p-group, metabelian, nilpotent (class 3), monomial
Aliases: Q8⋊5M4(2), C42.51D4, C42.612C23, Q8⋊C8⋊33C2, (C4×Q8).18C4, C22.22C4≀C2, C42.65(C2×C4), C4⋊C8.196C22, (C4×C8).313C22, (C22×C4).659D4, C4.23(C2×M4(2)), (C22×Q8).21C4, (C4×M4(2)).17C2, (C4×Q8).254C22, C4.128(C8.C22), C42.6C4.15C2, (C2×C42).168C22, C23.172(C22⋊C4), C2.17(C24.4C4), C2.5(C23.38D4), (C2×C4×Q8).6C2, C2.11(C2×C4≀C2), (C2×C4⋊C4).43C4, C4⋊C4.184(C2×C4), (C2×C4).1140(C2×D4), (C2×Q8).178(C2×C4), (C2×C4).81(C22⋊C4), (C2×C4).317(C22×C4), (C22×C4).190(C2×C4), C22.167(C2×C22⋊C4), SmallGroup(128,223)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for Q8⋊5M4(2)
G = < a,b,c,d | a4=c8=d2=1, b2=a2, bab-1=a-1, ac=ca, ad=da, cbc-1=a-1b, bd=db, dcd=c5 >
Subgroups: 212 in 126 conjugacy classes, 52 normal (26 characteristic)
C1, C2, C2, C4, C4, C22, C22, C22, C8, C2×C4, C2×C4, Q8, Q8, C23, C42, C42, C4⋊C4, C4⋊C4, C2×C8, M4(2), C22×C4, C22×C4, C2×Q8, C2×Q8, C4×C8, C8⋊C4, C22⋊C8, C4⋊C8, C2×C42, C2×C42, C2×C4⋊C4, C2×C4⋊C4, C4×Q8, C4×Q8, C2×M4(2), C22×Q8, Q8⋊C8, C4×M4(2), C42.6C4, C2×C4×Q8, Q8⋊5M4(2)
Quotients: C1, C2, C4, C22, C2×C4, D4, C23, C22⋊C4, M4(2), C22×C4, C2×D4, C4≀C2, C2×C22⋊C4, C2×M4(2), C8.C22, C24.4C4, C23.38D4, C2×C4≀C2, Q8⋊5M4(2)
►On 64 points
(1 26 14 49)(2 27 15 50)(3 28 16 51)(4 29 9 52)(5 30 10 53)(6 31 11 54)(7 32 12 55)(8 25 13 56)(17 63 48 36)(18 64 41 37)(19 57 42 38)(20 58 43 39)(21 59 44 40)(22 60 45 33)(23 61 46 34)(24 62 47 35)
(1 35 14 62)(2 17 15 48)(3 64 16 37)(4 42 9 19)(5 39 10 58)(6 21 11 44)(7 60 12 33)(8 46 13 23)(18 51 41 28)(20 30 43 53)(22 55 45 32)(24 26 47 49)(25 61 56 34)(27 36 50 63)(29 57 52 38)(31 40 54 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 10)(2 15)(3 12)(4 9)(5 14)(6 11)(7 16)(8 13)(17 48)(18 45)(19 42)(20 47)(21 44)(22 41)(23 46)(24 43)(25 56)(26 53)(27 50)(28 55)(29 52)(30 49)(31 54)(32 51)(33 64)(34 61)(35 58)(36 63)(37 60)(38 57)(39 62)(40 59)
G:=sub<Sym(64)| (1,26,14,49)(2,27,15,50)(3,28,16,51)(4,29,9,52)(5,30,10,53)(6,31,11,54)(7,32,12,55)(8,25,13,56)(17,63,48,36)(18,64,41,37)(19,57,42,38)(20,58,43,39)(21,59,44,40)(22,60,45,33)(23,61,46,34)(24,62,47,35), (1,35,14,62)(2,17,15,48)(3,64,16,37)(4,42,9,19)(5,39,10,58)(6,21,11,44)(7,60,12,33)(8,46,13,23)(18,51,41,28)(20,30,43,53)(22,55,45,32)(24,26,47,49)(25,61,56,34)(27,36,50,63)(29,57,52,38)(31,40,54,59), (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), (1,10)(2,15)(3,12)(4,9)(5,14)(6,11)(7,16)(8,13)(17,48)(18,45)(19,42)(20,47)(21,44)(22,41)(23,46)(24,43)(25,56)(26,53)(27,50)(28,55)(29,52)(30,49)(31,54)(32,51)(33,64)(34,61)(35,58)(36,63)(37,60)(38,57)(39,62)(40,59)>;
G:=Group( (1,26,14,49)(2,27,15,50)(3,28,16,51)(4,29,9,52)(5,30,10,53)(6,31,11,54)(7,32,12,55)(8,25,13,56)(17,63,48,36)(18,64,41,37)(19,57,42,38)(20,58,43,39)(21,59,44,40)(22,60,45,33)(23,61,46,34)(24,62,47,35), (1,35,14,62)(2,17,15,48)(3,64,16,37)(4,42,9,19)(5,39,10,58)(6,21,11,44)(7,60,12,33)(8,46,13,23)(18,51,41,28)(20,30,43,53)(22,55,45,32)(24,26,47,49)(25,61,56,34)(27,36,50,63)(29,57,52,38)(31,40,54,59), (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), (1,10)(2,15)(3,12)(4,9)(5,14)(6,11)(7,16)(8,13)(17,48)(18,45)(19,42)(20,47)(21,44)(22,41)(23,46)(24,43)(25,56)(26,53)(27,50)(28,55)(29,52)(30,49)(31,54)(32,51)(33,64)(34,61)(35,58)(36,63)(37,60)(38,57)(39,62)(40,59) );
G=PermutationGroup([[(1,26,14,49),(2,27,15,50),(3,28,16,51),(4,29,9,52),(5,30,10,53),(6,31,11,54),(7,32,12,55),(8,25,13,56),(17,63,48,36),(18,64,41,37),(19,57,42,38),(20,58,43,39),(21,59,44,40),(22,60,45,33),(23,61,46,34),(24,62,47,35)], [(1,35,14,62),(2,17,15,48),(3,64,16,37),(4,42,9,19),(5,39,10,58),(6,21,11,44),(7,60,12,33),(8,46,13,23),(18,51,41,28),(20,30,43,53),(22,55,45,32),(24,26,47,49),(25,61,56,34),(27,36,50,63),(29,57,52,38),(31,40,54,59)], [(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)], [(1,10),(2,15),(3,12),(4,9),(5,14),(6,11),(7,16),(8,13),(17,48),(18,45),(19,42),(20,47),(21,44),(22,41),(23,46),(24,43),(25,56),(26,53),(27,50),(28,55),(29,52),(30,49),(31,54),(32,51),(33,64),(34,61),(35,58),(36,63),(37,60),(38,57),(39,62),(40,59)]])
38 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 4A | 4B | 4C | 4D | 4E | ··· | 4J | 4K | ··· | 4T | 8A | ··· | 8H | 8I | 8J | 8K | 8L |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 4 | ··· | 4 | 8 | ··· | 8 | 8 | 8 | 8 | 8 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 1 | 1 | 1 | 1 | 2 | ··· | 2 | 4 | ··· | 4 | 4 | ··· | 4 | 8 | 8 | 8 | 8 |
38 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 4 |
| type | + | + | + | + | + | + | + | - | |||||
| image | C1 | C2 | C2 | C2 | C2 | C4 | C4 | C4 | D4 | D4 | M4(2) | C4≀C2 | C8.C22 |
| kernel | Q8⋊5M4(2) | Q8⋊C8 | C4×M4(2) | C42.6C4 | C2×C4×Q8 | C2×C4⋊C4 | C4×Q8 | C22×Q8 | C42 | C22×C4 | Q8 | C22 | C4 |
| # reps | 1 | 4 | 1 | 1 | 1 | 2 | 4 | 2 | 2 | 2 | 8 | 8 | 2 |
Matrix representation of Q8⋊5M4(2) ►in GL4(𝔽17) generated by
| 16 | 2 | 0 | 0 |
| 16 | 1 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 7 | 0 | 0 |
| 12 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 5 | 12 | 0 | 0 |
| 11 | 0 | 0 | 0 |
| 0 | 0 | 8 | 15 |
| 0 | 0 | 13 | 9 |
| 16 | 0 | 0 | 0 |
| 0 | 16 | 0 | 0 |
| 0 | 0 | 16 | 0 |
| 0 | 0 | 9 | 1 |
G:=sub<GL(4,GF(17))| [16,16,0,0,2,1,0,0,0,0,1,0,0,0,0,1],[0,12,0,0,7,0,0,0,0,0,1,0,0,0,0,1],[5,11,0,0,12,0,0,0,0,0,8,13,0,0,15,9],[16,0,0,0,0,16,0,0,0,0,16,9,0,0,0,1] >;
Q8⋊5M4(2) in GAP, Magma, Sage, TeX
Q_8\rtimes_5M_4(2)
% in TeX
G:=Group("Q8:5M4(2)"); // GroupNames label
G:=SmallGroup(128,223);
// by ID
G=gap.SmallGroup(128,223);
# by ID
G:=PCGroup([7,-2,2,2,-2,2,-2,2,112,141,232,1430,387,184,1123,570,136,172]);
// Polycyclic
G:=Group<a,b,c,d|a^4=c^8=d^2=1,b^2=a^2,b*a*b^-1=a^-1,a*c=c*a,a*d=d*a,c*b*c^-1=a^-1*b,b*d=d*b,d*c*d=c^5>;
// generators/relations