metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C22.4D12, C23.21D6, D6⋊C4⋊7C2, C6.6(C2×D4), (C2×C4).9D6, (C2×C6).4D4, C22⋊C4⋊6S3, C4⋊Dic3⋊5C2, C2.8(C2×D12), C6.23(C4○D4), (C2×C12).3C22, (C2×C6).27C23, (C22×Dic3)⋊2C2, C3⋊2(C22.D4), C2.10(D4⋊2S3), (C22×S3).5C22, (C22×C6).16C22, C22.45(C22×S3), (C2×Dic3).28C22, (C3×C22⋊C4)⋊4C2, (C2×C3⋊D4).5C2, SmallGroup(96,93)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C22.D12
G = < a,b,c,d | a2=b2=c12=1, d2=b, cac-1=ab=ba, ad=da, bc=cb, bd=db, dcd-1=bc-1 >
Subgroups: 186 in 78 conjugacy classes, 33 normal (15 characteristic)
C1, C2, C2 [×2], C2 [×3], C3, C4 [×5], C22, C22 [×2], C22 [×5], S3, C6, C6 [×2], C6 [×2], C2×C4 [×2], C2×C4 [×5], D4 [×2], C23, C23, Dic3 [×3], C12 [×2], D6 [×3], C2×C6, C2×C6 [×2], C2×C6 [×2], C22⋊C4, C22⋊C4 [×2], C4⋊C4 [×2], C22×C4, C2×D4, C2×Dic3, C2×Dic3 [×2], C2×Dic3 [×2], C3⋊D4 [×2], C2×C12 [×2], C22×S3, C22×C6, C22.D4, C4⋊Dic3 [×2], D6⋊C4 [×2], C3×C22⋊C4, C22×Dic3, C2×C3⋊D4, C22.D12
Quotients: C1, C2 [×7], C22 [×7], S3, D4 [×2], C23, D6 [×3], C2×D4, C4○D4 [×2], D12 [×2], C22×S3, C22.D4, C2×D12, D4⋊2S3 [×2], C22.D12
Character table of C22.D12
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 6A | 6B | 6C | 6D | 6E | 12A | 12B | 12C | 12D | |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 12 | 2 | 4 | 4 | 6 | 6 | 6 | 6 | 12 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | |
| ρ1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | trivial |
| ρ2 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | 1 | -1 | 1 | 1 | -1 | 1 | -1 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | 1 | 1 | linear of order 2 |
| ρ3 | 1 | 1 | 1 | 1 | 1 | 1 | -1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | -1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | linear of order 2 |
| ρ4 | 1 | 1 | 1 | 1 | -1 | -1 | 1 | 1 | -1 | 1 | -1 | 1 | -1 | 1 | -1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | 1 | 1 | linear of order 2 |
| ρ5 | 1 | 1 | 1 | 1 | -1 | -1 | 1 | 1 | 1 | -1 | 1 | -1 | 1 | -1 | -1 | 1 | 1 | 1 | -1 | -1 | 1 | 1 | -1 | -1 | linear of order 2 |
| ρ6 | 1 | 1 | 1 | 1 | 1 | 1 | -1 | 1 | -1 | -1 | 1 | 1 | 1 | 1 | -1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | linear of order 2 |
| ρ7 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | 1 | 1 | -1 | -1 | 1 | -1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | 1 | 1 | -1 | -1 | linear of order 2 |
| ρ8 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | -1 | -1 | 1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | linear of order 2 |
| ρ9 | 2 | 2 | 2 | 2 | 2 | 2 | 0 | -1 | -2 | -2 | 0 | 0 | 0 | 0 | 0 | -1 | -1 | -1 | -1 | -1 | 1 | 1 | 1 | 1 | orthogonal lifted from D6 |
| ρ10 | 2 | 2 | 2 | 2 | -2 | -2 | 0 | -1 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | -1 | -1 | -1 | 1 | 1 | 1 | 1 | -1 | -1 | orthogonal lifted from D6 |
| ρ11 | 2 | 2 | 2 | 2 | -2 | -2 | 0 | -1 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | -1 | -1 | -1 | 1 | 1 | -1 | -1 | 1 | 1 | orthogonal lifted from D6 |
| ρ12 | 2 | -2 | -2 | 2 | -2 | 2 | 0 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -2 | 2 | -2 | 2 | -2 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ13 | 2 | 2 | 2 | 2 | 2 | 2 | 0 | -1 | 2 | 2 | 0 | 0 | 0 | 0 | 0 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | orthogonal lifted from S3 |
| ρ14 | 2 | -2 | -2 | 2 | 2 | -2 | 0 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -2 | 2 | -2 | -2 | 2 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ15 | 2 | -2 | -2 | 2 | -2 | 2 | 0 | -1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | -1 | 1 | -1 | 1 | √3 | -√3 | -√3 | √3 | orthogonal lifted from D12 |
| ρ16 | 2 | -2 | -2 | 2 | 2 | -2 | 0 | -1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | -1 | 1 | 1 | -1 | √3 | -√3 | √3 | -√3 | orthogonal lifted from D12 |
| ρ17 | 2 | -2 | -2 | 2 | 2 | -2 | 0 | -1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | -1 | 1 | 1 | -1 | -√3 | √3 | -√3 | √3 | orthogonal lifted from D12 |
| ρ18 | 2 | -2 | -2 | 2 | -2 | 2 | 0 | -1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | -1 | 1 | -1 | 1 | -√3 | √3 | √3 | -√3 | orthogonal lifted from D12 |
| ρ19 | 2 | 2 | -2 | -2 | 0 | 0 | 0 | 2 | 0 | 0 | -2i | 0 | 2i | 0 | 0 | -2 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from C4○D4 |
| ρ20 | 2 | -2 | 2 | -2 | 0 | 0 | 0 | 2 | 0 | 0 | 0 | 2i | 0 | -2i | 0 | 2 | -2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from C4○D4 |
| ρ21 | 2 | 2 | -2 | -2 | 0 | 0 | 0 | 2 | 0 | 0 | 2i | 0 | -2i | 0 | 0 | -2 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from C4○D4 |
| ρ22 | 2 | -2 | 2 | -2 | 0 | 0 | 0 | 2 | 0 | 0 | 0 | -2i | 0 | 2i | 0 | 2 | -2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from C4○D4 |
| ρ23 | 4 | 4 | -4 | -4 | 0 | 0 | 0 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 2 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | symplectic lifted from D4⋊2S3, Schur index 2 |
| ρ24 | 4 | -4 | 4 | -4 | 0 | 0 | 0 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -2 | 2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | symplectic lifted from D4⋊2S3, Schur index 2 |
►On 48 points
(1 13)(2 35)(3 15)(4 25)(5 17)(6 27)(7 19)(8 29)(9 21)(10 31)(11 23)(12 33)(14 42)(16 44)(18 46)(20 48)(22 38)(24 40)(26 45)(28 47)(30 37)(32 39)(34 41)(36 43)
(1 41)(2 42)(3 43)(4 44)(5 45)(6 46)(7 47)(8 48)(9 37)(10 38)(11 39)(12 40)(13 34)(14 35)(15 36)(16 25)(17 26)(18 27)(19 28)(20 29)(21 30)(22 31)(23 32)(24 33)
(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 40 41 12)(2 11 42 39)(3 38 43 10)(4 9 44 37)(5 48 45 8)(6 7 46 47)(13 24 34 33)(14 32 35 23)(15 22 36 31)(16 30 25 21)(17 20 26 29)(18 28 27 19)
G:=sub<Sym(48)| (1,13)(2,35)(3,15)(4,25)(5,17)(6,27)(7,19)(8,29)(9,21)(10,31)(11,23)(12,33)(14,42)(16,44)(18,46)(20,48)(22,38)(24,40)(26,45)(28,47)(30,37)(32,39)(34,41)(36,43), (1,41)(2,42)(3,43)(4,44)(5,45)(6,46)(7,47)(8,48)(9,37)(10,38)(11,39)(12,40)(13,34)(14,35)(15,36)(16,25)(17,26)(18,27)(19,28)(20,29)(21,30)(22,31)(23,32)(24,33), (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), (1,40,41,12)(2,11,42,39)(3,38,43,10)(4,9,44,37)(5,48,45,8)(6,7,46,47)(13,24,34,33)(14,32,35,23)(15,22,36,31)(16,30,25,21)(17,20,26,29)(18,28,27,19)>;
G:=Group( (1,13)(2,35)(3,15)(4,25)(5,17)(6,27)(7,19)(8,29)(9,21)(10,31)(11,23)(12,33)(14,42)(16,44)(18,46)(20,48)(22,38)(24,40)(26,45)(28,47)(30,37)(32,39)(34,41)(36,43), (1,41)(2,42)(3,43)(4,44)(5,45)(6,46)(7,47)(8,48)(9,37)(10,38)(11,39)(12,40)(13,34)(14,35)(15,36)(16,25)(17,26)(18,27)(19,28)(20,29)(21,30)(22,31)(23,32)(24,33), (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), (1,40,41,12)(2,11,42,39)(3,38,43,10)(4,9,44,37)(5,48,45,8)(6,7,46,47)(13,24,34,33)(14,32,35,23)(15,22,36,31)(16,30,25,21)(17,20,26,29)(18,28,27,19) );
G=PermutationGroup([(1,13),(2,35),(3,15),(4,25),(5,17),(6,27),(7,19),(8,29),(9,21),(10,31),(11,23),(12,33),(14,42),(16,44),(18,46),(20,48),(22,38),(24,40),(26,45),(28,47),(30,37),(32,39),(34,41),(36,43)], [(1,41),(2,42),(3,43),(4,44),(5,45),(6,46),(7,47),(8,48),(9,37),(10,38),(11,39),(12,40),(13,34),(14,35),(15,36),(16,25),(17,26),(18,27),(19,28),(20,29),(21,30),(22,31),(23,32),(24,33)], [(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)], [(1,40,41,12),(2,11,42,39),(3,38,43,10),(4,9,44,37),(5,48,45,8),(6,7,46,47),(13,24,34,33),(14,32,35,23),(15,22,36,31),(16,30,25,21),(17,20,26,29),(18,28,27,19)])
Matrix representation of C22.D12 ►in GL6(𝔽13)
| 12 | 0 | 0 | 0 | 0 | 0 |
| 0 | 12 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 5 |
| 0 | 0 | 0 | 0 | 8 | 0 |
| 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 | 12 | 0 |
| 0 | 0 | 0 | 0 | 0 | 12 |
| 9 | 3 | 0 | 0 | 0 | 0 |
| 3 | 4 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 1 | 0 | 0 |
| 0 | 0 | 12 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 9 | 3 | 0 | 0 | 0 | 0 |
| 8 | 4 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 1 | 0 | 0 |
| 0 | 0 | 0 | 12 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 12 | 0 |
G:=sub<GL(6,GF(13))| [12,0,0,0,0,0,0,12,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,8,0,0,0,0,5,0],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,12,0,0,0,0,0,0,12],[9,3,0,0,0,0,3,4,0,0,0,0,0,0,1,12,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0],[9,8,0,0,0,0,3,4,0,0,0,0,0,0,1,0,0,0,0,0,1,12,0,0,0,0,0,0,0,12,0,0,0,0,1,0] >;
C22.D12 in GAP, Magma, Sage, TeX
C_2^2.D_{12} % in TeX
G:=Group("C2^2.D12"); // GroupNames label
G:=SmallGroup(96,93);
// by ID
G=gap.SmallGroup(96,93);
# by ID
G:=PCGroup([6,-2,-2,-2,-2,-2,-3,217,218,188,122,2309]);
// Polycyclic
G:=Group<a,b,c,d|a^2=b^2=c^12=1,d^2=b,c*a*c^-1=a*b=b*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d^-1=b*c^-1>;
// generators/relations