metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D8⋊2Dic3, C24.40D4, Q16⋊2Dic3, C12.37SD16, (C3×D8)⋊4C4, (C2×C6).4D8, (C3×Q16)⋊4C4, C4○D8.2S3, (C2×C8).51D6, C3⋊3(D8⋊2C4), C24.27(C2×C4), C8⋊Dic3⋊23C2, C12.C8⋊8C2, C8.3(C2×Dic3), (C2×C12).118D4, C8.30(C3⋊D4), C4.12(D4.S3), C22.3(D4⋊S3), C6.30(D4⋊C4), C12.17(C22⋊C4), (C2×C24).154C22, C4.5(C6.D4), C2.10(D4⋊Dic3), (C3×C4○D8).5C2, (C2×C4).26(C3⋊D4), SmallGroup(192,125)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D8⋊2Dic3
G = < a,b,c,d | a8=b2=c6=1, d2=c3, bab=a-1, ac=ca, dad-1=a3, cbc-1=a4b, dbd-1=a5b, dcd-1=c-1 >
Subgroups: 168 in 58 conjugacy classes, 27 normal (all characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, C6, C6, C8, C2×C4, C2×C4, D4, Q8, Dic3, C12, C12, C2×C6, C2×C6, C16, C4⋊C4, C2×C8, D8, SD16, Q16, C4○D4, C24, C2×Dic3, C2×C12, C2×C12, C3×D4, C3×Q8, C4.Q8, M5(2), C4○D8, C3⋊C16, C4⋊Dic3, C2×C24, C3×D8, C3×SD16, C3×Q16, C3×C4○D4, D8⋊2C4, C12.C8, C8⋊Dic3, C3×C4○D8, D8⋊2Dic3
Quotients: C1, C2, C4, C22, S3, C2×C4, D4, Dic3, D6, C22⋊C4, D8, SD16, C2×Dic3, C3⋊D4, D4⋊C4, D4⋊S3, D4.S3, C6.D4, D8⋊2C4, D4⋊Dic3, D8⋊2Dic3
Character table of D8⋊2Dic3
| class | 1 | 2A | 2B | 2C | 3 | 4A | 4B | 4C | 4D | 4E | 6A | 6B | 6C | 6D | 8A | 8B | 8C | 12A | 12B | 12C | 12D | 12E | 16A | 16B | 16C | 16D | 24A | 24B | 24C | 24D | |
| size | 1 | 1 | 2 | 8 | 2 | 2 | 2 | 8 | 24 | 24 | 2 | 4 | 8 | 8 | 2 | 2 | 4 | 2 | 2 | 4 | 8 | 8 | 12 | 12 | 12 | 12 | 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 | 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 | -i | i | 1 | -1 | -1 | -1 | -1 | -1 | 1 | -1 | -1 | 1 | 1 | 1 | i | -i | i | -i | 1 | -1 | 1 | -1 | linear of order 4 |
| ρ6 | 1 | 1 | -1 | -1 | 1 | -1 | 1 | 1 | i | -i | 1 | -1 | -1 | -1 | -1 | -1 | 1 | -1 | -1 | 1 | 1 | 1 | -i | i | -i | i | 1 | -1 | 1 | -1 | linear of order 4 |
| ρ7 | 1 | 1 | -1 | 1 | 1 | -1 | 1 | -1 | -i | i | 1 | -1 | 1 | 1 | -1 | -1 | 1 | -1 | -1 | 1 | -1 | -1 | -i | i | -i | i | 1 | -1 | 1 | -1 | linear of order 4 |
| ρ8 | 1 | 1 | -1 | 1 | 1 | -1 | 1 | -1 | i | -i | 1 | -1 | 1 | 1 | -1 | -1 | 1 | -1 | -1 | 1 | -1 | -1 | i | -i | i | -i | 1 | -1 | 1 | -1 | linear of order 4 |
| ρ9 | 2 | 2 | 2 | -2 | -1 | 2 | 2 | -2 | 0 | 0 | -1 | -1 | 1 | 1 | 2 | 2 | 2 | -1 | -1 | -1 | 1 | 1 | 0 | 0 | 0 | 0 | -1 | -1 | -1 | -1 | orthogonal lifted from D6 |
| ρ10 | 2 | 2 | -2 | 0 | 2 | -2 | 2 | 0 | 0 | 0 | 2 | -2 | 0 | 0 | 2 | 2 | -2 | -2 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | -2 | 2 | -2 | 2 | orthogonal lifted from D4 |
| ρ11 | 2 | 2 | 2 | 0 | 2 | 2 | 2 | 0 | 0 | 0 | 2 | 2 | 0 | 0 | -2 | -2 | -2 | 2 | 2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | -2 | -2 | -2 | -2 | orthogonal lifted from D4 |
| ρ12 | 2 | 2 | 2 | 2 | -1 | 2 | 2 | 2 | 0 | 0 | -1 | -1 | -1 | -1 | 2 | 2 | 2 | -1 | -1 | -1 | -1 | -1 | 0 | 0 | 0 | 0 | -1 | -1 | -1 | -1 | orthogonal lifted from S3 |
| ρ13 | 2 | 2 | 2 | 0 | 2 | -2 | -2 | 0 | 0 | 0 | 2 | 2 | 0 | 0 | 0 | 0 | 0 | -2 | -2 | -2 | 0 | 0 | -√2 | -√2 | √2 | √2 | 0 | 0 | 0 | 0 | orthogonal lifted from D8 |
| ρ14 | 2 | 2 | 2 | 0 | 2 | -2 | -2 | 0 | 0 | 0 | 2 | 2 | 0 | 0 | 0 | 0 | 0 | -2 | -2 | -2 | 0 | 0 | √2 | √2 | -√2 | -√2 | 0 | 0 | 0 | 0 | orthogonal lifted from D8 |
| ρ15 | 2 | 2 | -2 | -2 | -1 | -2 | 2 | 2 | 0 | 0 | -1 | 1 | 1 | 1 | -2 | -2 | 2 | 1 | 1 | -1 | -1 | -1 | 0 | 0 | 0 | 0 | -1 | 1 | -1 | 1 | symplectic lifted from Dic3, Schur index 2 |
| ρ16 | 2 | 2 | -2 | 2 | -1 | -2 | 2 | -2 | 0 | 0 | -1 | 1 | -1 | -1 | -2 | -2 | 2 | 1 | 1 | -1 | 1 | 1 | 0 | 0 | 0 | 0 | -1 | 1 | -1 | 1 | symplectic lifted from Dic3, Schur index 2 |
| ρ17 | 2 | 2 | -2 | 0 | -1 | -2 | 2 | 0 | 0 | 0 | -1 | 1 | √-3 | -√-3 | 2 | 2 | -2 | 1 | 1 | -1 | -√-3 | √-3 | 0 | 0 | 0 | 0 | 1 | -1 | 1 | -1 | complex lifted from C3⋊D4 |
| ρ18 | 2 | 2 | -2 | 0 | -1 | -2 | 2 | 0 | 0 | 0 | -1 | 1 | -√-3 | √-3 | 2 | 2 | -2 | 1 | 1 | -1 | √-3 | -√-3 | 0 | 0 | 0 | 0 | 1 | -1 | 1 | -1 | complex lifted from C3⋊D4 |
| ρ19 | 2 | 2 | -2 | 0 | 2 | 2 | -2 | 0 | 0 | 0 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | 2 | 2 | -2 | 0 | 0 | -√-2 | √-2 | √-2 | -√-2 | 0 | 0 | 0 | 0 | complex lifted from SD16 |
| ρ20 | 2 | 2 | -2 | 0 | 2 | 2 | -2 | 0 | 0 | 0 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | 2 | 2 | -2 | 0 | 0 | √-2 | -√-2 | -√-2 | √-2 | 0 | 0 | 0 | 0 | complex lifted from SD16 |
| ρ21 | 2 | 2 | 2 | 0 | -1 | 2 | 2 | 0 | 0 | 0 | -1 | -1 | √-3 | -√-3 | -2 | -2 | -2 | -1 | -1 | -1 | √-3 | -√-3 | 0 | 0 | 0 | 0 | 1 | 1 | 1 | 1 | complex lifted from C3⋊D4 |
| ρ22 | 2 | 2 | 2 | 0 | -1 | 2 | 2 | 0 | 0 | 0 | -1 | -1 | -√-3 | √-3 | -2 | -2 | -2 | -1 | -1 | -1 | -√-3 | √-3 | 0 | 0 | 0 | 0 | 1 | 1 | 1 | 1 | complex lifted from C3⋊D4 |
| ρ23 | 4 | 4 | 4 | 0 | -2 | -4 | -4 | 0 | 0 | 0 | -2 | -2 | 0 | 0 | 0 | 0 | 0 | 2 | 2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4⋊S3, Schur index 2 |
| ρ24 | 4 | 4 | -4 | 0 | -2 | 4 | -4 | 0 | 0 | 0 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | -2 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | symplectic lifted from D4.S3, Schur index 2 |
| ρ25 | 4 | -4 | 0 | 0 | 4 | 0 | 0 | 0 | 0 | 0 | -4 | 0 | 0 | 0 | 2√-2 | -2√-2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -2√-2 | 0 | 2√-2 | complex lifted from D8⋊2C4 |
| ρ26 | 4 | -4 | 0 | 0 | 4 | 0 | 0 | 0 | 0 | 0 | -4 | 0 | 0 | 0 | -2√-2 | 2√-2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 2√-2 | 0 | -2√-2 | complex lifted from D8⋊2C4 |
| ρ27 | 4 | -4 | 0 | 0 | -2 | 0 | 0 | 0 | 0 | 0 | 2 | 0 | 0 | 0 | -2√-2 | 2√-2 | 0 | -2√3 | 2√3 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -√-6 | -√-2 | √-6 | √-2 | complex faithful |
| ρ28 | 4 | -4 | 0 | 0 | -2 | 0 | 0 | 0 | 0 | 0 | 2 | 0 | 0 | 0 | 2√-2 | -2√-2 | 0 | -2√3 | 2√3 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | √-6 | √-2 | -√-6 | -√-2 | complex faithful |
| ρ29 | 4 | -4 | 0 | 0 | -2 | 0 | 0 | 0 | 0 | 0 | 2 | 0 | 0 | 0 | 2√-2 | -2√-2 | 0 | 2√3 | -2√3 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -√-6 | √-2 | √-6 | -√-2 | complex faithful |
| ρ30 | 4 | -4 | 0 | 0 | -2 | 0 | 0 | 0 | 0 | 0 | 2 | 0 | 0 | 0 | -2√-2 | 2√-2 | 0 | 2√3 | -2√3 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | √-6 | -√-2 | -√-6 | √-2 | complex faithful |
►On 48 points
Generators in S48
(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 16)(6 15)(7 14)(8 13)(17 31)(18 30)(19 29)(20 28)(21 27)(22 26)(23 25)(24 32)(33 46)(34 45)(35 44)(36 43)(37 42)(38 41)(39 48)(40 47)
(1 26 34)(2 27 35)(3 28 36)(4 29 37)(5 30 38)(6 31 39)(7 32 40)(8 25 33)(9 23 42 13 19 46)(10 24 43 14 20 47)(11 17 44 15 21 48)(12 18 45 16 22 41)
(2 4)(3 7)(6 8)(9 10 13 14)(11 16 15 12)(17 45 21 41)(18 48 22 44)(19 43 23 47)(20 46 24 42)(25 39)(26 34)(27 37)(28 40)(29 35)(30 38)(31 33)(32 36)
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)(5,16)(6,15)(7,14)(8,13)(17,31)(18,30)(19,29)(20,28)(21,27)(22,26)(23,25)(24,32)(33,46)(34,45)(35,44)(36,43)(37,42)(38,41)(39,48)(40,47), (1,26,34)(2,27,35)(3,28,36)(4,29,37)(5,30,38)(6,31,39)(7,32,40)(8,25,33)(9,23,42,13,19,46)(10,24,43,14,20,47)(11,17,44,15,21,48)(12,18,45,16,22,41), (2,4)(3,7)(6,8)(9,10,13,14)(11,16,15,12)(17,45,21,41)(18,48,22,44)(19,43,23,47)(20,46,24,42)(25,39)(26,34)(27,37)(28,40)(29,35)(30,38)(31,33)(32,36)>;
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)(5,16)(6,15)(7,14)(8,13)(17,31)(18,30)(19,29)(20,28)(21,27)(22,26)(23,25)(24,32)(33,46)(34,45)(35,44)(36,43)(37,42)(38,41)(39,48)(40,47), (1,26,34)(2,27,35)(3,28,36)(4,29,37)(5,30,38)(6,31,39)(7,32,40)(8,25,33)(9,23,42,13,19,46)(10,24,43,14,20,47)(11,17,44,15,21,48)(12,18,45,16,22,41), (2,4)(3,7)(6,8)(9,10,13,14)(11,16,15,12)(17,45,21,41)(18,48,22,44)(19,43,23,47)(20,46,24,42)(25,39)(26,34)(27,37)(28,40)(29,35)(30,38)(31,33)(32,36) );
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),(5,16),(6,15),(7,14),(8,13),(17,31),(18,30),(19,29),(20,28),(21,27),(22,26),(23,25),(24,32),(33,46),(34,45),(35,44),(36,43),(37,42),(38,41),(39,48),(40,47)], [(1,26,34),(2,27,35),(3,28,36),(4,29,37),(5,30,38),(6,31,39),(7,32,40),(8,25,33),(9,23,42,13,19,46),(10,24,43,14,20,47),(11,17,44,15,21,48),(12,18,45,16,22,41)], [(2,4),(3,7),(6,8),(9,10,13,14),(11,16,15,12),(17,45,21,41),(18,48,22,44),(19,43,23,47),(20,46,24,42),(25,39),(26,34),(27,37),(28,40),(29,35),(30,38),(31,33),(32,36)]])
Matrix representation of D8⋊2Dic3 ►in GL4(𝔽97) generated by
| 61 | 89 | 0 | 0 |
| 8 | 53 | 0 | 0 |
| 73 | 51 | 36 | 89 |
| 33 | 94 | 8 | 44 |
| 57 | 43 | 69 | 52 |
| 83 | 49 | 17 | 69 |
| 89 | 78 | 32 | 56 |
| 53 | 70 | 32 | 56 |
| 96 | 1 | 0 | 0 |
| 96 | 0 | 0 | 0 |
| 22 | 12 | 0 | 96 |
| 17 | 78 | 1 | 1 |
| 0 | 1 | 0 | 0 |
| 1 | 0 | 0 | 0 |
| 53 | 14 | 44 | 8 |
| 79 | 20 | 61 | 53 |
G:=sub<GL(4,GF(97))| [61,8,73,33,89,53,51,94,0,0,36,8,0,0,89,44],[57,83,89,53,43,49,78,70,69,17,32,32,52,69,56,56],[96,96,22,17,1,0,12,78,0,0,0,1,0,0,96,1],[0,1,53,79,1,0,14,20,0,0,44,61,0,0,8,53] >;
D8⋊2Dic3 in GAP, Magma, Sage, TeX
D_8\rtimes_2{\rm Dic}_3 % in TeX
G:=Group("D8:2Dic3"); // GroupNames label
G:=SmallGroup(192,125);
// by ID
G=gap.SmallGroup(192,125);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,28,141,387,675,794,80,1684,851,102,6278]);
// Polycyclic
G:=Group<a,b,c,d|a^8=b^2=c^6=1,d^2=c^3,b*a*b=a^-1,a*c=c*a,d*a*d^-1=a^3,c*b*c^-1=a^4*b,d*b*d^-1=a^5*b,d*c*d^-1=c^-1>;
// generators/relations
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