Aliases: Q8⋊23- 1+2, C32.SL2(𝔽3), Q8⋊C9⋊2C3, (C3×C6).1A4, C6.10(C3×A4), C2.(C32.A4), (C3×Q8).4C32, (Q8×C32).2C3, C3.4(C3×SL2(𝔽3)), SmallGroup(216,41)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for Q8⋊3- 1+2
G = < a,b,c,d | a4=c9=d3=1, b2=a2, bab-1=a-1, cac-1=b, ad=da, cbc-1=ab, bd=db, dcd-1=c4 >
Smallest permutation representation of Q8⋊3- 1+2
►On 72 points
Generators in S72
(1 18 71 28)(2 24 72 41)(3 61 64 47)(4 12 65 31)(5 27 66 44)(6 55 67 50)(7 15 68 34)(8 21 69 38)(9 58 70 53)(10 60 29 46)(11 42 30 25)(13 63 32 49)(14 45 33 19)(16 57 35 52)(17 39 36 22)(20 51 37 56)(23 54 40 59)(26 48 43 62)
(1 23 71 40)(2 60 72 46)(3 11 64 30)(4 26 65 43)(5 63 66 49)(6 14 67 33)(7 20 68 37)(8 57 69 52)(9 17 70 36)(10 41 29 24)(12 62 31 48)(13 44 32 27)(15 56 34 51)(16 38 35 21)(18 59 28 54)(19 50 45 55)(22 53 39 58)(25 47 42 61)
(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 65 66 67 68 69 70 71 72)
(2 8 5)(3 6 9)(10 16 13)(11 14 17)(19 22 25)(21 27 24)(29 35 32)(30 33 36)(38 44 41)(39 42 45)(46 52 49)(47 50 53)(55 58 61)(57 63 60)(64 67 70)(66 72 69)
G:=sub<Sym(72)| (1,18,71,28)(2,24,72,41)(3,61,64,47)(4,12,65,31)(5,27,66,44)(6,55,67,50)(7,15,68,34)(8,21,69,38)(9,58,70,53)(10,60,29,46)(11,42,30,25)(13,63,32,49)(14,45,33,19)(16,57,35,52)(17,39,36,22)(20,51,37,56)(23,54,40,59)(26,48,43,62), (1,23,71,40)(2,60,72,46)(3,11,64,30)(4,26,65,43)(5,63,66,49)(6,14,67,33)(7,20,68,37)(8,57,69,52)(9,17,70,36)(10,41,29,24)(12,62,31,48)(13,44,32,27)(15,56,34,51)(16,38,35,21)(18,59,28,54)(19,50,45,55)(22,53,39,58)(25,47,42,61), (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)(64,65,66,67,68,69,70,71,72), (2,8,5)(3,6,9)(10,16,13)(11,14,17)(19,22,25)(21,27,24)(29,35,32)(30,33,36)(38,44,41)(39,42,45)(46,52,49)(47,50,53)(55,58,61)(57,63,60)(64,67,70)(66,72,69)>;
G:=Group( (1,18,71,28)(2,24,72,41)(3,61,64,47)(4,12,65,31)(5,27,66,44)(6,55,67,50)(7,15,68,34)(8,21,69,38)(9,58,70,53)(10,60,29,46)(11,42,30,25)(13,63,32,49)(14,45,33,19)(16,57,35,52)(17,39,36,22)(20,51,37,56)(23,54,40,59)(26,48,43,62), (1,23,71,40)(2,60,72,46)(3,11,64,30)(4,26,65,43)(5,63,66,49)(6,14,67,33)(7,20,68,37)(8,57,69,52)(9,17,70,36)(10,41,29,24)(12,62,31,48)(13,44,32,27)(15,56,34,51)(16,38,35,21)(18,59,28,54)(19,50,45,55)(22,53,39,58)(25,47,42,61), (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)(64,65,66,67,68,69,70,71,72), (2,8,5)(3,6,9)(10,16,13)(11,14,17)(19,22,25)(21,27,24)(29,35,32)(30,33,36)(38,44,41)(39,42,45)(46,52,49)(47,50,53)(55,58,61)(57,63,60)(64,67,70)(66,72,69) );
G=PermutationGroup([[(1,18,71,28),(2,24,72,41),(3,61,64,47),(4,12,65,31),(5,27,66,44),(6,55,67,50),(7,15,68,34),(8,21,69,38),(9,58,70,53),(10,60,29,46),(11,42,30,25),(13,63,32,49),(14,45,33,19),(16,57,35,52),(17,39,36,22),(20,51,37,56),(23,54,40,59),(26,48,43,62)], [(1,23,71,40),(2,60,72,46),(3,11,64,30),(4,26,65,43),(5,63,66,49),(6,14,67,33),(7,20,68,37),(8,57,69,52),(9,17,70,36),(10,41,29,24),(12,62,31,48),(13,44,32,27),(15,56,34,51),(16,38,35,21),(18,59,28,54),(19,50,45,55),(22,53,39,58),(25,47,42,61)], [(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),(64,65,66,67,68,69,70,71,72)], [(2,8,5),(3,6,9),(10,16,13),(11,14,17),(19,22,25),(21,27,24),(29,35,32),(30,33,36),(38,44,41),(39,42,45),(46,52,49),(47,50,53),(55,58,61),(57,63,60),(64,67,70),(66,72,69)]])
Q8⋊3- 1+2 is a maximal subgroup of
C32.CSU2(𝔽3) C32.GL2(𝔽3) Q8⋊C9⋊4C6
31 conjugacy classes
| class | 1 | 2 | 3A | 3B | 3C | 3D | 4 | 6A | 6B | 6C | 6D | 9A | ··· | 9F | 12A | ··· | 12H | 18A | ··· | 18F |
| order | 1 | 2 | 3 | 3 | 3 | 3 | 4 | 6 | 6 | 6 | 6 | 9 | ··· | 9 | 12 | ··· | 12 | 18 | ··· | 18 |
| size | 1 | 1 | 1 | 1 | 3 | 3 | 6 | 1 | 1 | 3 | 3 | 12 | ··· | 12 | 6 | ··· | 6 | 12 | ··· | 12 |
31 irreducible representations
| dim | 1 | 1 | 1 | 2 | 2 | 2 | 3 | 3 | 3 | 3 | 6 |
| type | + | - | + | ||||||||
| image | C1 | C3 | C3 | SL2(𝔽3) | SL2(𝔽3) | C3×SL2(𝔽3) | A4 | 3- 1+2 | C3×A4 | C32.A4 | Q8⋊3- 1+2 |
| kernel | Q8⋊3- 1+2 | Q8⋊C9 | Q8×C32 | C32 | C32 | C3 | C3×C6 | Q8 | C6 | C2 | C1 |
| # reps | 1 | 6 | 2 | 1 | 2 | 6 | 1 | 2 | 2 | 6 | 2 |
Matrix representation of Q8⋊3- 1+2 ►in GL5(𝔽37)
| 0 | 36 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 36 | 0 | 0 |
| 0 | 0 | 0 | 36 | 0 |
| 0 | 0 | 36 | 0 | 1 |
| 8 | 3 | 0 | 0 | 0 |
| 3 | 29 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 1 | 36 | 0 |
| 0 | 0 | 1 | 0 | 36 |
| 31 | 3 | 0 | 0 | 0 |
| 2 | 5 | 0 | 0 | 0 |
| 0 | 0 | 1 | 35 | 0 |
| 0 | 0 | 6 | 36 | 26 |
| 0 | 0 | 0 | 36 | 0 |
| 26 | 0 | 0 | 0 | 0 |
| 0 | 26 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 6 | 26 | 0 |
| 0 | 0 | 14 | 0 | 10 |
G:=sub<GL(5,GF(37))| [0,1,0,0,0,36,0,0,0,0,0,0,36,0,36,0,0,0,36,0,0,0,0,0,1],[8,3,0,0,0,3,29,0,0,0,0,0,1,1,1,0,0,0,36,0,0,0,0,0,36],[31,2,0,0,0,3,5,0,0,0,0,0,1,6,0,0,0,35,36,36,0,0,0,26,0],[26,0,0,0,0,0,26,0,0,0,0,0,1,6,14,0,0,0,26,0,0,0,0,0,10] >;
Q8⋊3- 1+2 in GAP, Magma, Sage, TeX
Q_8\rtimes 3_-^{1+2} % in TeX
G:=Group("Q8:ES-(3,1)"); // GroupNames label
G:=SmallGroup(216,41);
// by ID
G=gap.SmallGroup(216,41);
# by ID
G:=PCGroup([6,-3,-3,-3,-2,2,-2,54,145,1299,117,2434,202,88]);
// Polycyclic
G:=Group<a,b,c,d|a^4=c^9=d^3=1,b^2=a^2,b*a*b^-1=a^-1,c*a*c^-1=b,a*d=d*a,c*b*c^-1=a*b,b*d=d*b,d*c*d^-1=c^4>;
// generators/relations
Export