direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C2×C4○D12, C6.4C24, D6.1C23, C23.31D6, D12⋊12C22, C12.43C23, Dic6⋊11C22, Dic3.2C23, (C2×C4)○D12, C4○(C2×D12), (C2×C4)⋊10D6, C4○(C2×Dic6), (C2×C4)○Dic6, C4○(C4○D12), C6⋊1(C4○D4), (C22×C4)⋊8S3, (C2×D12)⋊14C2, (C4×S3)⋊6C22, (C22×C12)⋊8C2, C3⋊D4⋊6C22, C2.5(S3×C23), (C2×C12)⋊13C22, (C2×Dic6)⋊15C2, (C2×C6).65C23, C4.43(C22×S3), C22.5(C22×S3), (C22×C6).46C22, (C22×S3).28C22, (C2×Dic3).43C22, C4○(C2×C3⋊D4), C3⋊1(C2×C4○D4), (S3×C2×C4)⋊15C2, (C2×C4)○(C2×D12), (C2×C4)○(C3⋊D4), (C2×C4)○(C2×Dic6), (C2×C3⋊D4)⋊12C2, (C2×C4)○(C2×C3⋊D4), SmallGroup(96,208)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C2×C4○D12
G = < a,b,c,d | a2=b4=d2=1, c6=b2, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd=b2c5 >
Subgroups: 322 in 164 conjugacy classes, 89 normal (17 characteristic)
C1, C2, C2, C2, C3, C4, C4, C22, C22, C22, S3, C6, C6, C6, C2×C4, C2×C4, C2×C4, D4, Q8, C23, C23, Dic3, C12, D6, D6, C2×C6, C2×C6, C2×C6, C22×C4, C22×C4, C2×D4, C2×Q8, C4○D4, Dic6, C4×S3, D12, C2×Dic3, C3⋊D4, C2×C12, C2×C12, C22×S3, C22×C6, C2×C4○D4, C2×Dic6, S3×C2×C4, C2×D12, C4○D12, C2×C3⋊D4, C22×C12, C2×C4○D12
Quotients: C1, C2, C22, S3, C23, D6, C4○D4, C24, C22×S3, C2×C4○D4, C4○D12, S3×C23, C2×C4○D12
►On 48 points
(1 14)(2 15)(3 16)(4 17)(5 18)(6 19)(7 20)(8 21)(9 22)(10 23)(11 24)(12 13)(25 45)(26 46)(27 47)(28 48)(29 37)(30 38)(31 39)(32 40)(33 41)(34 42)(35 43)(36 44)
(1 39 7 45)(2 40 8 46)(3 41 9 47)(4 42 10 48)(5 43 11 37)(6 44 12 38)(13 30 19 36)(14 31 20 25)(15 32 21 26)(16 33 22 27)(17 34 23 28)(18 35 24 29)
(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 13)(2 24)(3 23)(4 22)(5 21)(6 20)(7 19)(8 18)(9 17)(10 16)(11 15)(12 14)(25 44)(26 43)(27 42)(28 41)(29 40)(30 39)(31 38)(32 37)(33 48)(34 47)(35 46)(36 45)
G:=sub<Sym(48)| (1,14)(2,15)(3,16)(4,17)(5,18)(6,19)(7,20)(8,21)(9,22)(10,23)(11,24)(12,13)(25,45)(26,46)(27,47)(28,48)(29,37)(30,38)(31,39)(32,40)(33,41)(34,42)(35,43)(36,44), (1,39,7,45)(2,40,8,46)(3,41,9,47)(4,42,10,48)(5,43,11,37)(6,44,12,38)(13,30,19,36)(14,31,20,25)(15,32,21,26)(16,33,22,27)(17,34,23,28)(18,35,24,29), (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), (1,13)(2,24)(3,23)(4,22)(5,21)(6,20)(7,19)(8,18)(9,17)(10,16)(11,15)(12,14)(25,44)(26,43)(27,42)(28,41)(29,40)(30,39)(31,38)(32,37)(33,48)(34,47)(35,46)(36,45)>;
G:=Group( (1,14)(2,15)(3,16)(4,17)(5,18)(6,19)(7,20)(8,21)(9,22)(10,23)(11,24)(12,13)(25,45)(26,46)(27,47)(28,48)(29,37)(30,38)(31,39)(32,40)(33,41)(34,42)(35,43)(36,44), (1,39,7,45)(2,40,8,46)(3,41,9,47)(4,42,10,48)(5,43,11,37)(6,44,12,38)(13,30,19,36)(14,31,20,25)(15,32,21,26)(16,33,22,27)(17,34,23,28)(18,35,24,29), (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), (1,13)(2,24)(3,23)(4,22)(5,21)(6,20)(7,19)(8,18)(9,17)(10,16)(11,15)(12,14)(25,44)(26,43)(27,42)(28,41)(29,40)(30,39)(31,38)(32,37)(33,48)(34,47)(35,46)(36,45) );
G=PermutationGroup([[(1,14),(2,15),(3,16),(4,17),(5,18),(6,19),(7,20),(8,21),(9,22),(10,23),(11,24),(12,13),(25,45),(26,46),(27,47),(28,48),(29,37),(30,38),(31,39),(32,40),(33,41),(34,42),(35,43),(36,44)], [(1,39,7,45),(2,40,8,46),(3,41,9,47),(4,42,10,48),(5,43,11,37),(6,44,12,38),(13,30,19,36),(14,31,20,25),(15,32,21,26),(16,33,22,27),(17,34,23,28),(18,35,24,29)], [(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)], [(1,13),(2,24),(3,23),(4,22),(5,21),(6,20),(7,19),(8,18),(9,17),(10,16),(11,15),(12,14),(25,44),(26,43),(27,42),(28,41),(29,40),(30,39),(31,38),(32,37),(33,48),(34,47),(35,46),(36,45)]])
C2×C4○D12 is a maximal subgroup of
D6⋊C8⋊C2 D12.32D4 D12⋊14D4 C4○D12⋊C4 C4.(C2×D12) C42⋊6D6 (C2×D12)⋊13C4 D12⋊17D4 D12.37D4 (C22×C8)⋊7S3 C23.28D12 D6⋊C8⋊40C2 M4(2).31D6 C23.54D12 M4(2)⋊24D6 C42.276D6 C24.38D6 C6.82+ 1+4 C6.2- 1+4 C6.2+ 1+4 C42.188D6 C42.91D6 C42⋊10D6 C42⋊11D6 C42.92D6 C42⋊14D6 C42.228D6 D12⋊23D4 D12⋊24D4 Dic6⋊23D4 Dic6⋊24D4 Dic6⋊20D4 C6.382+ 1+4 C6.722- 1+4 D12⋊20D4 C6.162- 1+4 C6.172- 1+4 D12⋊22D4 Dic6⋊22D4 C6.1212+ 1+4 C6.822- 1+4 M4(2)⋊26D6 C24.9C23 C24.83D6 C24.52D6 C6.442- 1+4 C12.C24 (C2×C12)⋊17D4 C6.1082- 1+4 C2×S3×C4○D4 C6.C25
C2×C4○D12 is a maximal quotient of
C2×C4×Dic6 C42.274D6 C2×C4×D12 C42.276D6 C42.277D6 C24.38D6 C24.41D6 C24.42D6 C6.2- 1+4 C6.102+ 1+4 C6.52- 1+4 C6.112+ 1+4 C6.62- 1+4 C42.89D6 C42⋊12D6 C42.93D6 C42.94D6 C42.95D6 C42.96D6 C42.97D6 C42.98D6 C42.99D6 C42.100D6 C42.102D6 C42.104D6 C42.105D6 C42.106D6 C42⋊14D6 C42.228D6 D12⋊23D4 D12⋊24D4 Dic6⋊23D4 Dic6⋊24D4 C42⋊18D6 C42.229D6 C42.113D6 C42.114D6 C42⋊19D6 C42.115D6 C42.116D6 C42.117D6 C42.118D6 C42.119D6 Dic6⋊10Q8 C42.122D6 C42.232D6 D12⋊10Q8 C42.131D6 C42.132D6 C42.133D6 C42.134D6 C42.135D6 C42.136D6 C2×C4×C3⋊D4 C24.83D6
36 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 2H | 2I | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 6A | ··· | 6G | 12A | ··· | 12H |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 6 | ··· | 6 | 12 | ··· | 12 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 6 | 6 | 6 | 6 | 2 | 1 | 1 | 1 | 1 | 2 | 2 | 6 | 6 | 6 | 6 | 2 | ··· | 2 | 2 | ··· | 2 |
36 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | + | + | + | ||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | S3 | D6 | D6 | C4○D4 | C4○D12 |
| kernel | C2×C4○D12 | C2×Dic6 | S3×C2×C4 | C2×D12 | C4○D12 | C2×C3⋊D4 | C22×C12 | C22×C4 | C2×C4 | C23 | C6 | C2 |
| # reps | 1 | 1 | 2 | 1 | 8 | 2 | 1 | 1 | 6 | 1 | 4 | 8 |
Matrix representation of C2×C4○D12 ►in GL3(𝔽13) generated by
| 12 | 0 | 0 |
| 0 | 1 | 0 |
| 0 | 0 | 1 |
| 1 | 0 | 0 |
| 0 | 8 | 0 |
| 0 | 0 | 8 |
| 1 | 0 | 0 |
| 0 | 7 | 10 |
| 0 | 3 | 10 |
| 1 | 0 | 0 |
| 0 | 7 | 10 |
| 0 | 3 | 6 |
G:=sub<GL(3,GF(13))| [12,0,0,0,1,0,0,0,1],[1,0,0,0,8,0,0,0,8],[1,0,0,0,7,3,0,10,10],[1,0,0,0,7,3,0,10,6] >;
C2×C4○D12 in GAP, Magma, Sage, TeX
C_2\times C_4\circ D_{12} % in TeX
G:=Group("C2xC4oD12"); // GroupNames label
G:=SmallGroup(96,208);
// by ID
G=gap.SmallGroup(96,208);
# by ID
G:=PCGroup([6,-2,-2,-2,-2,-2,-3,86,579,2309]);
// Polycyclic
G:=Group<a,b,c,d|a^2=b^4=d^2=1,c^6=b^2,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d=b^2*c^5>;
// generators/relations