metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D12.15D4, C42.86D6, Dic6.15D4, C4:Q8:8S3, C4.58(S3xD4), C12.41(C2xD4), (C2xC12).12D4, (C2xQ8).73D6, C6.54C22wrC2, C42:4S3:14C2, C12.10D4:6C2, Q8.11D6:3C2, C3:3(D4.10D4), (C6xQ8).67C22, C2.22(C23:2D6), (C4xC12).142C22, (C2xC12).413C23, C4oD12.22C22, Q8.15D6.2C2, C4.Dic3.15C22, (C3xC4:Q8):8C2, (C2xC6).544(C2xD4), (C2xC4).11(C3:D4), C22.34(C2xC3:D4), (C2xC4).119(C22xS3), SmallGroup(192,654)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D12.15D4
G = < a,b,c,d | a12=b2=d2=1, c4=a6, bab=cac-1=a-1, dad=a5, cbc-1=a7b, dbd=a10b, dcd=c3 >
Subgroups: 400 in 142 conjugacy classes, 39 normal (19 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, S3, C6, C6, C8, C2xC4, C2xC4, C2xC4, D4, Q8, Dic3, C12, C12, D6, C2xC6, C42, C4:C4, M4(2), SD16, Q16, C2xQ8, C2xQ8, C4oD4, C3:C8, Dic6, Dic6, C4xS3, D12, D12, C3:D4, C2xC12, C2xC12, C2xC12, C3xQ8, C4.10D4, C4wrC2, C4:Q8, C8.C22, 2- 1+4, C4.Dic3, Q8:2S3, C3:Q16, C4xC12, C3xC4:C4, C4oD12, C4oD12, S3xQ8, Q8:3S3, C6xQ8, D4.10D4, C42:4S3, C12.10D4, Q8.11D6, C3xC4:Q8, Q8.15D6, D12.15D4
Quotients: C1, C2, C22, S3, D4, C23, D6, C2xD4, C3:D4, C22xS3, C22wrC2, S3xD4, C2xC3:D4, D4.10D4, C23:2D6, D12.15D4
Character table of D12.15D4
| class | 1 | 2A | 2B | 2C | 2D | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 6A | 6B | 6C | 8A | 8B | 12A | 12B | 12C | 12D | 12E | 12F | 12G | 12H | 12I | 12J | |
| size | 1 | 1 | 2 | 12 | 12 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 8 | 12 | 12 | 2 | 2 | 2 | 24 | 24 | 4 | 4 | 4 | 4 | 4 | 4 | 8 | 8 | 8 | 8 | |
| 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 | 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 | 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 | 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 | 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 | 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 | 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 | 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 | 1 | -1 | 1 | -1 | 1 | -1 | linear of order 2 |
| ρ9 | 2 | 2 | -2 | 0 | -2 | 2 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | 2 | 0 | -2 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ10 | 2 | 2 | -2 | 0 | 2 | 2 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | -2 | 0 | -2 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ11 | 2 | 2 | -2 | 2 | 0 | 2 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | -2 | -2 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ12 | 2 | 2 | 2 | 0 | 0 | 2 | -2 | -2 | 2 | 0 | 0 | -2 | 0 | 0 | 0 | 2 | 2 | 2 | 0 | 0 | 0 | 0 | 0 | -2 | -2 | 0 | 0 | 2 | 0 | -2 | orthogonal lifted from D4 |
| ρ13 | 2 | 2 | 2 | 0 | 0 | 2 | -2 | -2 | -2 | 0 | 0 | 2 | 0 | 0 | 0 | 2 | 2 | 2 | 0 | 0 | 0 | 0 | 0 | -2 | -2 | 0 | 0 | -2 | 0 | 2 | orthogonal lifted from D4 |
| ρ14 | 2 | 2 | -2 | -2 | 0 | 2 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 2 | -2 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ15 | 2 | 2 | 2 | 0 | 0 | -1 | 2 | 2 | -2 | 2 | 2 | -2 | -2 | 0 | 0 | -1 | -1 | -1 | 0 | 0 | -1 | -1 | -1 | -1 | -1 | -1 | 1 | 1 | 1 | 1 | orthogonal lifted from D6 |
| ρ16 | 2 | 2 | 2 | 0 | 0 | -1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 0 | 0 | -1 | -1 | -1 | 0 | 0 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | orthogonal lifted from S3 |
| ρ17 | 2 | 2 | 2 | 0 | 0 | -1 | 2 | 2 | -2 | -2 | -2 | -2 | 2 | 0 | 0 | -1 | -1 | -1 | 0 | 0 | 1 | 1 | 1 | -1 | -1 | 1 | -1 | 1 | -1 | 1 | orthogonal lifted from D6 |
| ρ18 | 2 | 2 | 2 | 0 | 0 | -1 | 2 | 2 | 2 | -2 | -2 | 2 | -2 | 0 | 0 | -1 | -1 | -1 | 0 | 0 | 1 | 1 | 1 | -1 | -1 | 1 | 1 | -1 | 1 | -1 | orthogonal lifted from D6 |
| ρ19 | 2 | 2 | 2 | 0 | 0 | -1 | -2 | -2 | 2 | 0 | 0 | -2 | 0 | 0 | 0 | -1 | -1 | -1 | 0 | 0 | -√-3 | √-3 | -√-3 | 1 | 1 | √-3 | √-3 | -1 | -√-3 | 1 | complex lifted from C3:D4 |
| ρ20 | 2 | 2 | 2 | 0 | 0 | -1 | -2 | -2 | 2 | 0 | 0 | -2 | 0 | 0 | 0 | -1 | -1 | -1 | 0 | 0 | √-3 | -√-3 | √-3 | 1 | 1 | -√-3 | -√-3 | -1 | √-3 | 1 | complex lifted from C3:D4 |
| ρ21 | 2 | 2 | 2 | 0 | 0 | -1 | -2 | -2 | -2 | 0 | 0 | 2 | 0 | 0 | 0 | -1 | -1 | -1 | 0 | 0 | √-3 | -√-3 | √-3 | 1 | 1 | -√-3 | √-3 | 1 | -√-3 | -1 | complex lifted from C3:D4 |
| ρ22 | 2 | 2 | 2 | 0 | 0 | -1 | -2 | -2 | -2 | 0 | 0 | 2 | 0 | 0 | 0 | -1 | -1 | -1 | 0 | 0 | -√-3 | √-3 | -√-3 | 1 | 1 | √-3 | -√-3 | 1 | √-3 | -1 | complex lifted from C3:D4 |
| ρ23 | 4 | 4 | -4 | 0 | 0 | -2 | -4 | 4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 2 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from S3xD4 |
| ρ24 | 4 | 4 | -4 | 0 | 0 | -2 | 4 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 2 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from S3xD4 |
| ρ25 | 4 | -4 | 0 | 0 | 0 | 4 | 0 | 0 | 0 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | -4 | 0 | 0 | 0 | -2 | 2 | 2 | 0 | 0 | -2 | 0 | 0 | 0 | 0 | symplectic lifted from D4.10D4, Schur index 2 |
| ρ26 | 4 | -4 | 0 | 0 | 0 | 4 | 0 | 0 | 0 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | -4 | 0 | 0 | 0 | 2 | -2 | -2 | 0 | 0 | 2 | 0 | 0 | 0 | 0 | symplectic lifted from D4.10D4, Schur index 2 |
| ρ27 | 4 | -4 | 0 | 0 | 0 | -2 | 0 | 0 | 0 | 2 | -2 | 0 | 0 | 0 | 0 | -2√-3 | 2 | 2√-3 | 0 | 0 | -1-√-3 | 1-√-3 | 1+√-3 | 0 | 0 | -1+√-3 | 0 | 0 | 0 | 0 | complex faithful |
| ρ28 | 4 | -4 | 0 | 0 | 0 | -2 | 0 | 0 | 0 | -2 | 2 | 0 | 0 | 0 | 0 | -2√-3 | 2 | 2√-3 | 0 | 0 | 1+√-3 | -1+√-3 | -1-√-3 | 0 | 0 | 1-√-3 | 0 | 0 | 0 | 0 | complex faithful |
| ρ29 | 4 | -4 | 0 | 0 | 0 | -2 | 0 | 0 | 0 | -2 | 2 | 0 | 0 | 0 | 0 | 2√-3 | 2 | -2√-3 | 0 | 0 | 1-√-3 | -1-√-3 | -1+√-3 | 0 | 0 | 1+√-3 | 0 | 0 | 0 | 0 | complex faithful |
| ρ30 | 4 | -4 | 0 | 0 | 0 | -2 | 0 | 0 | 0 | 2 | -2 | 0 | 0 | 0 | 0 | 2√-3 | 2 | -2√-3 | 0 | 0 | -1+√-3 | 1+√-3 | 1-√-3 | 0 | 0 | -1-√-3 | 0 | 0 | 0 | 0 | complex faithful |
►On 48 points
(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)(2 39)(3 38)(4 37)(5 48)(6 47)(7 46)(8 45)(9 44)(10 43)(11 42)(12 41)(13 28)(14 27)(15 26)(16 25)(17 36)(18 35)(19 34)(20 33)(21 32)(22 31)(23 30)(24 29)
(1 47 4 44 7 41 10 38)(2 46 5 43 8 40 11 37)(3 45 6 42 9 39 12 48)(13 35 16 32 19 29 22 26)(14 34 17 31 20 28 23 25)(15 33 18 30 21 27 24 36)
(1 26)(2 31)(3 36)(4 29)(5 34)(6 27)(7 32)(8 25)(9 30)(10 35)(11 28)(12 33)(13 44)(14 37)(15 42)(16 47)(17 40)(18 45)(19 38)(20 43)(21 48)(22 41)(23 46)(24 39)
G:=sub<Sym(48)| (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)(2,39)(3,38)(4,37)(5,48)(6,47)(7,46)(8,45)(9,44)(10,43)(11,42)(12,41)(13,28)(14,27)(15,26)(16,25)(17,36)(18,35)(19,34)(20,33)(21,32)(22,31)(23,30)(24,29), (1,47,4,44,7,41,10,38)(2,46,5,43,8,40,11,37)(3,45,6,42,9,39,12,48)(13,35,16,32,19,29,22,26)(14,34,17,31,20,28,23,25)(15,33,18,30,21,27,24,36), (1,26)(2,31)(3,36)(4,29)(5,34)(6,27)(7,32)(8,25)(9,30)(10,35)(11,28)(12,33)(13,44)(14,37)(15,42)(16,47)(17,40)(18,45)(19,38)(20,43)(21,48)(22,41)(23,46)(24,39)>;
G:=Group( (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)(2,39)(3,38)(4,37)(5,48)(6,47)(7,46)(8,45)(9,44)(10,43)(11,42)(12,41)(13,28)(14,27)(15,26)(16,25)(17,36)(18,35)(19,34)(20,33)(21,32)(22,31)(23,30)(24,29), (1,47,4,44,7,41,10,38)(2,46,5,43,8,40,11,37)(3,45,6,42,9,39,12,48)(13,35,16,32,19,29,22,26)(14,34,17,31,20,28,23,25)(15,33,18,30,21,27,24,36), (1,26)(2,31)(3,36)(4,29)(5,34)(6,27)(7,32)(8,25)(9,30)(10,35)(11,28)(12,33)(13,44)(14,37)(15,42)(16,47)(17,40)(18,45)(19,38)(20,43)(21,48)(22,41)(23,46)(24,39) );
G=PermutationGroup([[(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),(2,39),(3,38),(4,37),(5,48),(6,47),(7,46),(8,45),(9,44),(10,43),(11,42),(12,41),(13,28),(14,27),(15,26),(16,25),(17,36),(18,35),(19,34),(20,33),(21,32),(22,31),(23,30),(24,29)], [(1,47,4,44,7,41,10,38),(2,46,5,43,8,40,11,37),(3,45,6,42,9,39,12,48),(13,35,16,32,19,29,22,26),(14,34,17,31,20,28,23,25),(15,33,18,30,21,27,24,36)], [(1,26),(2,31),(3,36),(4,29),(5,34),(6,27),(7,32),(8,25),(9,30),(10,35),(11,28),(12,33),(13,44),(14,37),(15,42),(16,47),(17,40),(18,45),(19,38),(20,43),(21,48),(22,41),(23,46),(24,39)]])
Matrix representation of D12.15D4 ►in GL4(F7) generated by
| 3 | 4 | 3 | 6 |
| 3 | 5 | 3 | 1 |
| 5 | 3 | 5 | 6 |
| 6 | 5 | 1 | 1 |
| 2 | 6 | 0 | 2 |
| 2 | 3 | 6 | 0 |
| 3 | 6 | 4 | 4 |
| 3 | 6 | 3 | 5 |
| 2 | 4 | 4 | 5 |
| 6 | 3 | 3 | 6 |
| 3 | 0 | 1 | 4 |
| 1 | 2 | 5 | 1 |
| 3 | 6 | 6 | 6 |
| 2 | 3 | 0 | 4 |
| 2 | 6 | 0 | 6 |
| 4 | 2 | 4 | 1 |
G:=sub<GL(4,GF(7))| [3,3,5,6,4,5,3,5,3,3,5,1,6,1,6,1],[2,2,3,3,6,3,6,6,0,6,4,3,2,0,4,5],[2,6,3,1,4,3,0,2,4,3,1,5,5,6,4,1],[3,2,2,4,6,3,6,2,6,0,0,4,6,4,6,1] >;
D12.15D4 in GAP, Magma, Sage, TeX
D_{12}._{15}D_4 % in TeX
G:=Group("D12.15D4"); // GroupNames label
G:=SmallGroup(192,654);
// by ID
G=gap.SmallGroup(192,654);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,232,254,219,184,1123,570,297,136,1684,6278]);
// Polycyclic
G:=Group<a,b,c,d|a^12=b^2=d^2=1,c^4=a^6,b*a*b=c*a*c^-1=a^-1,d*a*d=a^5,c*b*c^-1=a^7*b,d*b*d=a^10*b,d*c*d=c^3>;
// generators/relations
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