p-group, metabelian, nilpotent (class 3), monomial
Aliases: M4(2)⋊17D4, C8⋊D4⋊6C2, C8.24(C2×D4), C8⋊8D4⋊14C2, C8.D4⋊5C2, (C2×D4).222D4, C4⋊C4.32C23, (C2×Q8).177D4, C2.12(Q8○D8), C23.81(C2×D4), C8.18D4⋊26C2, C4.38(C4⋊D4), C4.Q8.7C22, (C2×C4).267C24, (C2×C8).259C23, (C2×D4).69C23, C4.161(C22×D4), (C2×Q8).57C23, C2.D8.78C22, C2.19(D4○SD16), C4⋊D4.22C22, C22.9(C4⋊D4), (C2×Q16).55C22, (C2×SD16).9C22, M4(2)⋊C4⋊15C2, C22⋊Q8.22C22, C23.24D4⋊41C2, (C22×C8).263C22, (C22×C4).989C23, C22.527(C22×D4), D4⋊C4.142C22, Q8⋊C4.134C22, (C22×Q8).284C22, C42⋊C2.114C22, C23.38C23⋊11C2, (C2×M4(2)).268C22, C22.31C24.7C2, (C2×C8○D4)⋊6C2, C4.34(C2×C4○D4), (C2×C4).135(C2×D4), C2.85(C2×C4⋊D4), (C2×Q8⋊C4)⋊56C2, (C2×C8.C22)⋊19C2, (C2×C4).288(C4○D4), (C2×C4⋊C4).596C22, (C2×C4○D4).130C22, SmallGroup(128,1795)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for M4(2)⋊17D4
G = < a,b,c,d | a8=b2=c4=d2=1, bab=a5, cac-1=a-1, dad=a3, bc=cb, dbd=a4b, dcd=c-1 >
Subgroups: 436 in 236 conjugacy classes, 100 normal (38 characteristic)
C1, C2, C2, C4, C4, C4, C22, C22, C22, C8, C8, C2×C4, C2×C4, C2×C4, D4, Q8, C23, C23, C23, C42, C22⋊C4, C4⋊C4, C4⋊C4, C4⋊C4, C2×C8, C2×C8, C2×C8, M4(2), M4(2), SD16, Q16, C22×C4, C22×C4, C22×C4, C2×D4, C2×D4, C2×D4, C2×Q8, C2×Q8, C2×Q8, C4○D4, D4⋊C4, Q8⋊C4, Q8⋊C4, C4.Q8, C2.D8, C2×C4⋊C4, C42⋊C2, C4⋊D4, C4⋊D4, C22⋊Q8, C22⋊Q8, C22.D4, C4.4D4, C4⋊Q8, C22×C8, C22×C8, C2×M4(2), C2×M4(2), C8○D4, C2×SD16, C2×Q16, C8.C22, C22×Q8, C2×C4○D4, C2×Q8⋊C4, C23.24D4, M4(2)⋊C4, C8⋊8D4, C8.18D4, C8⋊D4, C8.D4, C23.38C23, C22.31C24, C2×C8○D4, C2×C8.C22, M4(2)⋊17D4
Quotients: C1, C2, C22, D4, C23, C2×D4, C4○D4, C24, C4⋊D4, C22×D4, C2×C4○D4, C2×C4⋊D4, D4○SD16, Q8○D8, M4(2)⋊17D4
►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 26)(2 31)(3 28)(4 25)(5 30)(6 27)(7 32)(8 29)(9 36)(10 33)(11 38)(12 35)(13 40)(14 37)(15 34)(16 39)(17 54)(18 51)(19 56)(20 53)(21 50)(22 55)(23 52)(24 49)(41 60)(42 57)(43 62)(44 59)(45 64)(46 61)(47 58)(48 63)
(1 12 18 44)(2 11 19 43)(3 10 20 42)(4 9 21 41)(5 16 22 48)(6 15 23 47)(7 14 24 46)(8 13 17 45)(25 36 50 60)(26 35 51 59)(27 34 52 58)(28 33 53 57)(29 40 54 64)(30 39 55 63)(31 38 56 62)(32 37 49 61)
(2 4)(3 7)(6 8)(9 43)(10 46)(11 41)(12 44)(13 47)(14 42)(15 45)(16 48)(17 23)(19 21)(20 24)(25 27)(26 30)(29 31)(33 57)(34 60)(35 63)(36 58)(37 61)(38 64)(39 59)(40 62)(50 52)(51 55)(54 56)
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), (1,26)(2,31)(3,28)(4,25)(5,30)(6,27)(7,32)(8,29)(9,36)(10,33)(11,38)(12,35)(13,40)(14,37)(15,34)(16,39)(17,54)(18,51)(19,56)(20,53)(21,50)(22,55)(23,52)(24,49)(41,60)(42,57)(43,62)(44,59)(45,64)(46,61)(47,58)(48,63), (1,12,18,44)(2,11,19,43)(3,10,20,42)(4,9,21,41)(5,16,22,48)(6,15,23,47)(7,14,24,46)(8,13,17,45)(25,36,50,60)(26,35,51,59)(27,34,52,58)(28,33,53,57)(29,40,54,64)(30,39,55,63)(31,38,56,62)(32,37,49,61), (2,4)(3,7)(6,8)(9,43)(10,46)(11,41)(12,44)(13,47)(14,42)(15,45)(16,48)(17,23)(19,21)(20,24)(25,27)(26,30)(29,31)(33,57)(34,60)(35,63)(36,58)(37,61)(38,64)(39,59)(40,62)(50,52)(51,55)(54,56)>;
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), (1,26)(2,31)(3,28)(4,25)(5,30)(6,27)(7,32)(8,29)(9,36)(10,33)(11,38)(12,35)(13,40)(14,37)(15,34)(16,39)(17,54)(18,51)(19,56)(20,53)(21,50)(22,55)(23,52)(24,49)(41,60)(42,57)(43,62)(44,59)(45,64)(46,61)(47,58)(48,63), (1,12,18,44)(2,11,19,43)(3,10,20,42)(4,9,21,41)(5,16,22,48)(6,15,23,47)(7,14,24,46)(8,13,17,45)(25,36,50,60)(26,35,51,59)(27,34,52,58)(28,33,53,57)(29,40,54,64)(30,39,55,63)(31,38,56,62)(32,37,49,61), (2,4)(3,7)(6,8)(9,43)(10,46)(11,41)(12,44)(13,47)(14,42)(15,45)(16,48)(17,23)(19,21)(20,24)(25,27)(26,30)(29,31)(33,57)(34,60)(35,63)(36,58)(37,61)(38,64)(39,59)(40,62)(50,52)(51,55)(54,56) );
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)], [(1,26),(2,31),(3,28),(4,25),(5,30),(6,27),(7,32),(8,29),(9,36),(10,33),(11,38),(12,35),(13,40),(14,37),(15,34),(16,39),(17,54),(18,51),(19,56),(20,53),(21,50),(22,55),(23,52),(24,49),(41,60),(42,57),(43,62),(44,59),(45,64),(46,61),(47,58),(48,63)], [(1,12,18,44),(2,11,19,43),(3,10,20,42),(4,9,21,41),(5,16,22,48),(6,15,23,47),(7,14,24,46),(8,13,17,45),(25,36,50,60),(26,35,51,59),(27,34,52,58),(28,33,53,57),(29,40,54,64),(30,39,55,63),(31,38,56,62),(32,37,49,61)], [(2,4),(3,7),(6,8),(9,43),(10,46),(11,41),(12,44),(13,47),(14,42),(15,45),(16,48),(17,23),(19,21),(20,24),(25,27),(26,30),(29,31),(33,57),(34,60),(35,63),(36,58),(37,61),(38,64),(39,59),(40,62),(50,52),(51,55),(54,56)]])
32 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 2H | 4A | 4B | 4C | 4D | 4E | 4F | 4G | ··· | 4M | 8A | 8B | 8C | 8D | 8E | ··· | 8J |
| order | 1 | 2 | 2 | 2 | 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 | 4 | 4 | 8 | 2 | 2 | 2 | 2 | 4 | 4 | 8 | ··· | 8 | 2 | 2 | 2 | 2 | 4 | ··· | 4 |
32 irreducible representations
| dim | 1 | 1 | 1 | 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 | C2 | C2 | C2 | D4 | D4 | D4 | C4○D4 | D4○SD16 | Q8○D8 |
| kernel | M4(2)⋊17D4 | C2×Q8⋊C4 | C23.24D4 | M4(2)⋊C4 | C8⋊8D4 | C8.18D4 | C8⋊D4 | C8.D4 | C23.38C23 | C22.31C24 | C2×C8○D4 | C2×C8.C22 | M4(2) | C2×D4 | C2×Q8 | C2×C4 | C2 | C2 |
| # reps | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 1 | 1 | 1 | 1 | 4 | 3 | 1 | 4 | 2 | 2 |
Matrix representation of M4(2)⋊17D4 ►in GL6(𝔽17)
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 3 | 3 | 4 | 13 |
| 0 | 0 | 14 | 3 | 4 | 4 |
| 0 | 0 | 13 | 4 | 14 | 14 |
| 0 | 0 | 13 | 13 | 3 | 14 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 1 | 2 | 0 | 0 | 0 | 0 |
| 16 | 16 | 0 | 0 | 0 | 0 |
| 0 | 0 | 11 | 0 | 0 | 4 |
| 0 | 0 | 0 | 6 | 4 | 0 |
| 0 | 0 | 0 | 4 | 11 | 0 |
| 0 | 0 | 4 | 0 | 0 | 6 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 16 | 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 | 0 | 0 | 1 |
G:=sub<GL(6,GF(17))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,3,14,13,13,0,0,3,3,4,13,0,0,4,4,14,3,0,0,13,4,14,14],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,0,1,0,0],[1,16,0,0,0,0,2,16,0,0,0,0,0,0,11,0,0,4,0,0,0,6,4,0,0,0,0,4,11,0,0,0,4,0,0,6],[1,16,0,0,0,0,0,16,0,0,0,0,0,0,1,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,1] >;
M4(2)⋊17D4 in GAP, Magma, Sage, TeX
M_4(2)\rtimes_{17}D_4 % in TeX
G:=Group("M4(2):17D4"); // GroupNames label
G:=SmallGroup(128,1795);
// by ID
G=gap.SmallGroup(128,1795);
# by ID
G:=PCGroup([7,-2,2,2,2,-2,2,-2,448,253,120,758,1018,521,2804,172]);
// Polycyclic
G:=Group<a,b,c,d|a^8=b^2=c^4=d^2=1,b*a*b=a^5,c*a*c^-1=a^-1,d*a*d=a^3,b*c=c*b,d*b*d=a^4*b,d*c*d=c^-1>;
// generators/relations