p-group, metabelian, nilpotent (class 3), monomial
Aliases: M4(2)⋊4Q8, C42.382C23, C8⋊Q8⋊5C2, (C2×C8)⋊5Q8, C8.4(C2×Q8), C4⋊C4.244D4, C8⋊2Q8⋊27C2, C4.29(C4⋊Q8), C2.23(D4○D8), C2.23(Q8○D8), C22⋊C4.84D4, C8.5Q8⋊15C2, C4.15(C22×Q8), C4⋊C4.103C23, (C2×C4).362C24, (C4×C8).178C22, (C2×C8).273C23, C23.458(C2×D4), C4⋊Q8.113C22, C22.10(C4⋊Q8), C4.Q8.133C22, C8⋊C4.123C22, C2.D8.180C22, C8○2M4(2).10C2, (C22×C8).277C22, C22.622(C22×D4), C42.C2.18C22, M4(2)⋊C4.13C2, (C22×C4).1046C23, C23.25D4.20C2, C42⋊C2.147C22, (C2×M4(2)).276C22, C23.41C23.9C2, C2.32(C2×C4⋊Q8), (C2×C4).142(C2×D4), (C2×C4).144(C2×Q8), (C2×C2.D8).39C2, (C2×C4⋊C4).633C22, SmallGroup(128,1896)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for M4(2)⋊4Q8
G = < a,b,c,d | a8=b2=c4=1, d2=c2, bab=cac-1=a5, dad-1=a3, cbc-1=a4b, bd=db, dcd-1=c-1 >
Subgroups: 284 in 170 conjugacy classes, 108 normal (24 characteristic)
C1, C2, C2, C4, C4, C4, C22, C22, C22, C8, C2×C4, C2×C4, C2×C4, Q8, C23, C42, C42, C22⋊C4, C22⋊C4, C4⋊C4, C4⋊C4, C4⋊C4, C2×C8, C2×C8, M4(2), C22×C4, C22×C4, C2×Q8, C4×C8, C8⋊C4, C4.Q8, C2.D8, C2.D8, C2×C4⋊C4, C42⋊C2, C42⋊C2, C22⋊Q8, C42.C2, C42.C2, C4⋊Q8, C4⋊Q8, C22×C8, C2×M4(2), C8○2M4(2), C2×C2.D8, C23.25D4, M4(2)⋊C4, C8.5Q8, C8⋊2Q8, C8⋊Q8, C23.41C23, M4(2)⋊4Q8
Quotients: C1, C2, C22, D4, Q8, C23, C2×D4, C2×Q8, C24, C4⋊Q8, C22×D4, C22×Q8, C2×C4⋊Q8, D4○D8, Q8○D8, M4(2)⋊4Q8
►On 64 points
(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 43)(2 48)(3 45)(4 42)(5 47)(6 44)(7 41)(8 46)(9 23)(10 20)(11 17)(12 22)(13 19)(14 24)(15 21)(16 18)(25 55)(26 52)(27 49)(28 54)(29 51)(30 56)(31 53)(32 50)(33 61)(34 58)(35 63)(36 60)(37 57)(38 62)(39 59)(40 64)
(1 16 47 18)(2 13 48 23)(3 10 41 20)(4 15 42 17)(5 12 43 22)(6 9 44 19)(7 14 45 24)(8 11 46 21)(25 63 55 39)(26 60 56 36)(27 57 49 33)(28 62 50 38)(29 59 51 35)(30 64 52 40)(31 61 53 37)(32 58 54 34)
(1 50 47 28)(2 53 48 31)(3 56 41 26)(4 51 42 29)(5 54 43 32)(6 49 44 27)(7 52 45 30)(8 55 46 25)(9 57 19 33)(10 60 20 36)(11 63 21 39)(12 58 22 34)(13 61 23 37)(14 64 24 40)(15 59 17 35)(16 62 18 38)
G:=sub<Sym(64)| (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), (1,43)(2,48)(3,45)(4,42)(5,47)(6,44)(7,41)(8,46)(9,23)(10,20)(11,17)(12,22)(13,19)(14,24)(15,21)(16,18)(25,55)(26,52)(27,49)(28,54)(29,51)(30,56)(31,53)(32,50)(33,61)(34,58)(35,63)(36,60)(37,57)(38,62)(39,59)(40,64), (1,16,47,18)(2,13,48,23)(3,10,41,20)(4,15,42,17)(5,12,43,22)(6,9,44,19)(7,14,45,24)(8,11,46,21)(25,63,55,39)(26,60,56,36)(27,57,49,33)(28,62,50,38)(29,59,51,35)(30,64,52,40)(31,61,53,37)(32,58,54,34), (1,50,47,28)(2,53,48,31)(3,56,41,26)(4,51,42,29)(5,54,43,32)(6,49,44,27)(7,52,45,30)(8,55,46,25)(9,57,19,33)(10,60,20,36)(11,63,21,39)(12,58,22,34)(13,61,23,37)(14,64,24,40)(15,59,17,35)(16,62,18,38)>;
G:=Group( (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), (1,43)(2,48)(3,45)(4,42)(5,47)(6,44)(7,41)(8,46)(9,23)(10,20)(11,17)(12,22)(13,19)(14,24)(15,21)(16,18)(25,55)(26,52)(27,49)(28,54)(29,51)(30,56)(31,53)(32,50)(33,61)(34,58)(35,63)(36,60)(37,57)(38,62)(39,59)(40,64), (1,16,47,18)(2,13,48,23)(3,10,41,20)(4,15,42,17)(5,12,43,22)(6,9,44,19)(7,14,45,24)(8,11,46,21)(25,63,55,39)(26,60,56,36)(27,57,49,33)(28,62,50,38)(29,59,51,35)(30,64,52,40)(31,61,53,37)(32,58,54,34), (1,50,47,28)(2,53,48,31)(3,56,41,26)(4,51,42,29)(5,54,43,32)(6,49,44,27)(7,52,45,30)(8,55,46,25)(9,57,19,33)(10,60,20,36)(11,63,21,39)(12,58,22,34)(13,61,23,37)(14,64,24,40)(15,59,17,35)(16,62,18,38) );
G=PermutationGroup([[(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)], [(1,43),(2,48),(3,45),(4,42),(5,47),(6,44),(7,41),(8,46),(9,23),(10,20),(11,17),(12,22),(13,19),(14,24),(15,21),(16,18),(25,55),(26,52),(27,49),(28,54),(29,51),(30,56),(31,53),(32,50),(33,61),(34,58),(35,63),(36,60),(37,57),(38,62),(39,59),(40,64)], [(1,16,47,18),(2,13,48,23),(3,10,41,20),(4,15,42,17),(5,12,43,22),(6,9,44,19),(7,14,45,24),(8,11,46,21),(25,63,55,39),(26,60,56,36),(27,57,49,33),(28,62,50,38),(29,59,51,35),(30,64,52,40),(31,61,53,37),(32,58,54,34)], [(1,50,47,28),(2,53,48,31),(3,56,41,26),(4,51,42,29),(5,54,43,32),(6,49,44,27),(7,52,45,30),(8,55,46,25),(9,57,19,33),(10,60,20,36),(11,63,21,39),(12,58,22,34),(13,61,23,37),(14,64,24,40),(15,59,17,35),(16,62,18,38)]])
32 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | ··· | 4P | 8A | 8B | 8C | 8D | 8E | ··· | 8J |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 8 | 8 | 8 | 8 | 8 | ··· | 8 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 8 | ··· | 8 | 2 | 2 | 2 | 2 | 4 | ··· | 4 |
32 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | - | - | + | - |
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | D4 | D4 | Q8 | Q8 | D4○D8 | Q8○D8 |
| kernel | M4(2)⋊4Q8 | C8○2M4(2) | C2×C2.D8 | C23.25D4 | M4(2)⋊C4 | C8.5Q8 | C8⋊2Q8 | C8⋊Q8 | C23.41C23 | C22⋊C4 | C4⋊C4 | C2×C8 | M4(2) | C2 | C2 |
| # reps | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 4 | 2 | 2 | 2 | 4 | 4 | 2 | 2 |
Matrix representation of M4(2)⋊4Q8 ►in GL6(𝔽17)
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 2 | 0 |
| 0 | 0 | 0 | 9 | 5 | 16 |
| 0 | 0 | 2 | 0 | 0 | 0 |
| 0 | 0 | 6 | 0 | 11 | 8 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 16 | 0 |
| 0 | 0 | 0 | 1 | 0 | 16 |
| 10 | 13 | 0 | 0 | 0 | 0 |
| 4 | 7 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 | 0 | 15 |
| 0 | 0 | 16 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 16 |
| 0 | 13 | 0 | 0 | 0 | 0 |
| 13 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 9 | 16 | 0 | 0 |
| 0 | 0 | 14 | 8 | 0 | 0 |
| 0 | 0 | 0 | 1 | 8 | 15 |
| 0 | 0 | 7 | 8 | 7 | 9 |
G:=sub<GL(6,GF(17))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,2,6,0,0,0,9,0,0,0,0,2,5,0,11,0,0,0,16,0,8],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,1,0,0,0,0,16,0,0,0,0,0,0,16],[10,4,0,0,0,0,13,7,0,0,0,0,0,0,0,0,16,0,0,0,0,1,0,1,0,0,1,0,0,0,0,0,0,15,0,16],[0,13,0,0,0,0,13,0,0,0,0,0,0,0,9,14,0,7,0,0,16,8,1,8,0,0,0,0,8,7,0,0,0,0,15,9] >;
M4(2)⋊4Q8 in GAP, Magma, Sage, TeX
M_4(2)\rtimes_4Q_8
% in TeX
G:=Group("M4(2):4Q8"); // GroupNames label
G:=SmallGroup(128,1896);
// by ID
G=gap.SmallGroup(128,1896);
# by ID
G:=PCGroup([7,-2,2,2,2,-2,2,-2,112,253,120,758,520,1018,4037,124]);
// Polycyclic
G:=Group<a,b,c,d|a^8=b^2=c^4=1,d^2=c^2,b*a*b=c*a*c^-1=a^5,d*a*d^-1=a^3,c*b*c^-1=a^4*b,b*d=d*b,d*c*d^-1=c^-1>;
// generators/relations