direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C4×D12, C12⋊5D4, C42⋊5S3, C3⋊1(C4×D4), C4⋊2(C4×S3), C12⋊4(C2×C4), (C4×C12)⋊7C2, D6⋊1(C2×C4), C6.2(C2×D4), C4○2(D6⋊C4), D6⋊C4⋊17C2, (C2×C4).75D6, C2.1(C2×D12), C4○2(C4⋊Dic3), C4⋊Dic3⋊16C2, C6.4(C4○D4), C6.4(C22×C4), (C2×D12).10C2, C2.3(C4○D12), (C2×C6).14C23, (C2×C12).86C22, C22.11(C22×S3), (C22×S3).15C22, (C2×Dic3).25C22, (S3×C2×C4)⋊7C2, C2.6(S3×C2×C4), SmallGroup(96,80)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C4×D12
G = < a,b,c | a4=b12=c2=1, ab=ba, ac=ca, cbc=b-1 >
Subgroups: 218 in 94 conjugacy classes, 45 normal (21 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, S3, C6, C2×C4, C2×C4, D4, C23, Dic3, C12, C12, D6, D6, C2×C6, C42, C22⋊C4, C4⋊C4, C22×C4, C2×D4, C4×S3, D12, C2×Dic3, C2×C12, C22×S3, C4×D4, C4⋊Dic3, D6⋊C4, C4×C12, S3×C2×C4, C2×D12, C4×D12
Quotients: C1, C2, C4, C22, S3, C2×C4, D4, C23, D6, C22×C4, C2×D4, C4○D4, C4×S3, D12, C22×S3, C4×D4, S3×C2×C4, C2×D12, C4○D12, C4×D12
►On 48 points
(1 34 17 39)(2 35 18 40)(3 36 19 41)(4 25 20 42)(5 26 21 43)(6 27 22 44)(7 28 23 45)(8 29 24 46)(9 30 13 47)(10 31 14 48)(11 32 15 37)(12 33 16 38)
(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 3)(4 12)(5 11)(6 10)(7 9)(13 23)(14 22)(15 21)(16 20)(17 19)(25 33)(26 32)(27 31)(28 30)(34 36)(37 43)(38 42)(39 41)(44 48)(45 47)
G:=sub<Sym(48)| (1,34,17,39)(2,35,18,40)(3,36,19,41)(4,25,20,42)(5,26,21,43)(6,27,22,44)(7,28,23,45)(8,29,24,46)(9,30,13,47)(10,31,14,48)(11,32,15,37)(12,33,16,38), (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), (1,3)(4,12)(5,11)(6,10)(7,9)(13,23)(14,22)(15,21)(16,20)(17,19)(25,33)(26,32)(27,31)(28,30)(34,36)(37,43)(38,42)(39,41)(44,48)(45,47)>;
G:=Group( (1,34,17,39)(2,35,18,40)(3,36,19,41)(4,25,20,42)(5,26,21,43)(6,27,22,44)(7,28,23,45)(8,29,24,46)(9,30,13,47)(10,31,14,48)(11,32,15,37)(12,33,16,38), (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), (1,3)(4,12)(5,11)(6,10)(7,9)(13,23)(14,22)(15,21)(16,20)(17,19)(25,33)(26,32)(27,31)(28,30)(34,36)(37,43)(38,42)(39,41)(44,48)(45,47) );
G=PermutationGroup([[(1,34,17,39),(2,35,18,40),(3,36,19,41),(4,25,20,42),(5,26,21,43),(6,27,22,44),(7,28,23,45),(8,29,24,46),(9,30,13,47),(10,31,14,48),(11,32,15,37),(12,33,16,38)], [(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)], [(1,3),(4,12),(5,11),(6,10),(7,9),(13,23),(14,22),(15,21),(16,20),(17,19),(25,33),(26,32),(27,31),(28,30),(34,36),(37,43),(38,42),(39,41),(44,48),(45,47)]])
C4×D12 is a maximal subgroup of
C4.17D24 D12⋊2C8 C8⋊6D12 C8⋊9D12 C42.16D6 D24⋊C4 D12⋊C8 D6⋊3M4(2) C12⋊2M4(2) C12⋊SD16 D12⋊3Q8 C4⋊D24 D12.19D4 D12⋊4Q8 D12.3Q8 C42.48D6 C42.56D6 D12.23D4 D12.4Q8 C12⋊2D8 C12⋊5SD16 D12⋊5Q8 D12⋊6Q8 C42.276D6 C42.277D6 C42⋊9D6 C42.91D6 C42⋊10D6 C42⋊12D6 C42.93D6 C42.95D6 C42.99D6 C42.100D6 C4×S3×D4 C42⋊13D6 C42⋊14D6 C42.228D6 D12⋊23D4 D12⋊24D4 D4⋊5D12 D4⋊6D12 C42.113D6 C42.116D6 C42.117D6 C42.119D6 C42.126D6 Q8⋊6D12 Q8⋊7D12 D12⋊10Q8 C42.131D6 C42.132D6 C42.133D6 C42.135D6 C42.136D6 D12⋊10D4 Dic6⋊10D4 C42⋊22D6 C42.143D6 D12⋊7Q8 C42.150D6 C42.152D6 C42.153D6 C42⋊25D6 C42⋊26D6 C42.161D6 C42.163D6 D12⋊11D4 Dic6⋊11D4 D12⋊12D4 D12⋊8Q8 D12⋊9Q8 C42.177D6 C42.179D6 Dic3⋊4D12 Dic3⋊5D12 Dic5⋊4D12 D60⋊17C4
C4×D12 is a maximal quotient of
C2.(C4×D12) (C2×C4)⋊9D12 D6⋊C4⋊C4 D6⋊C4⋊3C4 C8⋊6D12 D24⋊11C4 C8⋊9D12 C42.16D6 D24⋊C4 Dic12⋊C4 D24⋊4C4 C12⋊4(C4⋊C4) (C2×C4)⋊6D12 (C2×C42)⋊3S3 Dic3⋊4D12 Dic3⋊5D12 Dic5⋊4D12 D60⋊17C4
36 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 4K | 4L | 6A | 6B | 6C | 12A | ··· | 12L |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 6 | 6 | 6 | 12 | ··· | 12 |
| size | 1 | 1 | 1 | 1 | 6 | 6 | 6 | 6 | 2 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 6 | 6 | 6 | 6 | 2 | 2 | 2 | 2 | ··· | 2 |
36 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | + | + | + | ||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C4 | S3 | D4 | D6 | C4○D4 | C4×S3 | D12 | C4○D12 |
| kernel | C4×D12 | C4⋊Dic3 | D6⋊C4 | C4×C12 | S3×C2×C4 | C2×D12 | D12 | C42 | C12 | C2×C4 | C6 | C4 | C4 | C2 |
| # reps | 1 | 1 | 2 | 1 | 2 | 1 | 8 | 1 | 2 | 3 | 2 | 4 | 4 | 4 |
Matrix representation of C4×D12 ►in GL3(𝔽13) generated by
| 8 | 0 | 0 |
| 0 | 8 | 0 |
| 0 | 0 | 8 |
| 12 | 0 | 0 |
| 0 | 3 | 10 |
| 0 | 3 | 6 |
| 12 | 0 | 0 |
| 0 | 1 | 1 |
| 0 | 0 | 12 |
G:=sub<GL(3,GF(13))| [8,0,0,0,8,0,0,0,8],[12,0,0,0,3,3,0,10,6],[12,0,0,0,1,0,0,1,12] >;
C4×D12 in GAP, Magma, Sage, TeX
C_4\times D_{12} % in TeX
G:=Group("C4xD12"); // GroupNames label
G:=SmallGroup(96,80);
// by ID
G=gap.SmallGroup(96,80);
# by ID
G:=PCGroup([6,-2,-2,-2,-2,-2,-3,217,103,50,2309]);
// Polycyclic
G:=Group<a,b,c|a^4=b^12=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^-1>;
// generators/relations