metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C12.19D4, Q8.11D6, C12.15C23, D12.10C22, Dic6.9C22, (C6xQ8):2C2, (C2xQ8):4S3, C3:Q16:5C2, C6.54(C2xD4), (C2xC4).20D6, (C2xC6).42D4, C3:C8.3C22, Q8:2S3:5C2, C4oD12.5C2, C3:4(C8.C22), C4.Dic3:7C2, C4.17(C3:D4), C4.15(C22xS3), (C3xQ8).6C22, (C2xC12).37C22, C22.11(C3:D4), C2.18(C2xC3:D4), SmallGroup(96,149)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for Q8.11D6
G = < a,b,c,d | a4=1, b2=c6=d2=a2, bab-1=dad-1=a-1, ac=ca, cbc-1=a2b, dbd-1=ab, dcd-1=c5 >
Subgroups: 130 in 60 conjugacy classes, 29 normal (19 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, S3, C6, C6, C8, C2xC4, C2xC4, D4, Q8, Q8, Dic3, C12, C12, D6, C2xC6, M4(2), SD16, Q16, C2xQ8, C4oD4, C3:C8, Dic6, C4xS3, D12, C3:D4, C2xC12, C2xC12, C3xQ8, C3xQ8, C8.C22, C4.Dic3, Q8:2S3, C3:Q16, C4oD12, C6xQ8, Q8.11D6
Quotients: C1, C2, C22, S3, D4, C23, D6, C2xD4, C3:D4, C22xS3, C8.C22, C2xC3:D4, Q8.11D6
Character table of Q8.11D6
| class | 1 | 2A | 2B | 2C | 3 | 4A | 4B | 4C | 4D | 4E | 6A | 6B | 6C | 8A | 8B | 12A | 12B | 12C | 12D | 12E | 12F | |
| size | 1 | 1 | 2 | 12 | 2 | 2 | 2 | 4 | 4 | 12 | 2 | 2 | 2 | 12 | 12 | 4 | 4 | 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 | trivial |
| ρ2 | 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 | 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 | 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 | 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 | 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 | 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 | linear of order 2 |
| ρ9 | 2 | 2 | 2 | 0 | -1 | 2 | 2 | 2 | 2 | 0 | -1 | -1 | -1 | 0 | 0 | -1 | -1 | -1 | -1 | -1 | -1 | orthogonal lifted from S3 |
| ρ10 | 2 | 2 | -2 | 0 | -1 | -2 | 2 | 2 | -2 | 0 | 1 | -1 | 1 | 0 | 0 | 1 | 1 | -1 | -1 | -1 | 1 | orthogonal lifted from D6 |
| ρ11 | 2 | 2 | 2 | 0 | 2 | -2 | -2 | 0 | 0 | 0 | 2 | 2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | -2 | -2 | orthogonal lifted from D4 |
| ρ12 | 2 | 2 | -2 | 0 | 2 | 2 | -2 | 0 | 0 | 0 | -2 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | -2 | 2 | orthogonal lifted from D4 |
| ρ13 | 2 | 2 | -2 | 0 | -1 | -2 | 2 | -2 | 2 | 0 | 1 | -1 | 1 | 0 | 0 | -1 | -1 | 1 | 1 | -1 | 1 | orthogonal lifted from D6 |
| ρ14 | 2 | 2 | 2 | 0 | -1 | 2 | 2 | -2 | -2 | 0 | -1 | -1 | -1 | 0 | 0 | 1 | 1 | 1 | 1 | -1 | -1 | orthogonal lifted from D6 |
| ρ15 | 2 | 2 | 2 | 0 | -1 | -2 | -2 | 0 | 0 | 0 | -1 | -1 | -1 | 0 | 0 | √-3 | -√-3 | -√-3 | √-3 | 1 | 1 | complex lifted from C3:D4 |
| ρ16 | 2 | 2 | -2 | 0 | -1 | 2 | -2 | 0 | 0 | 0 | 1 | -1 | 1 | 0 | 0 | √-3 | -√-3 | √-3 | -√-3 | 1 | -1 | complex lifted from C3:D4 |
| ρ17 | 2 | 2 | -2 | 0 | -1 | 2 | -2 | 0 | 0 | 0 | 1 | -1 | 1 | 0 | 0 | -√-3 | √-3 | -√-3 | √-3 | 1 | -1 | complex lifted from C3:D4 |
| ρ18 | 2 | 2 | 2 | 0 | -1 | -2 | -2 | 0 | 0 | 0 | -1 | -1 | -1 | 0 | 0 | -√-3 | √-3 | √-3 | -√-3 | 1 | 1 | complex lifted from C3:D4 |
| ρ19 | 4 | -4 | 0 | 0 | 4 | 0 | 0 | 0 | 0 | 0 | 0 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | symplectic lifted from C8.C22, Schur index 2 |
| ρ20 | 4 | -4 | 0 | 0 | -2 | 0 | 0 | 0 | 0 | 0 | -2√-3 | 2 | 2√-3 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | complex faithful |
| ρ21 | 4 | -4 | 0 | 0 | -2 | 0 | 0 | 0 | 0 | 0 | 2√-3 | 2 | -2√-3 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | complex faithful |
►On 48 points
(1 38 7 44)(2 39 8 45)(3 40 9 46)(4 41 10 47)(5 42 11 48)(6 43 12 37)(13 32 19 26)(14 33 20 27)(15 34 21 28)(16 35 22 29)(17 36 23 30)(18 25 24 31)
(1 32 7 26)(2 27 8 33)(3 34 9 28)(4 29 10 35)(5 36 11 30)(6 31 12 25)(13 44 19 38)(14 39 20 45)(15 46 21 40)(16 41 22 47)(17 48 23 42)(18 43 24 37)
(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 6 7 12)(2 11 8 5)(3 4 9 10)(13 31 19 25)(14 36 20 30)(15 29 21 35)(16 34 22 28)(17 27 23 33)(18 32 24 26)(37 44 43 38)(39 42 45 48)(40 47 46 41)
G:=sub<Sym(48)| (1,38,7,44)(2,39,8,45)(3,40,9,46)(4,41,10,47)(5,42,11,48)(6,43,12,37)(13,32,19,26)(14,33,20,27)(15,34,21,28)(16,35,22,29)(17,36,23,30)(18,25,24,31), (1,32,7,26)(2,27,8,33)(3,34,9,28)(4,29,10,35)(5,36,11,30)(6,31,12,25)(13,44,19,38)(14,39,20,45)(15,46,21,40)(16,41,22,47)(17,48,23,42)(18,43,24,37), (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), (1,6,7,12)(2,11,8,5)(3,4,9,10)(13,31,19,25)(14,36,20,30)(15,29,21,35)(16,34,22,28)(17,27,23,33)(18,32,24,26)(37,44,43,38)(39,42,45,48)(40,47,46,41)>;
G:=Group( (1,38,7,44)(2,39,8,45)(3,40,9,46)(4,41,10,47)(5,42,11,48)(6,43,12,37)(13,32,19,26)(14,33,20,27)(15,34,21,28)(16,35,22,29)(17,36,23,30)(18,25,24,31), (1,32,7,26)(2,27,8,33)(3,34,9,28)(4,29,10,35)(5,36,11,30)(6,31,12,25)(13,44,19,38)(14,39,20,45)(15,46,21,40)(16,41,22,47)(17,48,23,42)(18,43,24,37), (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), (1,6,7,12)(2,11,8,5)(3,4,9,10)(13,31,19,25)(14,36,20,30)(15,29,21,35)(16,34,22,28)(17,27,23,33)(18,32,24,26)(37,44,43,38)(39,42,45,48)(40,47,46,41) );
G=PermutationGroup([[(1,38,7,44),(2,39,8,45),(3,40,9,46),(4,41,10,47),(5,42,11,48),(6,43,12,37),(13,32,19,26),(14,33,20,27),(15,34,21,28),(16,35,22,29),(17,36,23,30),(18,25,24,31)], [(1,32,7,26),(2,27,8,33),(3,34,9,28),(4,29,10,35),(5,36,11,30),(6,31,12,25),(13,44,19,38),(14,39,20,45),(15,46,21,40),(16,41,22,47),(17,48,23,42),(18,43,24,37)], [(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)], [(1,6,7,12),(2,11,8,5),(3,4,9,10),(13,31,19,25),(14,36,20,30),(15,29,21,35),(16,34,22,28),(17,27,23,33),(18,32,24,26),(37,44,43,38),(39,42,45,48),(40,47,46,41)]])
Q8.11D6 is a maximal subgroup of
D12.6D4 D12.7D4 C42:7D6 D12.15D4 C24.44D4 C24.29D4 D12.39D4 D12.40D4 SD16:13D6 D12.30D4 S3xC8.C22 D24:C22 C12.C24 D12.34C23 D12.35C23 C36.C23 Q8.D18 D12.32D6 Dic6.29D6 D12.24D6 Dic6.22D6 C62.134D4 SL2(F3).D6 D12.37D10 C12.D20 D12.27D10 C60.39C23 Q8.11D30
Q8.11D6 is a maximal quotient of
C4:C4.225D6 C4oD12:C4 (C2xC6).40D8 C4:C4.231D6 Q8.5Dic6 C42.56D6 Q8.6D12 C42.59D6 (C2xQ8).51D6 D12.37D4 C3:C8:6D4 C3:C8.6D4 C42.76D6 C42.77D6 C12:5SD16 C42.80D6 D12:6Q8 C42.82D6 C12:Q16 Dic6:6Q8 (C6xQ8):6C4 (C3xQ8):13D4 (C2xC6):8Q16 C36.C23 D12.32D6 Dic6.29D6 D12.24D6 Dic6.22D6 C62.134D4 D12.37D10 C12.D20 D12.27D10 C60.39C23 Q8.11D30
Matrix representation of Q8.11D6 ►in GL4(F7) generated by
| 6 | 6 | 1 | 1 |
| 2 | 0 | 4 | 1 |
| 3 | 3 | 0 | 1 |
| 4 | 3 | 5 | 1 |
| 1 | 6 | 6 | 6 |
| 6 | 0 | 3 | 6 |
| 1 | 1 | 4 | 5 |
| 2 | 5 | 6 | 2 |
| 0 | 1 | 1 | 6 |
| 2 | 5 | 1 | 4 |
| 1 | 4 | 6 | 1 |
| 2 | 2 | 2 | 3 |
| 0 | 1 | 1 | 6 |
| 5 | 5 | 0 | 2 |
| 6 | 4 | 1 | 6 |
| 5 | 2 | 1 | 1 |
G:=sub<GL(4,GF(7))| [6,2,3,4,6,0,3,3,1,4,0,5,1,1,1,1],[1,6,1,2,6,0,1,5,6,3,4,6,6,6,5,2],[0,2,1,2,1,5,4,2,1,1,6,2,6,4,1,3],[0,5,6,5,1,5,4,2,1,0,1,1,6,2,6,1] >;
Q8.11D6 in GAP, Magma, Sage, TeX
Q_8._{11}D_6 % in TeX
G:=Group("Q8.11D6"); // GroupNames label
G:=SmallGroup(96,149);
// by ID
G=gap.SmallGroup(96,149);
# by ID
G:=PCGroup([6,-2,-2,-2,-2,-2,-3,103,218,188,86,579,159,69,2309]);
// Polycyclic
G:=Group<a,b,c,d|a^4=1,b^2=c^6=d^2=a^2,b*a*b^-1=d*a*d^-1=a^-1,a*c=c*a,c*b*c^-1=a^2*b,d*b*d^-1=a*b,d*c*d^-1=c^5>;
// generators/relations
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