p-group, metabelian, nilpotent (class 3), monomial
Aliases: Q8:2M4(2), C42.645C23, C8:C8:6C2, Q8:C8:40C2, (C8xQ8):26C2, C8:1C8:17C2, D4:C8.14C2, C2.9(C8oD8), (C2xC8).310D4, C4.Q8.10C4, D4:C4.5C4, C4.37(C8oD4), C8:6D4.12C2, Q8:C4.5C4, (C2xSD16).6C4, (C4xSD16).2C2, C2.15(C8:9D4), C4:C8.279C22, (C4xC8).317C22, (C4xD4).12C22, C22.136(C4xD4), C4.31(C2xM4(2)), C4.145(C8:C22), (C4xQ8).260C22, C2.6(SD16:C4), C4.139(C8.C22), (C2xC8).33(C2xC4), C4:C4.139(C2xC4), (C2xD4).57(C2xC4), (C2xC4).1481(C2xD4), (C2xQ8).138(C2xC4), (C2xC4).506(C4oD4), (C2xC4).337(C22xC4), SmallGroup(128,320)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for Q8:2M4(2)
G = < a,b,c,d | a4=c8=d2=1, b2=a2, bab-1=cac-1=dad=a-1, cbc-1=a-1b, dbd=ab, dcd=c5 >
Subgroups: 152 in 83 conjugacy classes, 42 normal (40 characteristic)
C1, C2, C2, C4, C4, C22, C22, C8, C2xC4, C2xC4, D4, Q8, Q8, C23, C42, C42, C22:C4, C4:C4, C4:C4, C2xC8, C2xC8, M4(2), SD16, C22xC4, C2xD4, C2xQ8, C4xC8, C4xC8, C22:C8, D4:C4, Q8:C4, C4:C8, C4:C8, C4.Q8, C4xD4, C4xQ8, C2xM4(2), C2xSD16, C8:C8, D4:C8, Q8:C8, C8:1C8, C8:6D4, C4xSD16, C8xQ8, Q8:2M4(2)
Quotients: C1, C2, C4, C22, C2xC4, D4, C23, M4(2), C22xC4, C2xD4, C4oD4, C4xD4, C2xM4(2), C8oD4, C8:C22, C8.C22, C8:9D4, SD16:C4, C8oD8, Q8:2M4(2)
►On 64 points
(1 16 27 43)(2 44 28 9)(3 10 29 45)(4 46 30 11)(5 12 31 47)(6 48 32 13)(7 14 25 41)(8 42 26 15)(17 39 53 60)(18 61 54 40)(19 33 55 62)(20 63 56 34)(21 35 49 64)(22 57 50 36)(23 37 51 58)(24 59 52 38)
(1 20 27 56)(2 64 28 35)(3 22 29 50)(4 58 30 37)(5 24 31 52)(6 60 32 39)(7 18 25 54)(8 62 26 33)(9 21 44 49)(10 36 45 57)(11 23 46 51)(12 38 47 59)(13 17 48 53)(14 40 41 61)(15 19 42 55)(16 34 43 63)
(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)(2 13)(3 10)(4 15)(5 12)(6 9)(7 14)(8 11)(17 49)(18 54)(19 51)(20 56)(21 53)(22 50)(23 55)(24 52)(25 41)(26 46)(27 43)(28 48)(29 45)(30 42)(31 47)(32 44)(33 37)(35 39)(58 62)(60 64)
G:=sub<Sym(64)| (1,16,27,43)(2,44,28,9)(3,10,29,45)(4,46,30,11)(5,12,31,47)(6,48,32,13)(7,14,25,41)(8,42,26,15)(17,39,53,60)(18,61,54,40)(19,33,55,62)(20,63,56,34)(21,35,49,64)(22,57,50,36)(23,37,51,58)(24,59,52,38), (1,20,27,56)(2,64,28,35)(3,22,29,50)(4,58,30,37)(5,24,31,52)(6,60,32,39)(7,18,25,54)(8,62,26,33)(9,21,44,49)(10,36,45,57)(11,23,46,51)(12,38,47,59)(13,17,48,53)(14,40,41,61)(15,19,42,55)(16,34,43,63), (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)(2,13)(3,10)(4,15)(5,12)(6,9)(7,14)(8,11)(17,49)(18,54)(19,51)(20,56)(21,53)(22,50)(23,55)(24,52)(25,41)(26,46)(27,43)(28,48)(29,45)(30,42)(31,47)(32,44)(33,37)(35,39)(58,62)(60,64)>;
G:=Group( (1,16,27,43)(2,44,28,9)(3,10,29,45)(4,46,30,11)(5,12,31,47)(6,48,32,13)(7,14,25,41)(8,42,26,15)(17,39,53,60)(18,61,54,40)(19,33,55,62)(20,63,56,34)(21,35,49,64)(22,57,50,36)(23,37,51,58)(24,59,52,38), (1,20,27,56)(2,64,28,35)(3,22,29,50)(4,58,30,37)(5,24,31,52)(6,60,32,39)(7,18,25,54)(8,62,26,33)(9,21,44,49)(10,36,45,57)(11,23,46,51)(12,38,47,59)(13,17,48,53)(14,40,41,61)(15,19,42,55)(16,34,43,63), (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)(2,13)(3,10)(4,15)(5,12)(6,9)(7,14)(8,11)(17,49)(18,54)(19,51)(20,56)(21,53)(22,50)(23,55)(24,52)(25,41)(26,46)(27,43)(28,48)(29,45)(30,42)(31,47)(32,44)(33,37)(35,39)(58,62)(60,64) );
G=PermutationGroup([[(1,16,27,43),(2,44,28,9),(3,10,29,45),(4,46,30,11),(5,12,31,47),(6,48,32,13),(7,14,25,41),(8,42,26,15),(17,39,53,60),(18,61,54,40),(19,33,55,62),(20,63,56,34),(21,35,49,64),(22,57,50,36),(23,37,51,58),(24,59,52,38)], [(1,20,27,56),(2,64,28,35),(3,22,29,50),(4,58,30,37),(5,24,31,52),(6,60,32,39),(7,18,25,54),(8,62,26,33),(9,21,44,49),(10,36,45,57),(11,23,46,51),(12,38,47,59),(13,17,48,53),(14,40,41,61),(15,19,42,55),(16,34,43,63)], [(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),(2,13),(3,10),(4,15),(5,12),(6,9),(7,14),(8,11),(17,49),(18,54),(19,51),(20,56),(21,53),(22,50),(23,55),(24,52),(25,41),(26,46),(27,43),(28,48),(29,45),(30,42),(31,47),(32,44),(33,37),(35,39),(58,62),(60,64)]])
38 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 4K | 4L | 4M | 8A | 8B | 8C | 8D | 8E | ··· | 8R | 8S | 8T |
| order | 1 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 8 | 8 | 8 | 8 | 8 | ··· | 8 | 8 | 8 |
| size | 1 | 1 | 1 | 1 | 8 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 8 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 8 | 8 |
38 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | - | ||||||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C4 | C4 | C4 | C4 | D4 | C4oD4 | M4(2) | C8oD4 | C8oD8 | C8:C22 | C8.C22 |
| kernel | Q8:2M4(2) | C8:C8 | D4:C8 | Q8:C8 | C8:1C8 | C8:6D4 | C4xSD16 | C8xQ8 | D4:C4 | Q8:C4 | C4.Q8 | C2xSD16 | C2xC8 | C2xC4 | Q8 | C4 | C2 | C4 | C4 |
| # reps | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 8 | 1 | 1 |
Matrix representation of Q8:2M4(2) ►in GL4(F17) generated by
| 16 | 15 | 0 | 0 |
| 1 | 1 | 0 | 0 |
| 0 | 0 | 16 | 0 |
| 0 | 0 | 0 | 16 |
| 0 | 10 | 0 | 0 |
| 5 | 0 | 0 | 0 |
| 0 | 0 | 0 | 2 |
| 0 | 0 | 9 | 0 |
| 8 | 0 | 0 | 0 |
| 9 | 9 | 0 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 0 | 4 | 0 |
| 1 | 2 | 0 | 0 |
| 0 | 16 | 0 | 0 |
| 0 | 0 | 16 | 0 |
| 0 | 0 | 0 | 1 |
G:=sub<GL(4,GF(17))| [16,1,0,0,15,1,0,0,0,0,16,0,0,0,0,16],[0,5,0,0,10,0,0,0,0,0,0,9,0,0,2,0],[8,9,0,0,0,9,0,0,0,0,0,4,0,0,1,0],[1,0,0,0,2,16,0,0,0,0,16,0,0,0,0,1] >;
Q8:2M4(2) in GAP, Magma, Sage, TeX
Q_8\rtimes_2M_4(2)
% in TeX
G:=Group("Q8:2M4(2)"); // GroupNames label
G:=SmallGroup(128,320);
// by ID
G=gap.SmallGroup(128,320);
# by ID
G:=PCGroup([7,-2,2,2,-2,2,-2,2,112,141,232,2102,268,1123,570,136,172]);
// Polycyclic
G:=Group<a,b,c,d|a^4=c^8=d^2=1,b^2=a^2,b*a*b^-1=c*a*c^-1=d*a*d=a^-1,c*b*c^-1=a^-1*b,d*b*d=a*b,d*c*d=c^5>;
// generators/relations