metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D12.39D4, M4(2)⋊6D6, Dic6.39D4, (C2×Q8)⋊7D6, D4○D12.2C2, C4○D4.23D6, (C3×D4).14D4, C4.106(S3×D4), C8.C22⋊1S3, (C3×Q8).14D4, D12⋊C4⋊7C2, (C6×Q8)⋊4C22, C6.65C22≀C2, C12.198(C2×D4), C3⋊4(D4.9D4), (C22×S3).6D4, C22.37(S3×D4), Q8⋊3Dic3⋊8C2, Q8.11D6⋊5C2, C12.46D4⋊7C2, C12.23D4⋊7C2, D4.11(C3⋊D4), (C2×C12).17C23, Q8.18(C3⋊D4), (C4×Dic3)⋊6C22, C4.Dic3⋊9C22, C2.33(C23⋊2D6), C4○D12.25C22, (C2×D12).130C22, (C3×M4(2))⋊13C22, (C2×C6).36(C2×D4), C4.54(C2×C3⋊D4), (C3×C8.C22)⋊5C2, (C2×C4).17(C22×S3), (C3×C4○D4).15C22, SmallGroup(192,761)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D12.39D4
G = < a,b,c,d | a12=b2=c4=d2=1, bab=dad=a-1, cac-1=a5, cbc-1=ab, dbd=a10b, dcd=a6c-1 >
Subgroups: 528 in 152 conjugacy classes, 39 normal (37 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, S3, C6, C6, C8, C2×C4, C2×C4, D4, D4, Q8, Q8, C23, Dic3, C12, C12, D6, C2×C6, C2×C6, C42, C22⋊C4, M4(2), M4(2), SD16, Q16, C2×D4, C2×Q8, C4○D4, C4○D4, C3⋊C8, C24, Dic6, C4×S3, D12, D12, C2×Dic3, C3⋊D4, C2×C12, C2×C12, C3×D4, C3×D4, C3×Q8, C3×Q8, C22×S3, C22×S3, C4.D4, C4≀C2, C4.4D4, C8.C22, C8.C22, 2+ 1+4, C4.Dic3, C4×Dic3, D6⋊C4, Q8⋊2S3, C3⋊Q16, C3×M4(2), C3×SD16, C3×Q16, C2×D12, C2×D12, C4○D12, C4○D12, S3×D4, Q8⋊3S3, C6×Q8, C3×C4○D4, D4.9D4, C12.46D4, D12⋊C4, Q8⋊3Dic3, Q8.11D6, C12.23D4, C3×C8.C22, D4○D12, D12.39D4
Quotients: C1, C2, C22, S3, D4, C23, D6, C2×D4, C3⋊D4, C22×S3, C22≀C2, S3×D4, C2×C3⋊D4, D4.9D4, C23⋊2D6, D12.39D4
Character table of D12.39D4
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 6A | 6B | 6C | 8A | 8B | 12A | 12B | 12C | 12D | 12E | 24A | 24B | |
| size | 1 | 1 | 2 | 4 | 12 | 12 | 12 | 2 | 2 | 2 | 4 | 8 | 12 | 12 | 12 | 2 | 4 | 8 | 8 | 24 | 4 | 4 | 8 | 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 | 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 | 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 | 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 | 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 | 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 | 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 | 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 | linear of order 2 |
| ρ9 | 2 | 2 | 2 | 0 | 0 | -2 | 2 | 2 | -2 | -2 | 0 | 0 | 0 | 0 | 0 | 2 | 2 | 0 | 0 | 0 | -2 | -2 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ10 | 2 | 2 | -2 | -2 | 0 | 0 | 0 | 2 | 2 | -2 | 2 | 0 | 0 | 0 | 0 | 2 | -2 | -2 | 0 | 0 | 2 | -2 | 0 | 0 | 2 | 0 | 0 | orthogonal lifted from D4 |
| ρ11 | 2 | 2 | 2 | 2 | 0 | 0 | 0 | -1 | 2 | 2 | 2 | -2 | 0 | 0 | 0 | -1 | -1 | -1 | -2 | 0 | -1 | -1 | 1 | 1 | -1 | 1 | 1 | orthogonal lifted from D6 |
| ρ12 | 2 | 2 | -2 | 0 | 2 | 0 | 0 | 2 | -2 | 2 | 0 | 0 | -2 | 0 | 0 | 2 | -2 | 0 | 0 | 0 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ13 | 2 | 2 | -2 | 2 | 0 | 0 | 0 | 2 | 2 | -2 | -2 | 0 | 0 | 0 | 0 | 2 | -2 | 2 | 0 | 0 | 2 | -2 | 0 | 0 | -2 | 0 | 0 | orthogonal lifted from D4 |
| ρ14 | 2 | 2 | 2 | 0 | 0 | 2 | -2 | 2 | -2 | -2 | 0 | 0 | 0 | 0 | 0 | 2 | 2 | 0 | 0 | 0 | -2 | -2 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ15 | 2 | 2 | 2 | -2 | 0 | 0 | 0 | -1 | 2 | 2 | -2 | -2 | 0 | 0 | 0 | -1 | -1 | 1 | 2 | 0 | -1 | -1 | 1 | 1 | 1 | -1 | -1 | orthogonal lifted from D6 |
| ρ16 | 2 | 2 | 2 | 2 | 0 | 0 | 0 | -1 | 2 | 2 | 2 | 2 | 0 | 0 | 0 | -1 | -1 | -1 | 2 | 0 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | orthogonal lifted from S3 |
| ρ17 | 2 | 2 | 2 | -2 | 0 | 0 | 0 | -1 | 2 | 2 | -2 | 2 | 0 | 0 | 0 | -1 | -1 | 1 | -2 | 0 | -1 | -1 | -1 | -1 | 1 | 1 | 1 | orthogonal lifted from D6 |
| ρ18 | 2 | 2 | -2 | 0 | -2 | 0 | 0 | 2 | -2 | 2 | 0 | 0 | 2 | 0 | 0 | 2 | -2 | 0 | 0 | 0 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ19 | 2 | 2 | -2 | -2 | 0 | 0 | 0 | -1 | 2 | -2 | 2 | 0 | 0 | 0 | 0 | -1 | 1 | 1 | 0 | 0 | -1 | 1 | √-3 | -√-3 | -1 | √-3 | -√-3 | complex lifted from C3⋊D4 |
| ρ20 | 2 | 2 | -2 | 2 | 0 | 0 | 0 | -1 | 2 | -2 | -2 | 0 | 0 | 0 | 0 | -1 | 1 | -1 | 0 | 0 | -1 | 1 | √-3 | -√-3 | 1 | -√-3 | √-3 | complex lifted from C3⋊D4 |
| ρ21 | 2 | 2 | -2 | 2 | 0 | 0 | 0 | -1 | 2 | -2 | -2 | 0 | 0 | 0 | 0 | -1 | 1 | -1 | 0 | 0 | -1 | 1 | -√-3 | √-3 | 1 | √-3 | -√-3 | complex lifted from C3⋊D4 |
| ρ22 | 2 | 2 | -2 | -2 | 0 | 0 | 0 | -1 | 2 | -2 | 2 | 0 | 0 | 0 | 0 | -1 | 1 | 1 | 0 | 0 | -1 | 1 | -√-3 | √-3 | -1 | -√-3 | √-3 | complex lifted from C3⋊D4 |
| ρ23 | 4 | 4 | -4 | 0 | 0 | 0 | 0 | -2 | -4 | 4 | 0 | 0 | 0 | 0 | 0 | -2 | 2 | 0 | 0 | 0 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from S3×D4 |
| ρ24 | 4 | 4 | 4 | 0 | 0 | 0 | 0 | -2 | -4 | -4 | 0 | 0 | 0 | 0 | 0 | -2 | -2 | 0 | 0 | 0 | 2 | 2 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from S3×D4 |
| ρ25 | 4 | -4 | 0 | 0 | 0 | 0 | 0 | 4 | 0 | 0 | 0 | 0 | 0 | -2i | 2i | -4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from D4.9D4 |
| ρ26 | 4 | -4 | 0 | 0 | 0 | 0 | 0 | 4 | 0 | 0 | 0 | 0 | 0 | 2i | -2i | -4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from D4.9D4 |
| ρ27 | 8 | -8 | 0 | 0 | 0 | 0 | 0 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal faithful, Schur index 2 |
►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 12)(2 11)(3 10)(4 9)(5 8)(6 7)(13 17)(14 16)(18 24)(19 23)(20 22)(25 32)(26 31)(27 30)(28 29)(33 36)(34 35)(37 43)(38 42)(39 41)(44 48)(45 47)
(1 20 26 48)(2 13 27 41)(3 18 28 46)(4 23 29 39)(5 16 30 44)(6 21 31 37)(7 14 32 42)(8 19 33 47)(9 24 34 40)(10 17 35 45)(11 22 36 38)(12 15 25 43)
(1 35)(2 34)(3 33)(4 32)(5 31)(6 30)(7 29)(8 28)(9 27)(10 26)(11 25)(12 36)(13 18)(14 17)(15 16)(19 24)(20 23)(21 22)(37 38)(39 48)(40 47)(41 46)(42 45)(43 44)
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), (1,12)(2,11)(3,10)(4,9)(5,8)(6,7)(13,17)(14,16)(18,24)(19,23)(20,22)(25,32)(26,31)(27,30)(28,29)(33,36)(34,35)(37,43)(38,42)(39,41)(44,48)(45,47), (1,20,26,48)(2,13,27,41)(3,18,28,46)(4,23,29,39)(5,16,30,44)(6,21,31,37)(7,14,32,42)(8,19,33,47)(9,24,34,40)(10,17,35,45)(11,22,36,38)(12,15,25,43), (1,35)(2,34)(3,33)(4,32)(5,31)(6,30)(7,29)(8,28)(9,27)(10,26)(11,25)(12,36)(13,18)(14,17)(15,16)(19,24)(20,23)(21,22)(37,38)(39,48)(40,47)(41,46)(42,45)(43,44)>;
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), (1,12)(2,11)(3,10)(4,9)(5,8)(6,7)(13,17)(14,16)(18,24)(19,23)(20,22)(25,32)(26,31)(27,30)(28,29)(33,36)(34,35)(37,43)(38,42)(39,41)(44,48)(45,47), (1,20,26,48)(2,13,27,41)(3,18,28,46)(4,23,29,39)(5,16,30,44)(6,21,31,37)(7,14,32,42)(8,19,33,47)(9,24,34,40)(10,17,35,45)(11,22,36,38)(12,15,25,43), (1,35)(2,34)(3,33)(4,32)(5,31)(6,30)(7,29)(8,28)(9,27)(10,26)(11,25)(12,36)(13,18)(14,17)(15,16)(19,24)(20,23)(21,22)(37,38)(39,48)(40,47)(41,46)(42,45)(43,44) );
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)], [(1,12),(2,11),(3,10),(4,9),(5,8),(6,7),(13,17),(14,16),(18,24),(19,23),(20,22),(25,32),(26,31),(27,30),(28,29),(33,36),(34,35),(37,43),(38,42),(39,41),(44,48),(45,47)], [(1,20,26,48),(2,13,27,41),(3,18,28,46),(4,23,29,39),(5,16,30,44),(6,21,31,37),(7,14,32,42),(8,19,33,47),(9,24,34,40),(10,17,35,45),(11,22,36,38),(12,15,25,43)], [(1,35),(2,34),(3,33),(4,32),(5,31),(6,30),(7,29),(8,28),(9,27),(10,26),(11,25),(12,36),(13,18),(14,17),(15,16),(19,24),(20,23),(21,22),(37,38),(39,48),(40,47),(41,46),(42,45),(43,44)]])
Matrix representation of D12.39D4 ►in GL8(𝔽73)
| 1 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 72 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 72 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 | 0 | 72 |
| 0 | 0 | 0 | 0 | 72 | 0 | 72 | 0 |
| 0 | 0 | 0 | 0 | 72 | 1 | 0 | 1 |
| 0 | 0 | 0 | 0 | 2 | 0 | 0 | 72 |
| 1 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 72 | 0 | 0 | 0 | 0 | 0 | 0 |
| 1 | 2 | 1 | 72 | 0 | 0 | 0 | 0 |
| 2 | 1 | 0 | 72 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 72 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 | 72 | 72 |
| 0 | 0 | 0 | 0 | 1 | 72 | 0 | 72 |
| 0 | 0 | 0 | 0 | 71 | 0 | 0 | 1 |
| 1 | 0 | 48 | 48 | 0 | 0 | 0 | 0 |
| 72 | 72 | 48 | 50 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 72 | 0 | 0 | 0 | 0 |
| 0 | 0 | 72 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 46 | 27 | 0 | 27 |
| 0 | 0 | 0 | 0 | 0 | 46 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 46 | 46 | 46 |
| 0 | 0 | 0 | 0 | 46 | 27 | 27 | 27 |
| 72 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 1 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 72 | 0 | 0 | 0 | 0 |
| 0 | 0 | 72 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 | 72 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 71 | 72 |
G:=sub<GL(8,GF(73))| [1,72,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,72,0,0,0,0,0,0,1,1,0,0,0,0,0,0,0,0,1,72,72,2,0,0,0,0,0,0,1,0,0,0,0,0,0,72,0,0,0,0,0,0,72,0,1,72],[1,0,1,2,0,0,0,0,1,72,2,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,72,72,0,0,0,0,0,0,0,0,72,1,1,71,0,0,0,0,0,0,72,0,0,0,0,0,0,72,0,0,0,0,0,0,0,72,72,1],[1,72,0,0,0,0,0,0,0,72,0,0,0,0,0,0,48,48,0,72,0,0,0,0,48,50,72,0,0,0,0,0,0,0,0,0,46,0,0,46,0,0,0,0,27,46,46,27,0,0,0,0,0,0,46,27,0,0,0,0,27,0,46,27],[72,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,72,0,0,0,0,0,0,72,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,72,1,1,71,0,0,0,0,0,0,0,72] >;
D12.39D4 in GAP, Magma, Sage, TeX
D_{12}._{39}D_4 % in TeX
G:=Group("D12.39D4"); // GroupNames label
G:=SmallGroup(192,761);
// by ID
G=gap.SmallGroup(192,761);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,254,219,184,570,1684,851,438,102,6278]);
// Polycyclic
G:=Group<a,b,c,d|a^12=b^2=c^4=d^2=1,b*a*b=d*a*d=a^-1,c*a*c^-1=a^5,c*b*c^-1=a*b,d*b*d=a^10*b,d*c*d=a^6*c^-1>;
// generators/relations
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