metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D8:2Dic3, C24.40D4, Q16:2Dic3, C12.37SD16, (C3xD8):4C4, (C2xC6).4D8, (C3xQ16):4C4, C4oD8.2S3, (C2xC8).51D6, C3:3(D8:2C4), C24.27(C2xC4), C8:Dic3:23C2, C12.C8:8C2, C8.3(C2xDic3), (C2xC12).118D4, C8.30(C3:D4), C4.12(D4.S3), C22.3(D4:S3), C6.30(D4:C4), C12.17(C22:C4), (C2xC24).154C22, C4.5(C6.D4), C2.10(D4:Dic3), (C3xC4oD8).5C2, (C2xC4).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, C2xC4, C2xC4, D4, Q8, Dic3, C12, C12, C2xC6, C2xC6, C16, C4:C4, C2xC8, D8, SD16, Q16, C4oD4, C24, C2xDic3, C2xC12, C2xC12, C3xD4, C3xQ8, C4.Q8, M5(2), C4oD8, C3:C16, C4:Dic3, C2xC24, C3xD8, C3xSD16, C3xQ16, C3xC4oD4, D8:2C4, C12.C8, C8:Dic3, C3xC4oD8, D8:2Dic3
Quotients: C1, C2, C4, C22, S3, C2xC4, D4, Dic3, D6, C22:C4, D8, SD16, C2xDic3, 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 |
(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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