p-group, metabelian, nilpotent (class 3), monomial, rational
Aliases: M4(2)⋊8D4, C42.249D4, C42.374C23, C8.1(C2×D4), C8⋊5D4⋊4C2, C8.2D4⋊3C2, C4⋊Q16⋊18C2, C4⋊1(C8.C22), (C4×M4(2))⋊5C2, C4.10(C22×D4), C4.39(C4⋊1D4), (C2×C4).350C24, (C4×C8).175C22, (C2×C8).268C23, (C22×C4).469D4, C23.684(C2×D4), C4⋊Q8.281C22, (C2×D4).116C23, (C2×Q8).104C23, (C2×Q16).63C22, C8⋊C4.118C22, C4⋊1D4.153C22, C22.16(C4⋊1D4), (C2×C42).856C22, (C2×SD16).20C22, C22.610(C22×D4), (C22×C4).1040C23, C4.4D4.142C22, (C22×Q8).309C22, (C2×M4(2)).270C22, C22.26C24.37C2, (C2×C4⋊Q8)⋊39C2, (C2×C4).859(C2×D4), C2.29(C2×C4⋊1D4), (C2×C8.C22)⋊22C2, C2.42(C2×C8.C22), (C2×C4○D4).156C22, SmallGroup(128,1884)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for M4(2)⋊8D4
G = < a,b,c,d | a8=b2=c4=d2=1, bab=a5, ac=ca, dad=a3, bc=cb, dbd=a4b, dcd=c-1 >
Subgroups: 516 in 276 conjugacy classes, 112 normal (16 characteristic)
C1, C2, C2, C2, C4, C4, C22, C22, C22, C8, C2×C4, C2×C4, C2×C4, D4, Q8, C23, C23, C42, C42, C22⋊C4, C4⋊C4, C2×C8, M4(2), SD16, Q16, C22×C4, C22×C4, C22×C4, C2×D4, C2×D4, C2×Q8, C2×Q8, C4○D4, C4×C8, C8⋊C4, C2×C42, C2×C4⋊C4, C4×D4, C4⋊D4, C4.4D4, C4⋊1D4, C4⋊Q8, C4⋊Q8, C4⋊Q8, C2×M4(2), C2×SD16, C2×Q16, C8.C22, C22×Q8, C2×C4○D4, C4×M4(2), C8⋊5D4, C4⋊Q16, C8.2D4, C2×C4⋊Q8, C22.26C24, C2×C8.C22, M4(2)⋊8D4
Quotients: C1, C2, C22, D4, C23, C2×D4, C24, C4⋊1D4, C8.C22, C22×D4, C2×C4⋊1D4, C2×C8.C22, M4(2)⋊8D4
►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 30)(2 27)(3 32)(4 29)(5 26)(6 31)(7 28)(8 25)(9 22)(10 19)(11 24)(12 21)(13 18)(14 23)(15 20)(16 17)(33 46)(34 43)(35 48)(36 45)(37 42)(38 47)(39 44)(40 41)(49 58)(50 63)(51 60)(52 57)(53 62)(54 59)(55 64)(56 61)
(1 11 60 34)(2 12 61 35)(3 13 62 36)(4 14 63 37)(5 15 64 38)(6 16 57 39)(7 9 58 40)(8 10 59 33)(17 52 44 31)(18 53 45 32)(19 54 46 25)(20 55 47 26)(21 56 48 27)(22 49 41 28)(23 50 42 29)(24 51 43 30)
(1 60)(2 63)(3 58)(4 61)(5 64)(6 59)(7 62)(8 57)(9 13)(10 16)(12 14)(17 23)(19 21)(20 24)(25 56)(26 51)(27 54)(28 49)(29 52)(30 55)(31 50)(32 53)(33 39)(35 37)(36 40)(42 44)(43 47)(46 48)
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), (1,30)(2,27)(3,32)(4,29)(5,26)(6,31)(7,28)(8,25)(9,22)(10,19)(11,24)(12,21)(13,18)(14,23)(15,20)(16,17)(33,46)(34,43)(35,48)(36,45)(37,42)(38,47)(39,44)(40,41)(49,58)(50,63)(51,60)(52,57)(53,62)(54,59)(55,64)(56,61), (1,11,60,34)(2,12,61,35)(3,13,62,36)(4,14,63,37)(5,15,64,38)(6,16,57,39)(7,9,58,40)(8,10,59,33)(17,52,44,31)(18,53,45,32)(19,54,46,25)(20,55,47,26)(21,56,48,27)(22,49,41,28)(23,50,42,29)(24,51,43,30), (1,60)(2,63)(3,58)(4,61)(5,64)(6,59)(7,62)(8,57)(9,13)(10,16)(12,14)(17,23)(19,21)(20,24)(25,56)(26,51)(27,54)(28,49)(29,52)(30,55)(31,50)(32,53)(33,39)(35,37)(36,40)(42,44)(43,47)(46,48)>;
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), (1,30)(2,27)(3,32)(4,29)(5,26)(6,31)(7,28)(8,25)(9,22)(10,19)(11,24)(12,21)(13,18)(14,23)(15,20)(16,17)(33,46)(34,43)(35,48)(36,45)(37,42)(38,47)(39,44)(40,41)(49,58)(50,63)(51,60)(52,57)(53,62)(54,59)(55,64)(56,61), (1,11,60,34)(2,12,61,35)(3,13,62,36)(4,14,63,37)(5,15,64,38)(6,16,57,39)(7,9,58,40)(8,10,59,33)(17,52,44,31)(18,53,45,32)(19,54,46,25)(20,55,47,26)(21,56,48,27)(22,49,41,28)(23,50,42,29)(24,51,43,30), (1,60)(2,63)(3,58)(4,61)(5,64)(6,59)(7,62)(8,57)(9,13)(10,16)(12,14)(17,23)(19,21)(20,24)(25,56)(26,51)(27,54)(28,49)(29,52)(30,55)(31,50)(32,53)(33,39)(35,37)(36,40)(42,44)(43,47)(46,48) );
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)], [(1,30),(2,27),(3,32),(4,29),(5,26),(6,31),(7,28),(8,25),(9,22),(10,19),(11,24),(12,21),(13,18),(14,23),(15,20),(16,17),(33,46),(34,43),(35,48),(36,45),(37,42),(38,47),(39,44),(40,41),(49,58),(50,63),(51,60),(52,57),(53,62),(54,59),(55,64),(56,61)], [(1,11,60,34),(2,12,61,35),(3,13,62,36),(4,14,63,37),(5,15,64,38),(6,16,57,39),(7,9,58,40),(8,10,59,33),(17,52,44,31),(18,53,45,32),(19,54,46,25),(20,55,47,26),(21,56,48,27),(22,49,41,28),(23,50,42,29),(24,51,43,30)], [(1,60),(2,63),(3,58),(4,61),(5,64),(6,59),(7,62),(8,57),(9,13),(10,16),(12,14),(17,23),(19,21),(20,24),(25,56),(26,51),(27,54),(28,49),(29,52),(30,55),(31,50),(32,53),(33,39),(35,37),(36,40),(42,44),(43,47),(46,48)]])
32 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 4A | ··· | 4H | 4I | 4J | 4K | ··· | 4P | 8A | ··· | 8H |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 4 | 4 | 4 | ··· | 4 | 8 | ··· | 8 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 8 | 8 | 2 | ··· | 2 | 4 | 4 | 8 | ··· | 8 | 4 | ··· | 4 |
32 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | - |
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | D4 | D4 | D4 | C8.C22 |
| kernel | M4(2)⋊8D4 | C4×M4(2) | C8⋊5D4 | C4⋊Q16 | C8.2D4 | C2×C4⋊Q8 | C22.26C24 | C2×C8.C22 | C42 | M4(2) | C22×C4 | C4 |
| # reps | 1 | 1 | 2 | 2 | 4 | 1 | 1 | 4 | 2 | 8 | 2 | 4 |
Matrix representation of M4(2)⋊8D4 ►in GL6(𝔽17)
| 16 | 0 | 0 | 0 | 0 | 0 |
| 0 | 16 | 0 | 0 | 0 | 0 |
| 0 | 0 | 7 | 7 | 9 | 9 |
| 0 | 0 | 10 | 16 | 0 | 8 |
| 0 | 0 | 1 | 10 | 0 | 6 |
| 0 | 0 | 16 | 6 | 10 | 11 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 1 | 1 | 16 | 15 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 16 |
| 1 | 15 | 0 | 0 | 0 | 0 |
| 1 | 16 | 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 | 1 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 1 | 16 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 16 | 0 | 0 |
| 0 | 0 | 0 | 0 | 16 | 0 |
| 0 | 0 | 0 | 16 | 1 | 1 |
G:=sub<GL(6,GF(17))| [16,0,0,0,0,0,0,16,0,0,0,0,0,0,7,10,1,16,0,0,7,16,10,6,0,0,9,0,0,10,0,0,9,8,6,11],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,1,0,0,0,0,1,0,0,0,0,1,16,0,0,0,0,0,15,0,16],[1,1,0,0,0,0,15,16,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,1,0,0,0,0,0,16,0,0,0,0,0,0,1,0,0,0,0,0,0,16,0,16,0,0,0,0,16,1,0,0,0,0,0,1] >;
M4(2)⋊8D4 in GAP, Magma, Sage, TeX
M_4(2)\rtimes_8D_4
% in TeX
G:=Group("M4(2):8D4"); // GroupNames label
G:=SmallGroup(128,1884);
// by ID
G=gap.SmallGroup(128,1884);
# by ID
G:=PCGroup([7,-2,2,2,2,-2,2,-2,448,253,120,758,184,1018,2804,172]);
// Polycyclic
G:=Group<a,b,c,d|a^8=b^2=c^4=d^2=1,b*a*b=a^5,a*c=c*a,d*a*d=a^3,b*c=c*b,d*b*d=a^4*b,d*c*d=c^-1>;
// generators/relations