Aliases: Q8⋊Dic6, C4.GL2(𝔽3), SL2(𝔽3)⋊1Q8, (C2×C4).8S4, (C4×Q8).2S3, (C2×Q8).2D6, C2.3(A4⋊Q8), Q8⋊Dic3.3C2, C22.32(C2×S4), C2.3(C4.S4), C2.3(C2×GL2(𝔽3)), (C4×SL2(𝔽3)).4C2, (C2×SL2(𝔽3)).2C22, SmallGroup(192,945)
Series: Derived ►Chief ►Lower central ►Upper central
| C1 — C2 — Q8 — C2×SL2(𝔽3) — Q8⋊Dic6 |
| SL2(𝔽3) — C2×SL2(𝔽3) — Q8⋊Dic6 |
Generators and relations for Q8⋊Dic6
G = < a,b,c,d | a4=c12=1, b2=a2, d2=c6, bab-1=dbd-1=a-1, cac-1=b, dad-1=a2b, cbc-1=ab, dcd-1=c-1 >
Subgroups: 237 in 65 conjugacy classes, 17 normal (13 characteristic)
C1, C2, C3, C4, C4, C22, C6, C8, C2×C4, C2×C4, Q8, Q8, Dic3, C12, C2×C6, C42, C4⋊C4, C2×C8, C2×Q8, C2×Q8, SL2(𝔽3), Dic6, C2×Dic3, C2×C12, Q8⋊C4, C4⋊C8, C4.Q8, C4×Q8, C4⋊Q8, C2×SL2(𝔽3), C2×Dic6, Q8⋊Q8, Q8⋊Dic3, C4×SL2(𝔽3), Q8⋊Dic6
Quotients: C1, C2, C22, S3, Q8, D6, Dic6, S4, GL2(𝔽3), C2×S4, A4⋊Q8, C2×GL2(𝔽3), C4.S4, Q8⋊Dic6
Character table of Q8⋊Dic6
| class | 1 | 2A | 2B | 2C | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 6A | 6B | 6C | 8A | 8B | 8C | 8D | 12A | 12B | 12C | 12D | |
| size | 1 | 1 | 1 | 1 | 8 | 2 | 2 | 6 | 6 | 12 | 24 | 24 | 8 | 8 | 8 | 12 | 12 | 12 | 12 | 8 | 8 | 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 | 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 | 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 | 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 | linear of order 2 |
| ρ5 | 2 | 2 | 2 | 2 | -1 | -2 | -2 | 2 | 2 | -2 | 0 | 0 | -1 | -1 | -1 | 0 | 0 | 0 | 0 | 1 | 1 | 1 | 1 | orthogonal lifted from D6 |
| ρ6 | 2 | 2 | 2 | 2 | -1 | 2 | 2 | 2 | 2 | 2 | 0 | 0 | -1 | -1 | -1 | 0 | 0 | 0 | 0 | -1 | -1 | -1 | -1 | orthogonal lifted from S3 |
| ρ7 | 2 | 2 | -2 | -2 | 2 | 0 | 0 | -2 | 2 | 0 | 0 | 0 | -2 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | symplectic lifted from Q8, Schur index 2 |
| ρ8 | 2 | 2 | -2 | -2 | -1 | 0 | 0 | -2 | 2 | 0 | 0 | 0 | 1 | 1 | -1 | 0 | 0 | 0 | 0 | √3 | √3 | -√3 | -√3 | symplectic lifted from Dic6, Schur index 2 |
| ρ9 | 2 | 2 | -2 | -2 | -1 | 0 | 0 | -2 | 2 | 0 | 0 | 0 | 1 | 1 | -1 | 0 | 0 | 0 | 0 | -√3 | -√3 | √3 | √3 | symplectic lifted from Dic6, Schur index 2 |
| ρ10 | 2 | -2 | 2 | -2 | -1 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | 1 | -1 | 1 | √-2 | -√-2 | -√-2 | √-2 | -1 | 1 | -1 | 1 | complex lifted from GL2(𝔽3) |
| ρ11 | 2 | -2 | 2 | -2 | -1 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | 1 | -1 | 1 | √-2 | √-2 | -√-2 | -√-2 | 1 | -1 | 1 | -1 | complex lifted from GL2(𝔽3) |
| ρ12 | 2 | -2 | 2 | -2 | -1 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | 1 | -1 | 1 | -√-2 | √-2 | √-2 | -√-2 | -1 | 1 | -1 | 1 | complex lifted from GL2(𝔽3) |
| ρ13 | 2 | -2 | 2 | -2 | -1 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | 1 | -1 | 1 | -√-2 | -√-2 | √-2 | √-2 | 1 | -1 | 1 | -1 | complex lifted from GL2(𝔽3) |
| ρ14 | 3 | 3 | 3 | 3 | 0 | 3 | 3 | -1 | -1 | -1 | -1 | -1 | 0 | 0 | 0 | 1 | 1 | 1 | 1 | 0 | 0 | 0 | 0 | orthogonal lifted from S4 |
| ρ15 | 3 | 3 | 3 | 3 | 0 | -3 | -3 | -1 | -1 | 1 | -1 | 1 | 0 | 0 | 0 | 1 | -1 | 1 | -1 | 0 | 0 | 0 | 0 | orthogonal lifted from C2×S4 |
| ρ16 | 3 | 3 | 3 | 3 | 0 | -3 | -3 | -1 | -1 | 1 | 1 | -1 | 0 | 0 | 0 | -1 | 1 | -1 | 1 | 0 | 0 | 0 | 0 | orthogonal lifted from C2×S4 |
| ρ17 | 3 | 3 | 3 | 3 | 0 | 3 | 3 | -1 | -1 | -1 | 1 | 1 | 0 | 0 | 0 | -1 | -1 | -1 | -1 | 0 | 0 | 0 | 0 | orthogonal lifted from S4 |
| ρ18 | 4 | -4 | 4 | -4 | 1 | 4 | -4 | 0 | 0 | 0 | 0 | 0 | -1 | 1 | -1 | 0 | 0 | 0 | 0 | -1 | 1 | -1 | 1 | orthogonal lifted from GL2(𝔽3) |
| ρ19 | 4 | -4 | 4 | -4 | 1 | -4 | 4 | 0 | 0 | 0 | 0 | 0 | -1 | 1 | -1 | 0 | 0 | 0 | 0 | 1 | -1 | 1 | -1 | orthogonal lifted from GL2(𝔽3) |
| ρ20 | 4 | -4 | -4 | 4 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -2 | 2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | symplectic lifted from C4.S4, Schur index 2 |
| ρ21 | 4 | -4 | -4 | 4 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | -1 | -1 | 0 | 0 | 0 | 0 | √3 | -√3 | -√3 | √3 | symplectic lifted from C4.S4, Schur index 2 |
| ρ22 | 4 | -4 | -4 | 4 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | -1 | -1 | 0 | 0 | 0 | 0 | -√3 | √3 | √3 | -√3 | symplectic lifted from C4.S4, Schur index 2 |
| ρ23 | 6 | 6 | -6 | -6 | 0 | 0 | 0 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | symplectic lifted from A4⋊Q8, Schur index 2 |
►On 64 points
Generators in S64
(1 23 7 37)(2 20 8 34)(3 17 5 31)(4 26 6 40)(9 55 16 48)(10 64 13 45)(11 61 14 42)(12 58 15 51)(18 22 32 36)(19 29 33 27)(21 25 35 39)(24 28 38 30)(41 56 60 49)(43 47 62 54)(44 59 63 52)(46 50 53 57)
(1 19 7 33)(2 28 8 30)(3 25 5 39)(4 22 6 36)(9 63 16 44)(10 60 13 41)(11 57 14 50)(12 54 15 47)(17 21 31 35)(18 40 32 26)(20 24 34 38)(23 27 37 29)(42 46 61 53)(43 58 62 51)(45 49 64 56)(48 52 55 59)
(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 49 50 51 52)(53 54 55 56 57 58 59 60 61 62 63 64)
(1 16 3 14)(2 15 4 13)(5 11 7 9)(6 10 8 12)(17 57 23 63)(18 56 24 62)(19 55 25 61)(20 54 26 60)(21 53 27 59)(22 64 28 58)(29 52 35 46)(30 51 36 45)(31 50 37 44)(32 49 38 43)(33 48 39 42)(34 47 40 41)
G:=sub<Sym(64)| (1,23,7,37)(2,20,8,34)(3,17,5,31)(4,26,6,40)(9,55,16,48)(10,64,13,45)(11,61,14,42)(12,58,15,51)(18,22,32,36)(19,29,33,27)(21,25,35,39)(24,28,38,30)(41,56,60,49)(43,47,62,54)(44,59,63,52)(46,50,53,57), (1,19,7,33)(2,28,8,30)(3,25,5,39)(4,22,6,36)(9,63,16,44)(10,60,13,41)(11,57,14,50)(12,54,15,47)(17,21,31,35)(18,40,32,26)(20,24,34,38)(23,27,37,29)(42,46,61,53)(43,58,62,51)(45,49,64,56)(48,52,55,59), (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)(41,42,43,44,45,46,47,48,49,50,51,52)(53,54,55,56,57,58,59,60,61,62,63,64), (1,16,3,14)(2,15,4,13)(5,11,7,9)(6,10,8,12)(17,57,23,63)(18,56,24,62)(19,55,25,61)(20,54,26,60)(21,53,27,59)(22,64,28,58)(29,52,35,46)(30,51,36,45)(31,50,37,44)(32,49,38,43)(33,48,39,42)(34,47,40,41)>;
G:=Group( (1,23,7,37)(2,20,8,34)(3,17,5,31)(4,26,6,40)(9,55,16,48)(10,64,13,45)(11,61,14,42)(12,58,15,51)(18,22,32,36)(19,29,33,27)(21,25,35,39)(24,28,38,30)(41,56,60,49)(43,47,62,54)(44,59,63,52)(46,50,53,57), (1,19,7,33)(2,28,8,30)(3,25,5,39)(4,22,6,36)(9,63,16,44)(10,60,13,41)(11,57,14,50)(12,54,15,47)(17,21,31,35)(18,40,32,26)(20,24,34,38)(23,27,37,29)(42,46,61,53)(43,58,62,51)(45,49,64,56)(48,52,55,59), (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)(41,42,43,44,45,46,47,48,49,50,51,52)(53,54,55,56,57,58,59,60,61,62,63,64), (1,16,3,14)(2,15,4,13)(5,11,7,9)(6,10,8,12)(17,57,23,63)(18,56,24,62)(19,55,25,61)(20,54,26,60)(21,53,27,59)(22,64,28,58)(29,52,35,46)(30,51,36,45)(31,50,37,44)(32,49,38,43)(33,48,39,42)(34,47,40,41) );
G=PermutationGroup([[(1,23,7,37),(2,20,8,34),(3,17,5,31),(4,26,6,40),(9,55,16,48),(10,64,13,45),(11,61,14,42),(12,58,15,51),(18,22,32,36),(19,29,33,27),(21,25,35,39),(24,28,38,30),(41,56,60,49),(43,47,62,54),(44,59,63,52),(46,50,53,57)], [(1,19,7,33),(2,28,8,30),(3,25,5,39),(4,22,6,36),(9,63,16,44),(10,60,13,41),(11,57,14,50),(12,54,15,47),(17,21,31,35),(18,40,32,26),(20,24,34,38),(23,27,37,29),(42,46,61,53),(43,58,62,51),(45,49,64,56),(48,52,55,59)], [(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),(41,42,43,44,45,46,47,48,49,50,51,52),(53,54,55,56,57,58,59,60,61,62,63,64)], [(1,16,3,14),(2,15,4,13),(5,11,7,9),(6,10,8,12),(17,57,23,63),(18,56,24,62),(19,55,25,61),(20,54,26,60),(21,53,27,59),(22,64,28,58),(29,52,35,46),(30,51,36,45),(31,50,37,44),(32,49,38,43),(33,48,39,42),(34,47,40,41)]])
Matrix representation of Q8⋊Dic6 ►in GL4(𝔽73) generated by
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 28 | 29 |
| 0 | 0 | 56 | 45 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 28 | 17 |
| 0 | 0 | 44 | 45 |
| 66 | 7 | 0 | 0 |
| 66 | 59 | 0 | 0 |
| 0 | 0 | 16 | 28 |
| 0 | 0 | 45 | 56 |
| 39 | 65 | 0 | 0 |
| 26 | 34 | 0 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 0 | 1 | 0 |
G:=sub<GL(4,GF(73))| [1,0,0,0,0,1,0,0,0,0,28,56,0,0,29,45],[1,0,0,0,0,1,0,0,0,0,28,44,0,0,17,45],[66,66,0,0,7,59,0,0,0,0,16,45,0,0,28,56],[39,26,0,0,65,34,0,0,0,0,0,1,0,0,1,0] >;
Q8⋊Dic6 in GAP, Magma, Sage, TeX
Q_8\rtimes {\rm Dic}_6 % in TeX
G:=Group("Q8:Dic6"); // GroupNames label
G:=SmallGroup(192,945);
// by ID
G=gap.SmallGroup(192,945);
# by ID
G:=PCGroup([7,-2,-2,-2,-3,-2,2,-2,28,85,36,451,1684,655,172,1013,404,285,124]);
// Polycyclic
G:=Group<a,b,c,d|a^4=c^12=1,b^2=a^2,d^2=c^6,b*a*b^-1=d*b*d^-1=a^-1,c*a*c^-1=b,d*a*d^-1=a^2*b,c*b*c^-1=a*b,d*c*d^-1=c^-1>;
// generators/relations
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