direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C2×S3×SD16, C24⋊6C23, C12.5C24, Dic6⋊2C23, D12.3C23, (C2×C8)⋊28D6, C3⋊C8⋊7C23, C8⋊6(C22×S3), (C2×Q8)⋊23D6, C6⋊2(C2×SD16), C4.42(S3×D4), (C4×S3).28D4, D6.64(C2×D4), C12.80(C2×D4), C4.5(S3×C23), (C3×Q8)⋊1C23, (S3×Q8)⋊5C22, Q8⋊2(C22×S3), C3⋊2(C22×SD16), (S3×C8)⋊17C22, (C2×C24)⋊18C22, (C6×SD16)⋊10C2, (C2×D4).181D6, D4.S3⋊9C22, (C6×Q8)⋊17C22, D4.3(C22×S3), (C3×D4).3C23, (S3×D4).5C22, C24⋊C2⋊17C22, (C4×S3).25C23, Dic3.12(C2×D4), Q8⋊2S3⋊7C22, C6.106(C22×D4), C22.138(S3×D4), (C2×C12).522C23, (C2×Dic3).122D4, (C3×SD16)⋊12C22, (C2×Dic6)⋊37C22, (C6×D4).163C22, (C22×S3).111D4, (C2×D12).177C22, (S3×C2×C8)⋊9C2, (C2×S3×Q8)⋊14C2, C2.79(C2×S3×D4), (C2×S3×D4).10C2, (C2×C3⋊C8)⋊36C22, (C2×C24⋊C2)⋊31C2, (C2×D4.S3)⋊27C2, (C2×C6).395(C2×D4), (C2×Q8⋊2S3)⋊25C2, (S3×C2×C4).257C22, (C2×C4).611(C22×S3), SmallGroup(192,1317)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C2×S3×SD16
G = < a,b,c,d,e | a2=b3=c2=d8=e2=1, ab=ba, ac=ca, ad=da, ae=ea, cbc=b-1, bd=db, be=eb, cd=dc, ce=ec, ede=d3 >
Subgroups: 888 in 298 conjugacy classes, 111 normal (33 characteristic)
C1, C2, C2, C2, C3, C4, C4, C22, C22, S3, S3, C6, C6, C6, C8, C8, C2×C4, C2×C4, D4, D4, Q8, Q8, C23, Dic3, Dic3, C12, C12, D6, D6, C2×C6, C2×C6, C2×C8, C2×C8, SD16, SD16, C22×C4, C2×D4, C2×D4, C2×Q8, C2×Q8, C24, C3⋊C8, C24, Dic6, Dic6, C4×S3, C4×S3, D12, D12, C2×Dic3, C2×Dic3, C3⋊D4, C2×C12, C2×C12, C3×D4, C3×D4, C3×Q8, C3×Q8, C22×S3, C22×S3, C22×C6, C22×C8, C2×SD16, C2×SD16, C22×D4, C22×Q8, S3×C8, C24⋊C2, C2×C3⋊C8, D4.S3, Q8⋊2S3, C2×C24, C3×SD16, C2×Dic6, C2×Dic6, S3×C2×C4, S3×C2×C4, C2×D12, S3×D4, S3×D4, S3×Q8, S3×Q8, C2×C3⋊D4, C6×D4, C6×Q8, S3×C23, C22×SD16, S3×C2×C8, C2×C24⋊C2, S3×SD16, C2×D4.S3, C2×Q8⋊2S3, C6×SD16, C2×S3×D4, C2×S3×Q8, C2×S3×SD16
Quotients: C1, C2, C22, S3, D4, C23, D6, SD16, C2×D4, C24, C22×S3, C2×SD16, C22×D4, S3×D4, S3×C23, C22×SD16, S3×SD16, C2×S3×D4, C2×S3×SD16
►On 48 points
(1 15)(2 16)(3 9)(4 10)(5 11)(6 12)(7 13)(8 14)(17 27)(18 28)(19 29)(20 30)(21 31)(22 32)(23 25)(24 26)(33 47)(34 48)(35 41)(36 42)(37 43)(38 44)(39 45)(40 46)
(1 36 29)(2 37 30)(3 38 31)(4 39 32)(5 40 25)(6 33 26)(7 34 27)(8 35 28)(9 44 21)(10 45 22)(11 46 23)(12 47 24)(13 48 17)(14 41 18)(15 42 19)(16 43 20)
(1 11)(2 12)(3 13)(4 14)(5 15)(6 16)(7 9)(8 10)(17 38)(18 39)(19 40)(20 33)(21 34)(22 35)(23 36)(24 37)(25 42)(26 43)(27 44)(28 45)(29 46)(30 47)(31 48)(32 41)
(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 15)(2 10)(3 13)(4 16)(5 11)(6 14)(7 9)(8 12)(17 31)(18 26)(19 29)(20 32)(21 27)(22 30)(23 25)(24 28)(33 41)(34 44)(35 47)(36 42)(37 45)(38 48)(39 43)(40 46)
G:=sub<Sym(48)| (1,15)(2,16)(3,9)(4,10)(5,11)(6,12)(7,13)(8,14)(17,27)(18,28)(19,29)(20,30)(21,31)(22,32)(23,25)(24,26)(33,47)(34,48)(35,41)(36,42)(37,43)(38,44)(39,45)(40,46), (1,36,29)(2,37,30)(3,38,31)(4,39,32)(5,40,25)(6,33,26)(7,34,27)(8,35,28)(9,44,21)(10,45,22)(11,46,23)(12,47,24)(13,48,17)(14,41,18)(15,42,19)(16,43,20), (1,11)(2,12)(3,13)(4,14)(5,15)(6,16)(7,9)(8,10)(17,38)(18,39)(19,40)(20,33)(21,34)(22,35)(23,36)(24,37)(25,42)(26,43)(27,44)(28,45)(29,46)(30,47)(31,48)(32,41), (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), (1,15)(2,10)(3,13)(4,16)(5,11)(6,14)(7,9)(8,12)(17,31)(18,26)(19,29)(20,32)(21,27)(22,30)(23,25)(24,28)(33,41)(34,44)(35,47)(36,42)(37,45)(38,48)(39,43)(40,46)>;
G:=Group( (1,15)(2,16)(3,9)(4,10)(5,11)(6,12)(7,13)(8,14)(17,27)(18,28)(19,29)(20,30)(21,31)(22,32)(23,25)(24,26)(33,47)(34,48)(35,41)(36,42)(37,43)(38,44)(39,45)(40,46), (1,36,29)(2,37,30)(3,38,31)(4,39,32)(5,40,25)(6,33,26)(7,34,27)(8,35,28)(9,44,21)(10,45,22)(11,46,23)(12,47,24)(13,48,17)(14,41,18)(15,42,19)(16,43,20), (1,11)(2,12)(3,13)(4,14)(5,15)(6,16)(7,9)(8,10)(17,38)(18,39)(19,40)(20,33)(21,34)(22,35)(23,36)(24,37)(25,42)(26,43)(27,44)(28,45)(29,46)(30,47)(31,48)(32,41), (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), (1,15)(2,10)(3,13)(4,16)(5,11)(6,14)(7,9)(8,12)(17,31)(18,26)(19,29)(20,32)(21,27)(22,30)(23,25)(24,28)(33,41)(34,44)(35,47)(36,42)(37,45)(38,48)(39,43)(40,46) );
G=PermutationGroup([[(1,15),(2,16),(3,9),(4,10),(5,11),(6,12),(7,13),(8,14),(17,27),(18,28),(19,29),(20,30),(21,31),(22,32),(23,25),(24,26),(33,47),(34,48),(35,41),(36,42),(37,43),(38,44),(39,45),(40,46)], [(1,36,29),(2,37,30),(3,38,31),(4,39,32),(5,40,25),(6,33,26),(7,34,27),(8,35,28),(9,44,21),(10,45,22),(11,46,23),(12,47,24),(13,48,17),(14,41,18),(15,42,19),(16,43,20)], [(1,11),(2,12),(3,13),(4,14),(5,15),(6,16),(7,9),(8,10),(17,38),(18,39),(19,40),(20,33),(21,34),(22,35),(23,36),(24,37),(25,42),(26,43),(27,44),(28,45),(29,46),(30,47),(31,48),(32,41)], [(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)], [(1,15),(2,10),(3,13),(4,16),(5,11),(6,14),(7,9),(8,12),(17,31),(18,26),(19,29),(20,32),(21,27),(22,30),(23,25),(24,28),(33,41),(34,44),(35,47),(36,42),(37,45),(38,48),(39,43),(40,46)]])
42 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 2H | 2I | 2J | 2K | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 6A | 6B | 6C | 6D | 6E | 8A | 8B | 8C | 8D | 8E | 8F | 8G | 8H | 12A | 12B | 12C | 12D | 24A | 24B | 24C | 24D |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 6 | 6 | 6 | 6 | 6 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 12 | 12 | 12 | 12 | 24 | 24 | 24 | 24 |
| size | 1 | 1 | 1 | 1 | 3 | 3 | 3 | 3 | 4 | 4 | 12 | 12 | 2 | 2 | 2 | 4 | 4 | 6 | 6 | 12 | 12 | 2 | 2 | 2 | 8 | 8 | 2 | 2 | 2 | 2 | 6 | 6 | 6 | 6 | 4 | 4 | 8 | 8 | 4 | 4 | 4 | 4 |
42 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | ||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | S3 | D4 | D4 | D4 | D6 | D6 | D6 | D6 | SD16 | S3×D4 | S3×D4 | S3×SD16 |
| kernel | C2×S3×SD16 | S3×C2×C8 | C2×C24⋊C2 | S3×SD16 | C2×D4.S3 | C2×Q8⋊2S3 | C6×SD16 | C2×S3×D4 | C2×S3×Q8 | C2×SD16 | C4×S3 | C2×Dic3 | C22×S3 | C2×C8 | SD16 | C2×D4 | C2×Q8 | D6 | C4 | C22 | C2 |
| # reps | 1 | 1 | 1 | 8 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 1 | 1 | 1 | 4 | 1 | 1 | 8 | 1 | 1 | 4 |
Matrix representation of C2×S3×SD16 ►in GL6(𝔽73)
| 72 | 0 | 0 | 0 | 0 | 0 |
| 0 | 72 | 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 | 1 |
| 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 | 0 | 72 |
| 0 | 0 | 0 | 0 | 1 | 72 |
| 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 | 0 | 1 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 67 | 67 | 0 | 0 | 0 | 0 |
| 6 | 67 | 0 | 0 | 0 | 0 |
| 0 | 0 | 67 | 6 | 0 | 0 |
| 0 | 0 | 67 | 67 | 0 | 0 |
| 0 | 0 | 0 | 0 | 72 | 0 |
| 0 | 0 | 0 | 0 | 0 | 72 |
| 72 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 72 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
G:=sub<GL(6,GF(73))| [72,0,0,0,0,0,0,72,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,72,72],[72,0,0,0,0,0,0,72,0,0,0,0,0,0,72,0,0,0,0,0,0,72,0,0,0,0,0,0,0,1,0,0,0,0,1,0],[67,6,0,0,0,0,67,67,0,0,0,0,0,0,67,67,0,0,0,0,6,67,0,0,0,0,0,0,72,0,0,0,0,0,0,72],[72,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,72,0,0,0,0,0,0,1,0,0,0,0,0,0,1] >;
C2×S3×SD16 in GAP, Magma, Sage, TeX
C_2\times S_3\times {\rm SD}_{16} % in TeX
G:=Group("C2xS3xSD16"); // GroupNames label
G:=SmallGroup(192,1317);
// by ID
G=gap.SmallGroup(192,1317);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,185,136,438,235,102,6278]);
// Polycyclic
G:=Group<a,b,c,d,e|a^2=b^3=c^2=d^8=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^3>;
// generators/relations