metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D12.14D4, C42.66D6, Dic6.14D4, C4.52(S3×D4), (C2×D4).52D6, C12.29(C2×D4), C4.4D4⋊5S3, (C2×C12).11D4, (C2×Q8).65D6, C6.52C22≀C2, D12⋊6C22⋊2C2, C3⋊3(D4.8D4), C42⋊4S3⋊12C2, C12.10D4⋊5C2, Q8.15D6⋊2C2, (C6×D4).68C22, (C6×Q8).59C22, C2.20(C23⋊2D6), (C4×C12).110C22, (C2×C12).380C23, C4○D12.20C22, C4.Dic3.14C22, (C2×C6).511(C2×D4), (C3×C4.4D4)⋊5C2, (C2×C4).10(C3⋊D4), C22.32(C2×C3⋊D4), (C2×C4).117(C22×S3), SmallGroup(192,621)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D12.14D4
G = < a,b,c,d | a12=b2=d2=1, c4=a6, bab=cac-1=a-1, dad=a5, cbc-1=a7b, dbd=a10b, dcd=a6c3 >
Subgroups: 432 in 146 conjugacy classes, 39 normal (19 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, S3, C6, C6, C8, C2×C4, C2×C4, C2×C4, D4, Q8, C23, Dic3, C12, C12, D6, C2×C6, C2×C6, C42, C22⋊C4, M4(2), D8, SD16, C2×D4, C2×Q8, C2×Q8, C4○D4, C3⋊C8, Dic6, Dic6, C4×S3, D12, D12, C3⋊D4, C2×C12, C2×C12, C2×C12, C3×D4, C3×Q8, C22×C6, C4.10D4, C4≀C2, C4.4D4, C8⋊C22, 2- 1+4, C4.Dic3, D4⋊S3, D4.S3, C4×C12, C3×C22⋊C4, C4○D12, C4○D12, S3×Q8, Q8⋊3S3, C6×D4, C6×Q8, D4.8D4, C42⋊4S3, C12.10D4, D12⋊6C22, C3×C4.4D4, Q8.15D6, D12.14D4
Quotients: C1, C2, C22, S3, D4, C23, D6, C2×D4, C3⋊D4, C22×S3, C22≀C2, S3×D4, C2×C3⋊D4, D4.8D4, C23⋊2D6, D12.14D4
Character table of D12.14D4
| class | 1 | 2A | 2B | 2C | 2D | 2E | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 6A | 6B | 6C | 6D | 6E | 8A | 8B | 12A | 12B | 12C | 12D | 12E | 12F | 12G | 12H | |
| size | 1 | 1 | 2 | 8 | 12 | 12 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 12 | 12 | 2 | 2 | 2 | 8 | 8 | 24 | 24 | 4 | 4 | 4 | 4 | 4 | 4 | 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 | -2 | 0 | 0 | -1 | 2 | 2 | 2 | -2 | 2 | -2 | 0 | 0 | -1 | -1 | -1 | 1 | 1 | 0 | 0 | -1 | -1 | -1 | -1 | -1 | -1 | 1 | 1 | orthogonal lifted from D6 |
| ρ10 | 2 | 2 | -2 | 0 | -2 | 0 | 2 | -2 | 2 | 0 | 0 | 0 | 0 | 2 | 0 | -2 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | -2 | 2 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ11 | 2 | 2 | -2 | 0 | 2 | 0 | 2 | -2 | 2 | 0 | 0 | 0 | 0 | -2 | 0 | -2 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | -2 | 2 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ12 | 2 | 2 | -2 | 0 | 0 | -2 | 2 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | 2 | -2 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 2 | -2 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ13 | 2 | 2 | 2 | 2 | 0 | 0 | -1 | 2 | 2 | -2 | -2 | -2 | -2 | 0 | 0 | -1 | -1 | -1 | -1 | -1 | 0 | 0 | 1 | 1 | -1 | -1 | 1 | 1 | 1 | 1 | orthogonal lifted from D6 |
| ρ14 | 2 | 2 | -2 | 0 | 0 | 2 | 2 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | -2 | -2 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 2 | -2 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ15 | 2 | 2 | 2 | 0 | 0 | 0 | 2 | -2 | -2 | 0 | 2 | 0 | -2 | 0 | 0 | 2 | 2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | -2 | -2 | 0 | 0 | 2 | -2 | orthogonal lifted from D4 |
| ρ16 | 2 | 2 | 2 | 0 | 0 | 0 | 2 | -2 | -2 | 0 | -2 | 0 | 2 | 0 | 0 | 2 | 2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | -2 | -2 | 0 | 0 | -2 | 2 | orthogonal lifted from D4 |
| ρ17 | 2 | 2 | 2 | 2 | 0 | 0 | -1 | 2 | 2 | 2 | 2 | 2 | 2 | 0 | 0 | -1 | -1 | -1 | -1 | -1 | 0 | 0 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | orthogonal lifted from S3 |
| ρ18 | 2 | 2 | 2 | -2 | 0 | 0 | -1 | 2 | 2 | -2 | 2 | -2 | 2 | 0 | 0 | -1 | -1 | -1 | 1 | 1 | 0 | 0 | 1 | 1 | -1 | -1 | 1 | 1 | -1 | -1 | orthogonal lifted from D6 |
| ρ19 | 2 | 2 | 2 | 0 | 0 | 0 | -1 | -2 | -2 | 0 | 2 | 0 | -2 | 0 | 0 | -1 | -1 | -1 | √-3 | -√-3 | 0 | 0 | -√-3 | √-3 | 1 | 1 | -√-3 | √-3 | -1 | 1 | complex lifted from C3⋊D4 |
| ρ20 | 2 | 2 | 2 | 0 | 0 | 0 | -1 | -2 | -2 | 0 | -2 | 0 | 2 | 0 | 0 | -1 | -1 | -1 | √-3 | -√-3 | 0 | 0 | √-3 | -√-3 | 1 | 1 | √-3 | -√-3 | 1 | -1 | complex lifted from C3⋊D4 |
| ρ21 | 2 | 2 | 2 | 0 | 0 | 0 | -1 | -2 | -2 | 0 | 2 | 0 | -2 | 0 | 0 | -1 | -1 | -1 | -√-3 | √-3 | 0 | 0 | √-3 | -√-3 | 1 | 1 | √-3 | -√-3 | -1 | 1 | complex lifted from C3⋊D4 |
| ρ22 | 2 | 2 | 2 | 0 | 0 | 0 | -1 | -2 | -2 | 0 | -2 | 0 | 2 | 0 | 0 | -1 | -1 | -1 | -√-3 | √-3 | 0 | 0 | -√-3 | √-3 | 1 | 1 | -√-3 | √-3 | 1 | -1 | complex lifted from C3⋊D4 |
| ρ23 | 4 | 4 | -4 | 0 | 0 | 0 | -2 | 4 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | 2 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | -2 | 2 | 0 | 0 | 0 | 0 | orthogonal lifted from S3×D4 |
| ρ24 | 4 | 4 | -4 | 0 | 0 | 0 | -2 | -4 | 4 | 0 | 0 | 0 | 0 | 0 | 0 | 2 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 2 | -2 | 0 | 0 | 0 | 0 | orthogonal lifted from S3×D4 |
| ρ25 | 4 | -4 | 0 | 0 | 0 | 0 | 4 | 0 | 0 | -2i | 0 | 2i | 0 | 0 | 0 | 0 | -4 | 0 | 0 | 0 | 0 | 0 | -2i | -2i | 0 | 0 | 2i | 2i | 0 | 0 | complex lifted from D4.8D4 |
| ρ26 | 4 | -4 | 0 | 0 | 0 | 0 | 4 | 0 | 0 | 2i | 0 | -2i | 0 | 0 | 0 | 0 | -4 | 0 | 0 | 0 | 0 | 0 | 2i | 2i | 0 | 0 | -2i | -2i | 0 | 0 | complex lifted from D4.8D4 |
| ρ27 | 4 | -4 | 0 | 0 | 0 | 0 | -2 | 0 | 0 | 2i | 0 | -2i | 0 | 0 | 0 | 2√-3 | 2 | -2√-3 | 0 | 0 | 0 | 0 | 2ζ4ζ32 | 2ζ4ζ3 | 0 | 0 | 2ζ43ζ32 | 2ζ43ζ3 | 0 | 0 | complex faithful |
| ρ28 | 4 | -4 | 0 | 0 | 0 | 0 | -2 | 0 | 0 | -2i | 0 | 2i | 0 | 0 | 0 | -2√-3 | 2 | 2√-3 | 0 | 0 | 0 | 0 | 2ζ43ζ3 | 2ζ43ζ32 | 0 | 0 | 2ζ4ζ3 | 2ζ4ζ32 | 0 | 0 | complex faithful |
| ρ29 | 4 | -4 | 0 | 0 | 0 | 0 | -2 | 0 | 0 | -2i | 0 | 2i | 0 | 0 | 0 | 2√-3 | 2 | -2√-3 | 0 | 0 | 0 | 0 | 2ζ43ζ32 | 2ζ43ζ3 | 0 | 0 | 2ζ4ζ32 | 2ζ4ζ3 | 0 | 0 | complex faithful |
| ρ30 | 4 | -4 | 0 | 0 | 0 | 0 | -2 | 0 | 0 | 2i | 0 | -2i | 0 | 0 | 0 | -2√-3 | 2 | 2√-3 | 0 | 0 | 0 | 0 | 2ζ4ζ3 | 2ζ4ζ32 | 0 | 0 | 2ζ43ζ3 | 2ζ43ζ32 | 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 43)(2 42)(3 41)(4 40)(5 39)(6 38)(7 37)(8 48)(9 47)(10 46)(11 45)(12 44)(13 32)(14 31)(15 30)(16 29)(17 28)(18 27)(19 26)(20 25)(21 36)(22 35)(23 34)(24 33)
(1 30 4 27 7 36 10 33)(2 29 5 26 8 35 11 32)(3 28 6 25 9 34 12 31)(13 47 16 44 19 41 22 38)(14 46 17 43 20 40 23 37)(15 45 18 42 21 39 24 48)
(1 27)(2 32)(3 25)(4 30)(5 35)(6 28)(7 33)(8 26)(9 31)(10 36)(11 29)(12 34)(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), (1,43)(2,42)(3,41)(4,40)(5,39)(6,38)(7,37)(8,48)(9,47)(10,46)(11,45)(12,44)(13,32)(14,31)(15,30)(16,29)(17,28)(18,27)(19,26)(20,25)(21,36)(22,35)(23,34)(24,33), (1,30,4,27,7,36,10,33)(2,29,5,26,8,35,11,32)(3,28,6,25,9,34,12,31)(13,47,16,44,19,41,22,38)(14,46,17,43,20,40,23,37)(15,45,18,42,21,39,24,48), (1,27)(2,32)(3,25)(4,30)(5,35)(6,28)(7,33)(8,26)(9,31)(10,36)(11,29)(12,34)(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), (1,43)(2,42)(3,41)(4,40)(5,39)(6,38)(7,37)(8,48)(9,47)(10,46)(11,45)(12,44)(13,32)(14,31)(15,30)(16,29)(17,28)(18,27)(19,26)(20,25)(21,36)(22,35)(23,34)(24,33), (1,30,4,27,7,36,10,33)(2,29,5,26,8,35,11,32)(3,28,6,25,9,34,12,31)(13,47,16,44,19,41,22,38)(14,46,17,43,20,40,23,37)(15,45,18,42,21,39,24,48), (1,27)(2,32)(3,25)(4,30)(5,35)(6,28)(7,33)(8,26)(9,31)(10,36)(11,29)(12,34)(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)], [(1,43),(2,42),(3,41),(4,40),(5,39),(6,38),(7,37),(8,48),(9,47),(10,46),(11,45),(12,44),(13,32),(14,31),(15,30),(16,29),(17,28),(18,27),(19,26),(20,25),(21,36),(22,35),(23,34),(24,33)], [(1,30,4,27,7,36,10,33),(2,29,5,26,8,35,11,32),(3,28,6,25,9,34,12,31),(13,47,16,44,19,41,22,38),(14,46,17,43,20,40,23,37),(15,45,18,42,21,39,24,48)], [(1,27),(2,32),(3,25),(4,30),(5,35),(6,28),(7,33),(8,26),(9,31),(10,36),(11,29),(12,34),(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 D12.14D4 ►in GL4(𝔽73) generated by
| 0 | 9 | 0 | 0 |
| 64 | 0 | 0 | 0 |
| 0 | 0 | 0 | 65 |
| 0 | 0 | 8 | 0 |
| 0 | 0 | 70 | 0 |
| 0 | 0 | 0 | 3 |
| 24 | 0 | 0 | 0 |
| 0 | 49 | 0 | 0 |
| 0 | 0 | 0 | 72 |
| 0 | 0 | 72 | 0 |
| 1 | 0 | 0 | 0 |
| 0 | 72 | 0 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 0 | 72 | 0 |
| 0 | 72 | 0 | 0 |
| 1 | 0 | 0 | 0 |
G:=sub<GL(4,GF(73))| [0,64,0,0,9,0,0,0,0,0,0,8,0,0,65,0],[0,0,24,0,0,0,0,49,70,0,0,0,0,3,0,0],[0,0,1,0,0,0,0,72,0,72,0,0,72,0,0,0],[0,0,0,1,0,0,72,0,0,72,0,0,1,0,0,0] >;
D12.14D4 in GAP, Magma, Sage, TeX
D_{12}._{14}D_4 % in TeX
G:=Group("D12.14D4"); // GroupNames label
G:=SmallGroup(192,621);
// by ID
G=gap.SmallGroup(192,621);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,232,254,219,1123,570,297,136,1684,6278]);
// 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=a^6*c^3>;
// generators/relations
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