p-group, metabelian, nilpotent (class 2), monomial
Aliases: Q8⋊7M4(2), C42.699C23, C4.1792+ 1+4, C4.1262- 1+4, Q8○3(C4⋊C8), (C8×Q8)⋊36C2, C8⋊6D4⋊46C2, (C4×D4).39C4, (C4×Q8).36C4, C4.50(C8○D4), C4⋊C8.371C22, (C2×C4).688C24, (C2×C8).484C23, (C4×C8).345C22, C42.236(C2×C4), C4.38(C2×M4(2)), C4⋊M4(2)⋊38C2, C42⋊C2.37C4, (C4×D4).305C22, (C4×Q8).336C22, C42.12C4⋊58C2, C22⋊C8.241C22, C22.210(C23×C4), C23.110(C22×C4), (C2×C42).795C22, (C22×C4).949C23, C2.25(C22×M4(2)), (C2×M4(2)).251C22, C2.46(C23.33C23), C2.36(C2×C8○D4), C4⋊C4.235(C2×C4), (C2×C4○D4).31C4, (C4×C4○D4).21C2, (C2×D4).239(C2×C4), C22⋊C4.81(C2×C4), (C2×Q8).231(C2×C4), (C22×C4).366(C2×C4), (C2×C4).299(C22×C4), SmallGroup(128,1723)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for Q8⋊7M4(2)
G = < a,b,c,d | a4=c8=d2=1, b2=a2, bab-1=dad=a-1, ac=ca, cbc-1=a2b, bd=db, dcd=c5 >
Subgroups: 276 in 197 conjugacy classes, 136 normal (18 characteristic)
C1, C2, C2, C4, C4, C4, C22, C22, C8, C2×C4, C2×C4, C2×C4, D4, Q8, C23, C42, C42, C22⋊C4, C4⋊C4, C2×C8, M4(2), C22×C4, C2×D4, C2×Q8, C4○D4, C4×C8, C22⋊C8, C4⋊C8, C4⋊C8, C2×C42, C42⋊C2, C4×D4, C4×Q8, C2×M4(2), C2×C4○D4, C4⋊M4(2), C42.12C4, C8⋊6D4, C8×Q8, C4×C4○D4, Q8⋊7M4(2)
Quotients: C1, C2, C4, C22, C2×C4, C23, M4(2), C22×C4, C24, C2×M4(2), C8○D4, C23×C4, 2+ 1+4, 2- 1+4, C23.33C23, C22×M4(2), C2×C8○D4, Q8⋊7M4(2)
►On 64 points
(1 51 64 28)(2 52 57 29)(3 53 58 30)(4 54 59 31)(5 55 60 32)(6 56 61 25)(7 49 62 26)(8 50 63 27)(9 21 39 46)(10 22 40 47)(11 23 33 48)(12 24 34 41)(13 17 35 42)(14 18 36 43)(15 19 37 44)(16 20 38 45)
(1 18 64 43)(2 44 57 19)(3 20 58 45)(4 46 59 21)(5 22 60 47)(6 48 61 23)(7 24 62 41)(8 42 63 17)(9 54 39 31)(10 32 40 55)(11 56 33 25)(12 26 34 49)(13 50 35 27)(14 28 36 51)(15 52 37 29)(16 30 38 53)
(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 40)(2 37)(3 34)(4 39)(5 36)(6 33)(7 38)(8 35)(9 59)(10 64)(11 61)(12 58)(13 63)(14 60)(15 57)(16 62)(17 50)(18 55)(19 52)(20 49)(21 54)(22 51)(23 56)(24 53)(25 48)(26 45)(27 42)(28 47)(29 44)(30 41)(31 46)(32 43)
G:=sub<Sym(64)| (1,51,64,28)(2,52,57,29)(3,53,58,30)(4,54,59,31)(5,55,60,32)(6,56,61,25)(7,49,62,26)(8,50,63,27)(9,21,39,46)(10,22,40,47)(11,23,33,48)(12,24,34,41)(13,17,35,42)(14,18,36,43)(15,19,37,44)(16,20,38,45), (1,18,64,43)(2,44,57,19)(3,20,58,45)(4,46,59,21)(5,22,60,47)(6,48,61,23)(7,24,62,41)(8,42,63,17)(9,54,39,31)(10,32,40,55)(11,56,33,25)(12,26,34,49)(13,50,35,27)(14,28,36,51)(15,52,37,29)(16,30,38,53), (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), (1,40)(2,37)(3,34)(4,39)(5,36)(6,33)(7,38)(8,35)(9,59)(10,64)(11,61)(12,58)(13,63)(14,60)(15,57)(16,62)(17,50)(18,55)(19,52)(20,49)(21,54)(22,51)(23,56)(24,53)(25,48)(26,45)(27,42)(28,47)(29,44)(30,41)(31,46)(32,43)>;
G:=Group( (1,51,64,28)(2,52,57,29)(3,53,58,30)(4,54,59,31)(5,55,60,32)(6,56,61,25)(7,49,62,26)(8,50,63,27)(9,21,39,46)(10,22,40,47)(11,23,33,48)(12,24,34,41)(13,17,35,42)(14,18,36,43)(15,19,37,44)(16,20,38,45), (1,18,64,43)(2,44,57,19)(3,20,58,45)(4,46,59,21)(5,22,60,47)(6,48,61,23)(7,24,62,41)(8,42,63,17)(9,54,39,31)(10,32,40,55)(11,56,33,25)(12,26,34,49)(13,50,35,27)(14,28,36,51)(15,52,37,29)(16,30,38,53), (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), (1,40)(2,37)(3,34)(4,39)(5,36)(6,33)(7,38)(8,35)(9,59)(10,64)(11,61)(12,58)(13,63)(14,60)(15,57)(16,62)(17,50)(18,55)(19,52)(20,49)(21,54)(22,51)(23,56)(24,53)(25,48)(26,45)(27,42)(28,47)(29,44)(30,41)(31,46)(32,43) );
G=PermutationGroup([[(1,51,64,28),(2,52,57,29),(3,53,58,30),(4,54,59,31),(5,55,60,32),(6,56,61,25),(7,49,62,26),(8,50,63,27),(9,21,39,46),(10,22,40,47),(11,23,33,48),(12,24,34,41),(13,17,35,42),(14,18,36,43),(15,19,37,44),(16,20,38,45)], [(1,18,64,43),(2,44,57,19),(3,20,58,45),(4,46,59,21),(5,22,60,47),(6,48,61,23),(7,24,62,41),(8,42,63,17),(9,54,39,31),(10,32,40,55),(11,56,33,25),(12,26,34,49),(13,50,35,27),(14,28,36,51),(15,52,37,29),(16,30,38,53)], [(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)], [(1,40),(2,37),(3,34),(4,39),(5,36),(6,33),(7,38),(8,35),(9,59),(10,64),(11,61),(12,58),(13,63),(14,60),(15,57),(16,62),(17,50),(18,55),(19,52),(20,49),(21,54),(22,51),(23,56),(24,53),(25,48),(26,45),(27,42),(28,47),(29,44),(30,41),(31,46),(32,43)]])
50 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 4A | 4B | 4C | 4D | 4E | ··· | 4T | 4U | 4V | 4W | 8A | ··· | 8H | 8I | ··· | 8T |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 4 | 4 | 4 | 8 | ··· | 8 | 8 | ··· | 8 |
| size | 1 | 1 | 1 | 1 | 4 | 4 | 4 | 1 | 1 | 1 | 1 | 2 | ··· | 2 | 4 | 4 | 4 | 2 | ··· | 2 | 4 | ··· | 4 |
50 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | - | ||||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C4 | C4 | C4 | C4 | M4(2) | C8○D4 | 2+ 1+4 | 2- 1+4 |
| kernel | Q8⋊7M4(2) | C4⋊M4(2) | C42.12C4 | C8⋊6D4 | C8×Q8 | C4×C4○D4 | C42⋊C2 | C4×D4 | C4×Q8 | C2×C4○D4 | Q8 | C4 | C4 | C4 |
| # reps | 1 | 3 | 3 | 6 | 2 | 1 | 6 | 6 | 2 | 2 | 8 | 8 | 1 | 1 |
Matrix representation of Q8⋊7M4(2) ►in GL4(𝔽17) generated by
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 13 | 0 |
| 0 | 0 | 13 | 4 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 4 | 9 |
| 0 | 0 | 0 | 13 |
| 16 | 2 | 0 | 0 |
| 10 | 1 | 0 | 0 |
| 0 | 0 | 2 | 0 |
| 0 | 0 | 2 | 15 |
| 16 | 0 | 0 | 0 |
| 16 | 1 | 0 | 0 |
| 0 | 0 | 1 | 15 |
| 0 | 0 | 0 | 16 |
G:=sub<GL(4,GF(17))| [1,0,0,0,0,1,0,0,0,0,13,13,0,0,0,4],[1,0,0,0,0,1,0,0,0,0,4,0,0,0,9,13],[16,10,0,0,2,1,0,0,0,0,2,2,0,0,0,15],[16,16,0,0,0,1,0,0,0,0,1,0,0,0,15,16] >;
Q8⋊7M4(2) in GAP, Magma, Sage, TeX
Q_8\rtimes_7M_4(2)
% in TeX
G:=Group("Q8:7M4(2)"); // GroupNames label
G:=SmallGroup(128,1723);
// by ID
G=gap.SmallGroup(128,1723);
# by ID
G:=PCGroup([7,-2,2,2,2,-2,2,-2,224,253,1430,891,675,80,124]);
// Polycyclic
G:=Group<a,b,c,d|a^4=c^8=d^2=1,b^2=a^2,b*a*b^-1=d*a*d=a^-1,a*c=c*a,c*b*c^-1=a^2*b,b*d=d*b,d*c*d=c^5>;
// generators/relations