direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C2×D4⋊D6, C12.33C24, D12.29C23, C3⋊C8⋊5C23, C4○D4⋊17D6, (C2×D4)⋊41D6, (C2×Q8)⋊33D6, C6⋊5(C8⋊C22), D4⋊5(C22×S3), (C3×D4)⋊5C23, (C3×Q8)⋊5C23, Q8⋊6(C22×S3), D4⋊S3⋊18C22, (C2×C12).217D4, C12.426(C2×D4), (C6×D4)⋊45C22, C4.33(S3×C23), (C6×Q8)⋊37C22, (C2×D12)⋊58C22, (C22×D12)⋊20C2, (C22×C6).122D4, C6.158(C22×D4), (C22×C4).297D6, (C2×C12).555C23, Q8⋊2S3⋊17C22, C23.75(C3⋊D4), C4.Dic3⋊36C22, (C22×C12).290C22, C3⋊6(C2×C8⋊C22), (C6×C4○D4)⋊2C2, (C2×C4○D4)⋊6S3, (C2×D4⋊S3)⋊31C2, (C2×C3⋊C8)⋊22C22, (C2×C6).75(C2×D4), C4.29(C2×C3⋊D4), (C2×Q8⋊2S3)⋊31C2, (C3×C4○D4)⋊16C22, (C2×C4).95(C3⋊D4), (C2×C4.Dic3)⋊30C2, C2.31(C22×C3⋊D4), (C2×C4).245(C22×S3), C22.118(C2×C3⋊D4), SmallGroup(192,1379)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C2×D4⋊D6
G = < a,b,c,d,e | a2=b4=c2=d6=e2=1, ab=ba, ac=ca, ad=da, ae=ea, cbc=ebe=b-1, bd=db, dcd-1=b2c, ece=b-1c, ede=d-1 >
Subgroups: 872 in 298 conjugacy classes, 111 normal (27 characteristic)
C1, C2, C2, C2, C3, C4, C4, C4, C22, C22, C22, S3, C6, C6, C6, C8, C2×C4, C2×C4, C2×C4, D4, D4, Q8, Q8, C23, C23, C12, C12, C12, D6, C2×C6, C2×C6, C2×C6, C2×C8, M4(2), D8, SD16, C22×C4, C22×C4, C2×D4, C2×D4, C2×Q8, C4○D4, C4○D4, C24, C3⋊C8, D12, D12, C2×C12, C2×C12, C2×C12, C3×D4, C3×D4, C3×Q8, C3×Q8, C22×S3, C22×C6, C22×C6, C2×M4(2), C2×D8, C2×SD16, C8⋊C22, C22×D4, C2×C4○D4, C2×C3⋊C8, C4.Dic3, D4⋊S3, Q8⋊2S3, C2×D12, C2×D12, C22×C12, C22×C12, C6×D4, C6×D4, C6×Q8, C3×C4○D4, C3×C4○D4, S3×C23, C2×C8⋊C22, C2×C4.Dic3, C2×D4⋊S3, C2×Q8⋊2S3, D4⋊D6, C22×D12, C6×C4○D4, C2×D4⋊D6
Quotients: C1, C2, C22, S3, D4, C23, D6, C2×D4, C24, C3⋊D4, C22×S3, C8⋊C22, C22×D4, C2×C3⋊D4, S3×C23, C2×C8⋊C22, D4⋊D6, C22×C3⋊D4, C2×D4⋊D6
►On 48 points
(1 15)(2 13)(3 14)(4 21)(5 19)(6 20)(7 22)(8 23)(9 24)(10 16)(11 17)(12 18)(25 37)(26 38)(27 39)(28 40)(29 41)(30 42)(31 46)(32 47)(33 48)(34 43)(35 44)(36 45)
(1 11 5 8)(2 12 6 9)(3 10 4 7)(13 18 20 24)(14 16 21 22)(15 17 19 23)(25 33 28 36)(26 34 29 31)(27 35 30 32)(37 48 40 45)(38 43 41 46)(39 44 42 47)
(1 32)(2 36)(3 34)(4 31)(5 35)(6 33)(7 29)(8 27)(9 25)(10 26)(11 30)(12 28)(13 45)(14 43)(15 47)(16 38)(17 42)(18 40)(19 44)(20 48)(21 46)(22 41)(23 39)(24 37)
(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 14)(2 13)(3 15)(4 19)(5 21)(6 20)(7 17)(8 16)(9 18)(10 23)(11 22)(12 24)(25 45)(26 44)(27 43)(28 48)(29 47)(30 46)(31 42)(32 41)(33 40)(34 39)(35 38)(36 37)
G:=sub<Sym(48)| (1,15)(2,13)(3,14)(4,21)(5,19)(6,20)(7,22)(8,23)(9,24)(10,16)(11,17)(12,18)(25,37)(26,38)(27,39)(28,40)(29,41)(30,42)(31,46)(32,47)(33,48)(34,43)(35,44)(36,45), (1,11,5,8)(2,12,6,9)(3,10,4,7)(13,18,20,24)(14,16,21,22)(15,17,19,23)(25,33,28,36)(26,34,29,31)(27,35,30,32)(37,48,40,45)(38,43,41,46)(39,44,42,47), (1,32)(2,36)(3,34)(4,31)(5,35)(6,33)(7,29)(8,27)(9,25)(10,26)(11,30)(12,28)(13,45)(14,43)(15,47)(16,38)(17,42)(18,40)(19,44)(20,48)(21,46)(22,41)(23,39)(24,37), (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), (1,14)(2,13)(3,15)(4,19)(5,21)(6,20)(7,17)(8,16)(9,18)(10,23)(11,22)(12,24)(25,45)(26,44)(27,43)(28,48)(29,47)(30,46)(31,42)(32,41)(33,40)(34,39)(35,38)(36,37)>;
G:=Group( (1,15)(2,13)(3,14)(4,21)(5,19)(6,20)(7,22)(8,23)(9,24)(10,16)(11,17)(12,18)(25,37)(26,38)(27,39)(28,40)(29,41)(30,42)(31,46)(32,47)(33,48)(34,43)(35,44)(36,45), (1,11,5,8)(2,12,6,9)(3,10,4,7)(13,18,20,24)(14,16,21,22)(15,17,19,23)(25,33,28,36)(26,34,29,31)(27,35,30,32)(37,48,40,45)(38,43,41,46)(39,44,42,47), (1,32)(2,36)(3,34)(4,31)(5,35)(6,33)(7,29)(8,27)(9,25)(10,26)(11,30)(12,28)(13,45)(14,43)(15,47)(16,38)(17,42)(18,40)(19,44)(20,48)(21,46)(22,41)(23,39)(24,37), (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), (1,14)(2,13)(3,15)(4,19)(5,21)(6,20)(7,17)(8,16)(9,18)(10,23)(11,22)(12,24)(25,45)(26,44)(27,43)(28,48)(29,47)(30,46)(31,42)(32,41)(33,40)(34,39)(35,38)(36,37) );
G=PermutationGroup([[(1,15),(2,13),(3,14),(4,21),(5,19),(6,20),(7,22),(8,23),(9,24),(10,16),(11,17),(12,18),(25,37),(26,38),(27,39),(28,40),(29,41),(30,42),(31,46),(32,47),(33,48),(34,43),(35,44),(36,45)], [(1,11,5,8),(2,12,6,9),(3,10,4,7),(13,18,20,24),(14,16,21,22),(15,17,19,23),(25,33,28,36),(26,34,29,31),(27,35,30,32),(37,48,40,45),(38,43,41,46),(39,44,42,47)], [(1,32),(2,36),(3,34),(4,31),(5,35),(6,33),(7,29),(8,27),(9,25),(10,26),(11,30),(12,28),(13,45),(14,43),(15,47),(16,38),(17,42),(18,40),(19,44),(20,48),(21,46),(22,41),(23,39),(24,37)], [(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)], [(1,14),(2,13),(3,15),(4,19),(5,21),(6,20),(7,17),(8,16),(9,18),(10,23),(11,22),(12,24),(25,45),(26,44),(27,43),(28,48),(29,47),(30,46),(31,42),(32,41),(33,40),(34,39),(35,38),(36,37)]])
42 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 | ··· | 6I | 8A | 8B | 8C | 8D | 12A | 12B | 12C | 12D | 12E | ··· | 12J |
| 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 | 8 | 8 | 8 | 8 | 12 | 12 | 12 | 12 | 12 | ··· | 12 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 4 | 4 | 12 | 12 | 12 | 12 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 2 | 2 | 2 | 4 | ··· | 4 | 12 | 12 | 12 | 12 | 2 | 2 | 2 | 2 | 4 | ··· | 4 |
42 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | ||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | S3 | D4 | D4 | D6 | D6 | D6 | D6 | C3⋊D4 | C3⋊D4 | C8⋊C22 | D4⋊D6 |
| kernel | C2×D4⋊D6 | C2×C4.Dic3 | C2×D4⋊S3 | C2×Q8⋊2S3 | D4⋊D6 | C22×D12 | C6×C4○D4 | C2×C4○D4 | C2×C12 | C22×C6 | C22×C4 | C2×D4 | C2×Q8 | C4○D4 | C2×C4 | C23 | C6 | C2 |
| # reps | 1 | 1 | 2 | 2 | 8 | 1 | 1 | 1 | 3 | 1 | 1 | 1 | 1 | 4 | 6 | 2 | 2 | 4 |
Matrix representation of C2×D4⋊D6 ►in 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 |
| 72 | 0 | 0 | 0 | 0 | 0 |
| 0 | 72 | 0 | 0 | 0 | 0 |
| 0 | 0 | 66 | 14 | 0 | 0 |
| 0 | 0 | 59 | 7 | 0 | 0 |
| 0 | 0 | 0 | 0 | 7 | 59 |
| 0 | 0 | 0 | 0 | 14 | 66 |
| 43 | 60 | 0 | 0 | 0 | 0 |
| 13 | 30 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 66 | 14 |
| 0 | 0 | 0 | 0 | 59 | 7 |
| 0 | 0 | 7 | 59 | 0 | 0 |
| 0 | 0 | 14 | 66 | 0 | 0 |
| 1 | 72 | 0 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 72 | 1 | 0 | 0 |
| 0 | 0 | 72 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 72 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 1 | 72 | 0 | 0 | 0 | 0 |
| 0 | 0 | 72 | 0 | 0 | 0 |
| 0 | 0 | 72 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 66 | 14 |
| 0 | 0 | 0 | 0 | 7 | 7 |
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],[72,0,0,0,0,0,0,72,0,0,0,0,0,0,66,59,0,0,0,0,14,7,0,0,0,0,0,0,7,14,0,0,0,0,59,66],[43,13,0,0,0,0,60,30,0,0,0,0,0,0,0,0,7,14,0,0,0,0,59,66,0,0,66,59,0,0,0,0,14,7,0,0],[1,1,0,0,0,0,72,0,0,0,0,0,0,0,72,72,0,0,0,0,1,0,0,0,0,0,0,0,1,1,0,0,0,0,72,0],[1,1,0,0,0,0,0,72,0,0,0,0,0,0,72,72,0,0,0,0,0,1,0,0,0,0,0,0,66,7,0,0,0,0,14,7] >;
C2×D4⋊D6 in GAP, Magma, Sage, TeX
C_2\times D_4\rtimes D_6
% in TeX
G:=Group("C2xD4:D6"); // GroupNames label
G:=SmallGroup(192,1379);
// by ID
G=gap.SmallGroup(192,1379);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,675,297,1684,235,102,6278]);
// Polycyclic
G:=Group<a,b,c,d,e|a^2=b^4=c^2=d^6=e^2=1,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,c*b*c=e*b*e=b^-1,b*d=d*b,d*c*d^-1=b^2*c,e*c*e=b^-1*c,e*d*e=d^-1>;
// generators/relations