direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C2×D6⋊C4, C23.36D6, C22.16D12, (C2×C4)⋊8D6, D6⋊6(C2×C4), (C22×C4)⋊3S3, C6.40(C2×D4), C2.3(C2×D12), (C2×C6).36D4, C6⋊1(C22⋊C4), (C22×S3)⋊3C4, (C22×C12)⋊1C2, (C2×C12)⋊10C22, (S3×C23).2C2, C22.17(C4×S3), (C2×C6).45C23, C6.18(C22×C4), (C22×Dic3)⋊3C2, (C2×Dic3)⋊6C22, C22.20(C3⋊D4), C22.23(C22×S3), (C22×C6).37C22, (C22×S3).23C22, C3⋊2(C2×C22⋊C4), C2.19(S3×C2×C4), C2.2(C2×C3⋊D4), (C2×C6).18(C2×C4), SmallGroup(96,134)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C2×D6⋊C4
G = < a,b,c,d | a2=b6=c2=d4=1, ab=ba, ac=ca, ad=da, cbc=b-1, bd=db, dcd-1=b3c >
Subgroups: 322 in 132 conjugacy classes, 57 normal (17 characteristic)
C1, C2, C2, C2, C3, C4, C22, C22, C22, S3, C6, C6, C2×C4, C2×C4, C23, C23, Dic3, C12, D6, D6, C2×C6, C2×C6, C22⋊C4, C22×C4, C22×C4, C24, C2×Dic3, C2×Dic3, C2×C12, C2×C12, C22×S3, C22×S3, C22×C6, C2×C22⋊C4, D6⋊C4, C22×Dic3, C22×C12, S3×C23, C2×D6⋊C4
Quotients: C1, C2, C4, C22, S3, C2×C4, D4, C23, D6, C22⋊C4, C22×C4, C2×D4, C4×S3, D12, C3⋊D4, C22×S3, C2×C22⋊C4, D6⋊C4, S3×C2×C4, C2×D12, C2×C3⋊D4, C2×D6⋊C4
►On 48 points
(1 32)(2 33)(3 34)(4 35)(5 36)(6 31)(7 28)(8 29)(9 30)(10 25)(11 26)(12 27)(13 46)(14 47)(15 48)(16 43)(17 44)(18 45)(19 40)(20 41)(21 42)(22 37)(23 38)(24 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 28)(2 27)(3 26)(4 25)(5 30)(6 29)(7 32)(8 31)(9 36)(10 35)(11 34)(12 33)(13 41)(14 40)(15 39)(16 38)(17 37)(18 42)(19 47)(20 46)(21 45)(22 44)(23 43)(24 48)
(1 23 11 17)(2 24 12 18)(3 19 7 13)(4 20 8 14)(5 21 9 15)(6 22 10 16)(25 43 31 37)(26 44 32 38)(27 45 33 39)(28 46 34 40)(29 47 35 41)(30 48 36 42)
G:=sub<Sym(48)| (1,32)(2,33)(3,34)(4,35)(5,36)(6,31)(7,28)(8,29)(9,30)(10,25)(11,26)(12,27)(13,46)(14,47)(15,48)(16,43)(17,44)(18,45)(19,40)(20,41)(21,42)(22,37)(23,38)(24,39), (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), (1,28)(2,27)(3,26)(4,25)(5,30)(6,29)(7,32)(8,31)(9,36)(10,35)(11,34)(12,33)(13,41)(14,40)(15,39)(16,38)(17,37)(18,42)(19,47)(20,46)(21,45)(22,44)(23,43)(24,48), (1,23,11,17)(2,24,12,18)(3,19,7,13)(4,20,8,14)(5,21,9,15)(6,22,10,16)(25,43,31,37)(26,44,32,38)(27,45,33,39)(28,46,34,40)(29,47,35,41)(30,48,36,42)>;
G:=Group( (1,32)(2,33)(3,34)(4,35)(5,36)(6,31)(7,28)(8,29)(9,30)(10,25)(11,26)(12,27)(13,46)(14,47)(15,48)(16,43)(17,44)(18,45)(19,40)(20,41)(21,42)(22,37)(23,38)(24,39), (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), (1,28)(2,27)(3,26)(4,25)(5,30)(6,29)(7,32)(8,31)(9,36)(10,35)(11,34)(12,33)(13,41)(14,40)(15,39)(16,38)(17,37)(18,42)(19,47)(20,46)(21,45)(22,44)(23,43)(24,48), (1,23,11,17)(2,24,12,18)(3,19,7,13)(4,20,8,14)(5,21,9,15)(6,22,10,16)(25,43,31,37)(26,44,32,38)(27,45,33,39)(28,46,34,40)(29,47,35,41)(30,48,36,42) );
G=PermutationGroup([[(1,32),(2,33),(3,34),(4,35),(5,36),(6,31),(7,28),(8,29),(9,30),(10,25),(11,26),(12,27),(13,46),(14,47),(15,48),(16,43),(17,44),(18,45),(19,40),(20,41),(21,42),(22,37),(23,38),(24,39)], [(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)], [(1,28),(2,27),(3,26),(4,25),(5,30),(6,29),(7,32),(8,31),(9,36),(10,35),(11,34),(12,33),(13,41),(14,40),(15,39),(16,38),(17,37),(18,42),(19,47),(20,46),(21,45),(22,44),(23,43),(24,48)], [(1,23,11,17),(2,24,12,18),(3,19,7,13),(4,20,8,14),(5,21,9,15),(6,22,10,16),(25,43,31,37),(26,44,32,38),(27,45,33,39),(28,46,34,40),(29,47,35,41),(30,48,36,42)]])
C2×D6⋊C4 is a maximal subgroup of
(C22×S3)⋊C8 C22.58(S3×D4) (C2×C4)⋊9D12 D6⋊C42 D6⋊(C4⋊C4) D6⋊C4⋊C4 D6⋊C4⋊5C4 D6⋊C4⋊3C4 (C2×C12)⋊5D4 C6.C22≀C2 (C22×S3)⋊Q8 (C2×C4).21D12 C6.(C4⋊D4) (C22×C4).37D6 (C2×C12).33D4 (C2×C4)⋊6D12 (C2×C42)⋊3S3 C24.59D6 C24.23D6 C24.24D6 C24.60D6 C24.25D6 C23⋊3D12 C24.27D6 C4⋊(D6⋊C4) (C2×D12)⋊10C4 D6⋊C4⋊6C4 D6⋊C4⋊7C4 (C2×C4)⋊3D12 (C2×C12).289D4 (C2×C12).290D4 (C2×C12).56D4 C24.76D6 C24.32D6 (C22×Q8)⋊9S3 C2×C4×D12 C2×S3×C22⋊C4 C42⋊12D6 C42⋊13D6 D4⋊5D12 C42⋊18D6 C42⋊19D6 C6.372+ 1+4 C6.402+ 1+4 C6.462+ 1+4 C6.512+ 1+4 C6.532+ 1+4 C6.562+ 1+4 C6.1212+ 1+4 C6.1222+ 1+4 C2×C4×C3⋊D4 C6.1452+ 1+4
C2×D6⋊C4 is a maximal quotient of
(C2×Dic6)⋊7C4 (C2×C4)⋊6D12 C24.56D6 C24.59D6 C24.60D6 C4○D12⋊C4 C4.(D6⋊C4) C4⋊(D6⋊C4) (C2×D12)⋊10C4 C4⋊C4⋊36D6 C4.(C2×D12) C4⋊C4.237D6 C42⋊6D6 (C2×D12)⋊13C4 (C22×C8)⋊7S3 C23.28D12 C23.51D12 D6⋊6M4(2) D6⋊C8⋊40C2 C23.53D12 M4(2).31D6 C23.54D12 M4(2)⋊24D6 C24.76D6
36 conjugacy classes
| class | 1 | 2A | ··· | 2G | 2H | 2I | 2J | 2K | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 6A | ··· | 6G | 12A | ··· | 12H |
| order | 1 | 2 | ··· | 2 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 6 | ··· | 6 | 12 | ··· | 12 |
| size | 1 | 1 | ··· | 1 | 6 | 6 | 6 | 6 | 2 | 2 | 2 | 2 | 2 | 6 | 6 | 6 | 6 | 2 | ··· | 2 | 2 | ··· | 2 |
36 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | + | + | + | |||
| image | C1 | C2 | C2 | C2 | C2 | C4 | S3 | D4 | D6 | D6 | C4×S3 | D12 | C3⋊D4 |
| kernel | C2×D6⋊C4 | D6⋊C4 | C22×Dic3 | C22×C12 | S3×C23 | C22×S3 | C22×C4 | C2×C6 | C2×C4 | C23 | C22 | C22 | C22 |
| # reps | 1 | 4 | 1 | 1 | 1 | 8 | 1 | 4 | 2 | 1 | 4 | 4 | 4 |
Matrix representation of C2×D6⋊C4 ►in GL6(𝔽13)
| 12 | 0 | 0 | 0 | 0 | 0 |
| 0 | 12 | 0 | 0 | 0 | 0 |
| 0 | 0 | 12 | 0 | 0 | 0 |
| 0 | 0 | 0 | 12 | 0 | 0 |
| 0 | 0 | 0 | 0 | 12 | 0 |
| 0 | 0 | 0 | 0 | 0 | 12 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 12 | 12 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 12 | 12 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 12 |
| 0 | 0 | 0 | 0 | 1 | 1 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 12 | 0 | 0 |
| 0 | 0 | 12 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 5 | 0 | 0 | 0 | 0 | 0 |
| 0 | 5 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 10 | 7 |
| 0 | 0 | 0 | 0 | 6 | 3 |
G:=sub<GL(6,GF(13))| [12,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,12],[0,12,0,0,0,0,1,12,0,0,0,0,0,0,0,12,0,0,0,0,1,12,0,0,0,0,0,0,0,1,0,0,0,0,12,1],[0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,12,0,0,0,0,12,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0],[5,0,0,0,0,0,0,5,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,10,6,0,0,0,0,7,3] >;
C2×D6⋊C4 in GAP, Magma, Sage, TeX
C_2\times D_6\rtimes C_4
% in TeX
G:=Group("C2xD6:C4"); // GroupNames label
G:=SmallGroup(96,134);
// by ID
G=gap.SmallGroup(96,134);
# by ID
G:=PCGroup([6,-2,-2,-2,-2,-2,-3,362,50,2309]);
// Polycyclic
G:=Group<a,b,c,d|a^2=b^6=c^2=d^4=1,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c=b^-1,b*d=d*b,d*c*d^-1=b^3*c>;
// generators/relations