metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D4.8D6, C12.57D4, Q8.13D6, C12.18C23, D12.12C22, Dic6.11C22, C4○(D4⋊S3), D4⋊S3⋊7C2, C4○D4⋊4S3, C3⋊5(C4○D8), (C2×C6).9D4, C4○D12⋊4C2, C4○(D4.S3), D4.S3⋊7C2, C4○(C3⋊Q16), C3⋊Q16⋊7C2, (C2×C4).59D6, C6.60(C2×D4), C4○(Q8⋊2S3), Q8⋊2S3⋊7C2, C3⋊C8.10C22, C4.32(C3⋊D4), C4.18(C22×S3), (C3×D4).8C22, (C3×Q8).8C22, (C2×C12).43C22, C22.1(C3⋊D4), (C2×C3⋊C8)⋊8C2, (C3×C4○D4)⋊2C2, C2.24(C2×C3⋊D4), SmallGroup(96,157)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for Q8.13D6
G = < a,b,c,d | a4=1, b2=c6=d2=a2, bab-1=dad-1=a-1, ac=ca, bc=cb, dbd-1=a-1b, dcd-1=c5 >
Subgroups: 138 in 62 conjugacy classes, 29 normal (all characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, S3, C6, C6, C8, C2×C4, C2×C4, D4, D4, Q8, Q8, Dic3, C12, C12, D6, C2×C6, C2×C6, C2×C8, D8, SD16, Q16, C4○D4, C4○D4, C3⋊C8, Dic6, C4×S3, D12, C3⋊D4, C2×C12, C2×C12, C3×D4, C3×D4, C3×Q8, C4○D8, C2×C3⋊C8, D4⋊S3, D4.S3, Q8⋊2S3, C3⋊Q16, C4○D12, C3×C4○D4, Q8.13D6
Quotients: C1, C2, C22, S3, D4, C23, D6, C2×D4, C3⋊D4, C22×S3, C4○D8, C2×C3⋊D4, Q8.13D6
Character table of Q8.13D6
| class | 1 | 2A | 2B | 2C | 2D | 3 | 4A | 4B | 4C | 4D | 4E | 6A | 6B | 6C | 6D | 8A | 8B | 8C | 8D | 12A | 12B | 12C | 12D | 12E | |
| size | 1 | 1 | 2 | 4 | 12 | 2 | 1 | 1 | 2 | 4 | 12 | 2 | 4 | 4 | 4 | 6 | 6 | 6 | 6 | 2 | 2 | 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 | 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 | 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 | 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 | 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 | 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 | 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 | 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 | linear of order 2 |
| ρ9 | 2 | 2 | 2 | 0 | 0 | 2 | -2 | -2 | -2 | 0 | 0 | 2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | -2 | -2 | 0 | 0 | -2 | orthogonal lifted from D4 |
| ρ10 | 2 | 2 | 2 | -2 | 0 | -1 | 2 | 2 | 2 | -2 | 0 | -1 | -1 | 1 | 1 | 0 | 0 | 0 | 0 | -1 | -1 | 1 | 1 | -1 | orthogonal lifted from D6 |
| ρ11 | 2 | 2 | 2 | 2 | 0 | -1 | 2 | 2 | 2 | 2 | 0 | -1 | -1 | -1 | -1 | 0 | 0 | 0 | 0 | -1 | -1 | -1 | -1 | -1 | orthogonal lifted from S3 |
| ρ12 | 2 | 2 | -2 | 0 | 0 | 2 | 2 | 2 | -2 | 0 | 0 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 2 | 2 | 0 | 0 | -2 | orthogonal lifted from D4 |
| ρ13 | 2 | 2 | -2 | 2 | 0 | -1 | -2 | -2 | 2 | -2 | 0 | -1 | 1 | -1 | -1 | 0 | 0 | 0 | 0 | 1 | 1 | 1 | 1 | -1 | orthogonal lifted from D6 |
| ρ14 | 2 | 2 | -2 | -2 | 0 | -1 | -2 | -2 | 2 | 2 | 0 | -1 | 1 | 1 | 1 | 0 | 0 | 0 | 0 | 1 | 1 | -1 | -1 | -1 | orthogonal lifted from D6 |
| ρ15 | 2 | 2 | 2 | 0 | 0 | -1 | -2 | -2 | -2 | 0 | 0 | -1 | -1 | √-3 | -√-3 | 0 | 0 | 0 | 0 | 1 | 1 | -√-3 | √-3 | 1 | complex lifted from C3⋊D4 |
| ρ16 | 2 | 2 | -2 | 0 | 0 | -1 | 2 | 2 | -2 | 0 | 0 | -1 | 1 | -√-3 | √-3 | 0 | 0 | 0 | 0 | -1 | -1 | -√-3 | √-3 | 1 | complex lifted from C3⋊D4 |
| ρ17 | 2 | 2 | -2 | 0 | 0 | -1 | 2 | 2 | -2 | 0 | 0 | -1 | 1 | √-3 | -√-3 | 0 | 0 | 0 | 0 | -1 | -1 | √-3 | -√-3 | 1 | complex lifted from C3⋊D4 |
| ρ18 | 2 | 2 | 2 | 0 | 0 | -1 | -2 | -2 | -2 | 0 | 0 | -1 | -1 | -√-3 | √-3 | 0 | 0 | 0 | 0 | 1 | 1 | √-3 | -√-3 | 1 | complex lifted from C3⋊D4 |
| ρ19 | 2 | -2 | 0 | 0 | 0 | 2 | 2i | -2i | 0 | 0 | 0 | -2 | 0 | 0 | 0 | -√-2 | -√2 | √-2 | √2 | 2i | -2i | 0 | 0 | 0 | complex lifted from C4○D8 |
| ρ20 | 2 | -2 | 0 | 0 | 0 | 2 | 2i | -2i | 0 | 0 | 0 | -2 | 0 | 0 | 0 | √-2 | √2 | -√-2 | -√2 | 2i | -2i | 0 | 0 | 0 | complex lifted from C4○D8 |
| ρ21 | 2 | -2 | 0 | 0 | 0 | 2 | -2i | 2i | 0 | 0 | 0 | -2 | 0 | 0 | 0 | -√-2 | √2 | √-2 | -√2 | -2i | 2i | 0 | 0 | 0 | complex lifted from C4○D8 |
| ρ22 | 2 | -2 | 0 | 0 | 0 | 2 | -2i | 2i | 0 | 0 | 0 | -2 | 0 | 0 | 0 | √-2 | -√2 | -√-2 | √2 | -2i | 2i | 0 | 0 | 0 | complex lifted from C4○D8 |
| ρ23 | 4 | -4 | 0 | 0 | 0 | -2 | -4i | 4i | 0 | 0 | 0 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 2i | -2i | 0 | 0 | 0 | complex faithful |
| ρ24 | 4 | -4 | 0 | 0 | 0 | -2 | 4i | -4i | 0 | 0 | 0 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -2i | 2i | 0 | 0 | 0 | complex faithful |
►On 48 points
(1 25 7 31)(2 26 8 32)(3 27 9 33)(4 28 10 34)(5 29 11 35)(6 30 12 36)(13 48 19 42)(14 37 20 43)(15 38 21 44)(16 39 22 45)(17 40 23 46)(18 41 24 47)
(1 37 7 43)(2 38 8 44)(3 39 9 45)(4 40 10 46)(5 41 11 47)(6 42 12 48)(13 30 19 36)(14 31 20 25)(15 32 21 26)(16 33 22 27)(17 34 23 28)(18 35 24 29)
(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 43 19 37)(14 48 20 42)(15 41 21 47)(16 46 22 40)(17 39 23 45)(18 44 24 38)(25 36 31 30)(26 29 32 35)(27 34 33 28)
G:=sub<Sym(48)| (1,25,7,31)(2,26,8,32)(3,27,9,33)(4,28,10,34)(5,29,11,35)(6,30,12,36)(13,48,19,42)(14,37,20,43)(15,38,21,44)(16,39,22,45)(17,40,23,46)(18,41,24,47), (1,37,7,43)(2,38,8,44)(3,39,9,45)(4,40,10,46)(5,41,11,47)(6,42,12,48)(13,30,19,36)(14,31,20,25)(15,32,21,26)(16,33,22,27)(17,34,23,28)(18,35,24,29), (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)(13,43,19,37)(14,48,20,42)(15,41,21,47)(16,46,22,40)(17,39,23,45)(18,44,24,38)(25,36,31,30)(26,29,32,35)(27,34,33,28)>;
G:=Group( (1,25,7,31)(2,26,8,32)(3,27,9,33)(4,28,10,34)(5,29,11,35)(6,30,12,36)(13,48,19,42)(14,37,20,43)(15,38,21,44)(16,39,22,45)(17,40,23,46)(18,41,24,47), (1,37,7,43)(2,38,8,44)(3,39,9,45)(4,40,10,46)(5,41,11,47)(6,42,12,48)(13,30,19,36)(14,31,20,25)(15,32,21,26)(16,33,22,27)(17,34,23,28)(18,35,24,29), (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)(13,43,19,37)(14,48,20,42)(15,41,21,47)(16,46,22,40)(17,39,23,45)(18,44,24,38)(25,36,31,30)(26,29,32,35)(27,34,33,28) );
G=PermutationGroup([[(1,25,7,31),(2,26,8,32),(3,27,9,33),(4,28,10,34),(5,29,11,35),(6,30,12,36),(13,48,19,42),(14,37,20,43),(15,38,21,44),(16,39,22,45),(17,40,23,46),(18,41,24,47)], [(1,37,7,43),(2,38,8,44),(3,39,9,45),(4,40,10,46),(5,41,11,47),(6,42,12,48),(13,30,19,36),(14,31,20,25),(15,32,21,26),(16,33,22,27),(17,34,23,28),(18,35,24,29)], [(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),(13,43,19,37),(14,48,20,42),(15,41,21,47),(16,46,22,40),(17,39,23,45),(18,44,24,38),(25,36,31,30),(26,29,32,35),(27,34,33,28)]])
Q8.13D6 is a maximal subgroup of
M4(2).22D6 C42.196D6 C24.100D4 C24.54D4 S3×C4○D8 SD16⋊D6 D8⋊5D6 D8⋊6D6 C24.C23 SD16.D6 C12.C24 D12.32C23 D12.33C23 D12.34C23 D12.35C23 D4.9D18 C12.11S4 D12.30D6 D12.27D6 D12.22D6 Dic6.20D6 D12.12D6 D12.13D6 C62.74D4 C12.14S4 D20.34D6 D20.31D6 D12.24D10 C60.16C23 D20.14D6 D20.D6 D4.8D30
Q8.13D6 is a maximal quotient of
C4⋊C4.233D6 C4⋊C4.234D6 C4.(C2×D12) C4⋊C4.236D6 D4.3Dic6 C4×D4⋊S3 D4.1D12 C4×D4.S3 Q8.5Dic6 C4×Q8⋊2S3 Q8.6D12 C4×C3⋊Q16 C6.Q16⋊C2 D12⋊17D4 C3⋊C8⋊22D4 C3⋊C8⋊23D4 (C2×Q8).51D6 D12.37D4 C3⋊C8⋊24D4 C3⋊C8.29D4 C42.61D6 C42.213D6 D12.23D4 C42.214D6 Dic6.4Q8 C42.215D6 D12.4Q8 C42.216D6 C4○D4⋊4Dic3 (C3×D4)⋊14D4 (C3×D4).32D4 D4.9D18 D12.30D6 D12.27D6 D12.22D6 Dic6.20D6 D12.12D6 D12.13D6 C62.74D4 D20.34D6 D20.31D6 D12.24D10 C60.16C23 D20.14D6 D20.D6 D4.8D30
Matrix representation of Q8.13D6 ►in GL4(𝔽5) generated by
| 4 | 0 | 0 | 1 |
| 0 | 0 | 4 | 1 |
| 3 | 1 | 0 | 1 |
| 3 | 0 | 0 | 1 |
| 4 | 0 | 1 | 0 |
| 3 | 0 | 1 | 1 |
| 3 | 0 | 1 | 0 |
| 0 | 4 | 1 | 0 |
| 0 | 1 | 4 | 1 |
| 1 | 2 | 0 | 2 |
| 0 | 0 | 0 | 2 |
| 0 | 0 | 3 | 2 |
| 2 | 0 | 0 | 3 |
| 4 | 3 | 0 | 3 |
| 0 | 0 | 2 | 3 |
| 0 | 0 | 0 | 3 |
G:=sub<GL(4,GF(5))| [4,0,3,3,0,0,1,0,0,4,0,0,1,1,1,1],[4,3,3,0,0,0,0,4,1,1,1,1,0,1,0,0],[0,1,0,0,1,2,0,0,4,0,0,3,1,2,2,2],[2,4,0,0,0,3,0,0,0,0,2,0,3,3,3,3] >;
Q8.13D6 in GAP, Magma, Sage, TeX
Q_8._{13}D_6 % in TeX
G:=Group("Q8.13D6"); // GroupNames label
G:=SmallGroup(96,157);
// by ID
G=gap.SmallGroup(96,157);
# by ID
G:=PCGroup([6,-2,-2,-2,-2,-2,-3,103,218,579,159,69,2309]);
// 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,b*c=c*b,d*b*d^-1=a^-1*b,d*c*d^-1=c^5>;
// generators/relations
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