p-group, metabelian, nilpotent (class 3), monomial
Aliases: Q8⋊3M4(2), C42.49D4, C42.608C23, Q8⋊C8⋊30C2, (C4×C8).6C22, (C4×Q8).15C4, (C2×C4).54Q16, C4.52(C2×Q16), C42.61(C2×C4), C4.94(C2×SD16), C4⋊C8.193C22, (C2×C4).114SD16, (C22×C4).731D4, C4.19(C2×M4(2)), (C22×Q8).20C4, C4.24(Q8⋊C4), (C4×Q8).251C22, C4⋊M4(2).12C2, (C2×C42).164C22, C23.170(C22⋊C4), C42.12C4.18C2, C2.13(C24.4C4), C22.20(Q8⋊C4), C2.10(C42⋊C22), (C2×C4×Q8).5C2, (C2×C4⋊C4).41C4, C4⋊C4.180(C2×C4), C2.5(C2×Q8⋊C4), (C2×C4).1450(C2×D4), (C2×Q8).175(C2×C4), (C2×C4).79(C22⋊C4), (C2×C4).313(C22×C4), (C22×C4).186(C2×C4), C22.163(C2×C22⋊C4), SmallGroup(128,219)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for Q8⋊M4(2)
G = < a,b,c,d | a4=c8=d2=1, b2=a2, bab-1=cac-1=a-1, ad=da, cbc-1=a-1b, bd=db, dcd=c5 >
Subgroups: 212 in 124 conjugacy classes, 58 normal (28 characteristic)
C1, C2, C2, C4, C4, C4, C22, C22, C22, C8, C2×C4, 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, C22⋊C8, C4⋊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), C42.12C4, C2×C4×Q8, Q8⋊M4(2)
Quotients: C1, C2, C4, C22, C2×C4, D4, C23, C22⋊C4, M4(2), SD16, Q16, C22×C4, C2×D4, Q8⋊C4, C2×C22⋊C4, C2×M4(2), C2×SD16, C2×Q16, C24.4C4, C2×Q8⋊C4, C42⋊C22, Q8⋊M4(2)
►On 64 points
(1 31 51 19)(2 20 52 32)(3 25 53 21)(4 22 54 26)(5 27 55 23)(6 24 56 28)(7 29 49 17)(8 18 50 30)(9 46 63 40)(10 33 64 47)(11 48 57 34)(12 35 58 41)(13 42 59 36)(14 37 60 43)(15 44 61 38)(16 39 62 45)
(1 63 51 9)(2 33 52 47)(3 57 53 11)(4 35 54 41)(5 59 55 13)(6 37 56 43)(7 61 49 15)(8 39 50 45)(10 32 64 20)(12 26 58 22)(14 28 60 24)(16 30 62 18)(17 38 29 44)(19 40 31 46)(21 34 25 48)(23 36 27 42)
(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)
(2 6)(4 8)(10 14)(12 16)(18 22)(20 24)(26 30)(28 32)(33 37)(35 39)(41 45)(43 47)(50 54)(52 56)(58 62)(60 64)
G:=sub<Sym(64)| (1,31,51,19)(2,20,52,32)(3,25,53,21)(4,22,54,26)(5,27,55,23)(6,24,56,28)(7,29,49,17)(8,18,50,30)(9,46,63,40)(10,33,64,47)(11,48,57,34)(12,35,58,41)(13,42,59,36)(14,37,60,43)(15,44,61,38)(16,39,62,45), (1,63,51,9)(2,33,52,47)(3,57,53,11)(4,35,54,41)(5,59,55,13)(6,37,56,43)(7,61,49,15)(8,39,50,45)(10,32,64,20)(12,26,58,22)(14,28,60,24)(16,30,62,18)(17,38,29,44)(19,40,31,46)(21,34,25,48)(23,36,27,42), (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), (2,6)(4,8)(10,14)(12,16)(18,22)(20,24)(26,30)(28,32)(33,37)(35,39)(41,45)(43,47)(50,54)(52,56)(58,62)(60,64)>;
G:=Group( (1,31,51,19)(2,20,52,32)(3,25,53,21)(4,22,54,26)(5,27,55,23)(6,24,56,28)(7,29,49,17)(8,18,50,30)(9,46,63,40)(10,33,64,47)(11,48,57,34)(12,35,58,41)(13,42,59,36)(14,37,60,43)(15,44,61,38)(16,39,62,45), (1,63,51,9)(2,33,52,47)(3,57,53,11)(4,35,54,41)(5,59,55,13)(6,37,56,43)(7,61,49,15)(8,39,50,45)(10,32,64,20)(12,26,58,22)(14,28,60,24)(16,30,62,18)(17,38,29,44)(19,40,31,46)(21,34,25,48)(23,36,27,42), (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), (2,6)(4,8)(10,14)(12,16)(18,22)(20,24)(26,30)(28,32)(33,37)(35,39)(41,45)(43,47)(50,54)(52,56)(58,62)(60,64) );
G=PermutationGroup([[(1,31,51,19),(2,20,52,32),(3,25,53,21),(4,22,54,26),(5,27,55,23),(6,24,56,28),(7,29,49,17),(8,18,50,30),(9,46,63,40),(10,33,64,47),(11,48,57,34),(12,35,58,41),(13,42,59,36),(14,37,60,43),(15,44,61,38),(16,39,62,45)], [(1,63,51,9),(2,33,52,47),(3,57,53,11),(4,35,54,41),(5,59,55,13),(6,37,56,43),(7,61,49,15),(8,39,50,45),(10,32,64,20),(12,26,58,22),(14,28,60,24),(16,30,62,18),(17,38,29,44),(19,40,31,46),(21,34,25,48),(23,36,27,42)], [(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)], [(2,6),(4,8),(10,14),(12,16),(18,22),(20,24),(26,30),(28,32),(33,37),(35,39),(41,45),(43,47),(50,54),(52,56),(58,62),(60,64)]])
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 | 2 | 4 |
| type | + | + | + | + | + | + | + | - | ||||||
| image | C1 | C2 | C2 | C2 | C2 | C4 | C4 | C4 | D4 | D4 | SD16 | Q16 | M4(2) | C42⋊C22 |
| kernel | Q8⋊M4(2) | Q8⋊C8 | C4⋊M4(2) | C42.12C4 | C2×C4×Q8 | C2×C4⋊C4 | C4×Q8 | C22×Q8 | C42 | C22×C4 | C2×C4 | C2×C4 | Q8 | C2 |
| # reps | 1 | 4 | 1 | 1 | 1 | 2 | 4 | 2 | 2 | 2 | 4 | 4 | 8 | 2 |
Matrix representation of Q8⋊M4(2) ►in GL4(𝔽17) generated by
| 4 | 0 | 0 | 0 |
| 4 | 13 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 8 | 1 | 0 | 0 |
| 3 | 9 | 0 | 0 |
| 0 | 0 | 16 | 0 |
| 0 | 0 | 0 | 16 |
| 13 | 8 | 0 | 0 |
| 13 | 4 | 0 | 0 |
| 0 | 0 | 5 | 15 |
| 0 | 0 | 2 | 12 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 5 | 16 |
G:=sub<GL(4,GF(17))| [4,4,0,0,0,13,0,0,0,0,1,0,0,0,0,1],[8,3,0,0,1,9,0,0,0,0,16,0,0,0,0,16],[13,13,0,0,8,4,0,0,0,0,5,2,0,0,15,12],[1,0,0,0,0,1,0,0,0,0,1,5,0,0,0,16] >;
Q8⋊M4(2) in GAP, Magma, Sage, TeX
Q_8\rtimes M_4(2)
% in TeX
G:=Group("Q8:M4(2)"); // GroupNames label
G:=SmallGroup(128,219);
// by ID
G=gap.SmallGroup(128,219);
# by ID
G:=PCGroup([7,-2,2,2,-2,2,-2,2,112,141,232,1430,1123,570,136,172]);
// 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=a^-1,a*d=d*a,c*b*c^-1=a^-1*b,b*d=d*b,d*c*d=c^5>;
// generators/relations