(11,13)(18,24)(19,23)(20,22)(25,27)(28,32)(29,31)(34,40)(35,39)(36,38)(41,45)(42,44)(46,48), (1,34,13)(2,35,14)(3,36,15)(4,37,16)(5,38,9)(6,39,10)(7,40,11)(8,33,12)(17,30,47)(18,31,48)(19,32,41)(20,25,42)(21,26,43)(22,27,44)(23,28,45)(24,29,46), (1,18)(2,23)(3,20)(4,17)(5,22)(6,19)(7,24)(8,21)(9,27)(10,32)(11,29)(12,26)(13,31)(14,28)(15,25)(16,30)(33,43)(34,48)(35,45)(36,42)(37,47)(38,44)(39,41)(40,46) );
G=PermutationGroup([(1,22),(2,23),(3,24),(4,17),(5,18),(6,19),(7,20),(8,21),(9,48),(10,41),(11,42),(12,43),(13,44),(14,45),(15,46),(16,47),(25,40),(26,33),(27,34),(28,35),(29,36),(30,37),(31,38),(32,39)], [(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)], [(1,7),(2,6),(3,5),(9,15),(10,14),(11,13),(18,24),(19,23),(20,22),(25,27),(28,32),(29,31),(34,40),(35,39),(36,38),(41,45),(42,44),(46,48)], [(1,34,13),(2,35,14),(3,36,15),(4,37,16),(5,38,9),(6,39,10),(7,40,11),(8,33,12),(17,30,47),(18,31,48),(19,32,41),(20,25,42),(21,26,43),(22,27,44),(23,28,45),(24,29,46)], [(1,18),(2,23),(3,20),(4,17),(5,22),(6,19),(7,24),(8,21),(9,27),(10,32),(11,29),(12,26),(13,31),(14,28),(15,25),(16,30),(33,43),(34,48),(35,45),(36,42),(37,47),(38,44),(39,41),(40,46)])
Matrix representation ►G ⊆ GL6(𝔽73)
| 72 | 0 | 0 | 0 | 0 | 0 |
| 0 | 72 | 0 | 0 | 0 | 0 |
| 0 | 0 | 72 | 0 | 0 | 0 |
| 0 | 0 | 0 | 72 | 0 | 0 |
| 0 | 0 | 0 | 0 | 72 | 0 |
| 0 | 0 | 0 | 0 | 0 | 72 |
| 1 | 2 | 0 | 0 | 0 | 0 |
| 72 | 72 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 31 | 62 |
| 0 | 0 | 0 | 0 | 11 | 42 |
| 0 | 0 | 11 | 22 | 11 | 22 |
| 0 | 0 | 51 | 62 | 51 | 62 |
| 72 | 0 | 0 | 0 | 0 | 0 |
| 1 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 71 | 0 | 72 | 0 |
| 0 | 0 | 0 | 71 | 0 | 72 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 72 | 72 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 72 | 72 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 1 | 0 |
G:=sub<GL(6,GF(73))| [72,0,0,0,0,0,0,72,0,0,0,0,0,0,72,0,0,0,0,0,0,72,0,0,0,0,0,0,72,0,0,0,0,0,0,72],[1,72,0,0,0,0,2,72,0,0,0,0,0,0,0,0,11,51,0,0,0,0,22,62,0,0,31,11,11,51,0,0,62,42,22,62],[72,1,0,0,0,0,0,1,0,0,0,0,0,0,1,0,71,0,0,0,0,1,0,71,0,0,0,0,72,0,0,0,0,0,0,72],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,72,0,0,0,0,1,72,0,0,0,0,0,0,0,72,0,0,0,0,1,72],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0] >;
36 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 2H | 2I | 2J | 2K | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 6A | 6B | 6C | 6D | 6E | 6F | 6G | 8A | 8B | 8C | 8D | 12A | 12B | 24A | 24B | 24C | 24D |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 8 | 8 | 8 | 8 | 12 | 12 | 24 | 24 | 24 | 24 |
| size | 1 | 1 | 1 | 1 | 4 | 4 | 4 | 4 | 6 | 6 | 12 | 12 | 2 | 2 | 2 | 6 | 6 | 12 | 12 | 2 | 2 | 2 | 8 | 8 | 8 | 8 | 4 | 4 | 12 | 12 | 4 | 4 | 4 | 4 | 4 | 4 |
36 irreducible representations
| dim | 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 | C2 | C2 | C2 | S3 | D4 | D4 | D4 | D6 | D6 | D6 | C8⋊C22 | S3×D4 | S3×D4 | D8⋊S3 |
| kernel | C2×D8⋊S3 | C2×C8⋊S3 | C2×C24⋊C2 | D8⋊S3 | C2×D4⋊S3 | C2×D4.S3 | C6×D8 | C2×S3×D4 | C2×D4⋊2S3 | C2×D8 | C4×S3 | C2×Dic3 | C22×S3 | C2×C8 | D8 | C2×D4 | C6 | C4 | C22 | C2 |
| # reps | 1 | 1 | 1 | 8 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 1 | 1 | 1 | 4 | 2 | 2 | 1 | 1 | 4 |
In GAP, Magma, Sage, TeX
C_2\times D_8\rtimes S_3
% in TeX
G:=Group("C2xD8:S3"); // GroupNames label
G:=SmallGroup(192,1314);
// by ID
G=gap.SmallGroup(192,1314);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,1123,185,438,235,102,6278]);
// Polycyclic
G:=Group<a,b,c,d,e|a^2=b^8=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,e*b*e=b^5,c*d=d*c,c*e=e*c,e*d*e=d^-1>;
// generators/relations