p-group, metabelian, nilpotent (class 3), monomial
Aliases: M4(2)⋊3Q8, C42.381C23, C8⋊Q8⋊4C2, (C2×C8)⋊4Q8, C8.3(C2×Q8), C8⋊3Q8⋊4C2, C4⋊C4.243D4, C4.28(C4⋊Q8), C22⋊C4.83D4, C8.5Q8⋊14C2, C22.9(C4⋊Q8), C4.14(C22×Q8), C4⋊C4.102C23, (C4×C8).177C22, (C2×C4).361C24, (C2×C8).272C23, C23.457(C2×D4), C4⋊Q8.112C22, C2.36(D4○SD16), C8○2M4(2).9C2, C2.D8.217C22, C4.Q8.160C22, C8⋊C4.122C22, (C22×C8).276C22, C22.621(C22×D4), C42.C2.17C22, M4(2)⋊C4.12C2, (C22×C4).1045C23, C23.25D4.19C2, C42⋊C2.146C22, (C2×M4(2)).275C22, C23.41C23.8C2, C2.31(C2×C4⋊Q8), (C2×C4).141(C2×D4), (C2×C4).143(C2×Q8), (C2×C4.Q8).12C2, (C2×C4⋊C4).632C22, SmallGroup(128,1895)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for M4(2)⋊3Q8
G = < a,b,c,d | a8=b2=c4=1, d2=c2, bab=cac-1=a5, dad-1=a-1, cbc-1=a4b, bd=db, dcd-1=c-1 >
Subgroups: 284 in 170 conjugacy classes, 108 normal (20 characteristic)
C1, C2, 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, C2×C8, C2×C8, M4(2), C22×C4, C22×C4, C2×Q8, C4×C8, C8⋊C4, C4.Q8, C4.Q8, 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×C4.Q8, C23.25D4, M4(2)⋊C4, C8⋊3Q8, C8.5Q8, C8⋊Q8, C23.41C23, M4(2)⋊3Q8
Quotients: C1, C2, C22, D4, Q8, C23, C2×D4, C2×Q8, C24, C4⋊Q8, C22×D4, C22×Q8, C2×C4⋊Q8, D4○SD16, M4(2)⋊3Q8
►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 45)(2 42)(3 47)(4 44)(5 41)(6 46)(7 43)(8 48)(9 19)(10 24)(11 21)(12 18)(13 23)(14 20)(15 17)(16 22)(25 49)(26 54)(27 51)(28 56)(29 53)(30 50)(31 55)(32 52)(33 64)(34 61)(35 58)(36 63)(37 60)(38 57)(39 62)(40 59)
(1 12 45 22)(2 9 46 19)(3 14 47 24)(4 11 48 21)(5 16 41 18)(6 13 42 23)(7 10 43 20)(8 15 44 17)(25 59 53 40)(26 64 54 37)(27 61 55 34)(28 58 56 39)(29 63 49 36)(30 60 50 33)(31 57 51 38)(32 62 52 35)
(1 56 45 28)(2 55 46 27)(3 54 47 26)(4 53 48 25)(5 52 41 32)(6 51 42 31)(7 50 43 30)(8 49 44 29)(9 61 19 34)(10 60 20 33)(11 59 21 40)(12 58 22 39)(13 57 23 38)(14 64 24 37)(15 63 17 36)(16 62 18 35)
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), (1,45)(2,42)(3,47)(4,44)(5,41)(6,46)(7,43)(8,48)(9,19)(10,24)(11,21)(12,18)(13,23)(14,20)(15,17)(16,22)(25,49)(26,54)(27,51)(28,56)(29,53)(30,50)(31,55)(32,52)(33,64)(34,61)(35,58)(36,63)(37,60)(38,57)(39,62)(40,59), (1,12,45,22)(2,9,46,19)(3,14,47,24)(4,11,48,21)(5,16,41,18)(6,13,42,23)(7,10,43,20)(8,15,44,17)(25,59,53,40)(26,64,54,37)(27,61,55,34)(28,58,56,39)(29,63,49,36)(30,60,50,33)(31,57,51,38)(32,62,52,35), (1,56,45,28)(2,55,46,27)(3,54,47,26)(4,53,48,25)(5,52,41,32)(6,51,42,31)(7,50,43,30)(8,49,44,29)(9,61,19,34)(10,60,20,33)(11,59,21,40)(12,58,22,39)(13,57,23,38)(14,64,24,37)(15,63,17,36)(16,62,18,35)>;
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), (1,45)(2,42)(3,47)(4,44)(5,41)(6,46)(7,43)(8,48)(9,19)(10,24)(11,21)(12,18)(13,23)(14,20)(15,17)(16,22)(25,49)(26,54)(27,51)(28,56)(29,53)(30,50)(31,55)(32,52)(33,64)(34,61)(35,58)(36,63)(37,60)(38,57)(39,62)(40,59), (1,12,45,22)(2,9,46,19)(3,14,47,24)(4,11,48,21)(5,16,41,18)(6,13,42,23)(7,10,43,20)(8,15,44,17)(25,59,53,40)(26,64,54,37)(27,61,55,34)(28,58,56,39)(29,63,49,36)(30,60,50,33)(31,57,51,38)(32,62,52,35), (1,56,45,28)(2,55,46,27)(3,54,47,26)(4,53,48,25)(5,52,41,32)(6,51,42,31)(7,50,43,30)(8,49,44,29)(9,61,19,34)(10,60,20,33)(11,59,21,40)(12,58,22,39)(13,57,23,38)(14,64,24,37)(15,63,17,36)(16,62,18,35) );
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)], [(1,45),(2,42),(3,47),(4,44),(5,41),(6,46),(7,43),(8,48),(9,19),(10,24),(11,21),(12,18),(13,23),(14,20),(15,17),(16,22),(25,49),(26,54),(27,51),(28,56),(29,53),(30,50),(31,55),(32,52),(33,64),(34,61),(35,58),(36,63),(37,60),(38,57),(39,62),(40,59)], [(1,12,45,22),(2,9,46,19),(3,14,47,24),(4,11,48,21),(5,16,41,18),(6,13,42,23),(7,10,43,20),(8,15,44,17),(25,59,53,40),(26,64,54,37),(27,61,55,34),(28,58,56,39),(29,63,49,36),(30,60,50,33),(31,57,51,38),(32,62,52,35)], [(1,56,45,28),(2,55,46,27),(3,54,47,26),(4,53,48,25),(5,52,41,32),(6,51,42,31),(7,50,43,30),(8,49,44,29),(9,61,19,34),(10,60,20,33),(11,59,21,40),(12,58,22,39),(13,57,23,38),(14,64,24,37),(15,63,17,36),(16,62,18,35)]])
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 |
| type | + | + | + | + | + | + | + | + | + | + | + | - | - | |
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | D4 | D4 | Q8 | Q8 | D4○SD16 |
| kernel | M4(2)⋊3Q8 | C8○2M4(2) | C2×C4.Q8 | C23.25D4 | M4(2)⋊C4 | C8⋊3Q8 | C8.5Q8 | C8⋊Q8 | C23.41C23 | C22⋊C4 | C4⋊C4 | C2×C8 | M4(2) | C2 |
| # reps | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 4 | 2 | 2 | 2 | 4 | 4 | 4 |
Matrix representation of M4(2)⋊3Q8 ►in GL6(𝔽17)
| 16 | 0 | 0 | 0 | 0 | 0 |
| 0 | 16 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 16 | 7 | 10 |
| 0 | 0 | 0 | 16 | 0 | 10 |
| 0 | 0 | 12 | 12 | 0 | 0 |
| 0 | 0 | 12 | 5 | 0 | 1 |
| 1 | 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 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 10 | 0 | 1 |
| 0 | 16 | 0 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 7 | 1 | 15 |
| 0 | 0 | 16 | 0 | 0 | 0 |
| 0 | 0 | 8 | 8 | 12 | 10 |
| 16 | 10 | 0 | 0 | 0 | 0 |
| 10 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 10 | 16 | 0 | 0 |
| 0 | 0 | 16 | 7 | 0 | 0 |
| 0 | 0 | 0 | 7 | 8 | 15 |
| 0 | 0 | 5 | 10 | 7 | 9 |
G:=sub<GL(6,GF(17))| [16,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,12,12,0,0,16,16,12,5,0,0,7,0,0,0,0,0,10,10,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,10,0,0,0,0,1,0,0,0,0,0,0,1],[0,1,0,0,0,0,16,0,0,0,0,0,0,0,0,0,16,8,0,0,0,7,0,8,0,0,1,1,0,12,0,0,0,15,0,10],[16,10,0,0,0,0,10,1,0,0,0,0,0,0,10,16,0,5,0,0,16,7,7,10,0,0,0,0,8,7,0,0,0,0,15,9] >;
M4(2)⋊3Q8 in GAP, Magma, Sage, TeX
M_4(2)\rtimes_3Q_8
% in TeX
G:=Group("M4(2):3Q8"); // GroupNames label
G:=SmallGroup(128,1895);
// by ID
G=gap.SmallGroup(128,1895);
# by ID
G:=PCGroup([7,-2,2,2,2,-2,2,-2,112,253,120,758,184,1018,4037,124]);
// 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^-1,c*b*c^-1=a^4*b,b*d=d*b,d*c*d^-1=c^-1>;
// generators/relations