p-group, metabelian, nilpotent (class 3), monomial
Aliases: M5(2)⋊7C4, M4(2).3C8, C23.19M4(2), C4⋊C4.6C8, C4.8(C4×C8), (C2×C16)⋊10C4, C8.41(C4⋊C4), C4.17(C4⋊C8), (C2×C8).57Q8, (C2×C8).376D4, (C2×C4).56C42, C2.3(D4.C8), (C22×C16).2C2, C22.11(C4⋊C8), C4.32(C22⋊C8), C8.51(C22⋊C4), (C2×C4).40M4(2), C42⋊C2.14C4, C22.7(C8⋊C4), (C2×M5(2)).10C2, (C2×M4(2)).20C4, C8○2M4(2).13C2, C22.39(C22⋊C8), (C22×C8).572C22, C4.29(C2.C42), C2.17(C22.7C42), (C2×C4).47(C2×C8), (C2×C8).239(C2×C4), (C2×C4).107(C4⋊C4), (C22×C4).391(C2×C4), (C2×C4).386(C22⋊C4), SmallGroup(128,111)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for M5(2)⋊7C4
G = < a,b,c | a16=b2=c4=1, bab=a9, cac-1=a13b, cbc-1=a8b >
Subgroups: 96 in 68 conjugacy classes, 44 normal (34 characteristic)
C1, C2, C2, C2, C4, C4, C22, C22, C8, C8, C2×C4, C2×C4, C23, C16, C42, C22⋊C4, C4⋊C4, C2×C8, C2×C8, M4(2), M4(2), C22×C4, C4×C8, C8⋊C4, C2×C16, C2×C16, M5(2), M5(2), C42⋊C2, C22×C8, C2×M4(2), C8○2M4(2), C22×C16, C2×M5(2), M5(2)⋊7C4
Quotients: C1, C2, C4, C22, C8, C2×C4, D4, Q8, C42, C22⋊C4, C4⋊C4, C2×C8, M4(2), C2.C42, C4×C8, C8⋊C4, C22⋊C8, C4⋊C8, C22.7C42, D4.C8, M5(2)⋊7C4
►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)
(2 10)(4 12)(6 14)(8 16)(18 26)(20 28)(22 30)(24 32)(34 42)(36 44)(38 46)(40 48)(50 58)(52 60)(54 62)(56 64)
(1 24 45 54)(2 21 46 51)(3 26 47 56)(4 23 48 53)(5 28 33 58)(6 25 34 55)(7 30 35 60)(8 27 36 57)(9 32 37 62)(10 29 38 59)(11 18 39 64)(12 31 40 61)(13 20 41 50)(14 17 42 63)(15 22 43 52)(16 19 44 49)
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), (2,10)(4,12)(6,14)(8,16)(18,26)(20,28)(22,30)(24,32)(34,42)(36,44)(38,46)(40,48)(50,58)(52,60)(54,62)(56,64), (1,24,45,54)(2,21,46,51)(3,26,47,56)(4,23,48,53)(5,28,33,58)(6,25,34,55)(7,30,35,60)(8,27,36,57)(9,32,37,62)(10,29,38,59)(11,18,39,64)(12,31,40,61)(13,20,41,50)(14,17,42,63)(15,22,43,52)(16,19,44,49)>;
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), (2,10)(4,12)(6,14)(8,16)(18,26)(20,28)(22,30)(24,32)(34,42)(36,44)(38,46)(40,48)(50,58)(52,60)(54,62)(56,64), (1,24,45,54)(2,21,46,51)(3,26,47,56)(4,23,48,53)(5,28,33,58)(6,25,34,55)(7,30,35,60)(8,27,36,57)(9,32,37,62)(10,29,38,59)(11,18,39,64)(12,31,40,61)(13,20,41,50)(14,17,42,63)(15,22,43,52)(16,19,44,49) );
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)], [(2,10),(4,12),(6,14),(8,16),(18,26),(20,28),(22,30),(24,32),(34,42),(36,44),(38,46),(40,48),(50,58),(52,60),(54,62),(56,64)], [(1,24,45,54),(2,21,46,51),(3,26,47,56),(4,23,48,53),(5,28,33,58),(6,25,34,55),(7,30,35,60),(8,27,36,57),(9,32,37,62),(10,29,38,59),(11,18,39,64),(12,31,40,61),(13,20,41,50),(14,17,42,63),(15,22,43,52),(16,19,44,49)]])
56 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 8A | ··· | 8H | 8I | 8J | 8K | 8L | 8M | 8N | 8O | 8P | 16A | ··· | 16P | 16Q | ··· | 16X |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 8 | ··· | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 16 | ··· | 16 | 16 | ··· | 16 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 1 | 1 | 1 | 1 | 2 | 2 | 4 | 4 | 4 | 4 | 1 | ··· | 1 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 2 | ··· | 2 | 4 | ··· | 4 |
56 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | - | |||||||||
| image | C1 | C2 | C2 | C2 | C4 | C4 | C4 | C4 | C8 | C8 | D4 | Q8 | M4(2) | M4(2) | D4.C8 |
| kernel | M5(2)⋊7C4 | C8○2M4(2) | C22×C16 | C2×M5(2) | C2×C16 | M5(2) | C42⋊C2 | C2×M4(2) | C4⋊C4 | M4(2) | C2×C8 | C2×C8 | C2×C4 | C23 | C2 |
| # reps | 1 | 1 | 1 | 1 | 4 | 4 | 2 | 2 | 8 | 8 | 3 | 1 | 2 | 2 | 16 |
Matrix representation of M5(2)⋊7C4 ►in GL3(𝔽17) generated by
| 9 | 0 | 0 |
| 0 | 0 | 11 |
| 0 | 7 | 0 |
| 16 | 0 | 0 |
| 0 | 1 | 0 |
| 0 | 0 | 16 |
| 4 | 0 | 0 |
| 0 | 0 | 1 |
| 0 | 16 | 0 |
G:=sub<GL(3,GF(17))| [9,0,0,0,0,7,0,11,0],[16,0,0,0,1,0,0,0,16],[4,0,0,0,0,16,0,1,0] >;
M5(2)⋊7C4 in GAP, Magma, Sage, TeX
M_5(2)\rtimes_7C_4
% in TeX
G:=Group("M5(2):7C4"); // GroupNames label
G:=SmallGroup(128,111);
// by ID
G=gap.SmallGroup(128,111);
# by ID
G:=PCGroup([7,-2,2,-2,2,2,-2,-2,56,85,120,723,352,1018,136,124]);
// Polycyclic
G:=Group<a,b,c|a^16=b^2=c^4=1,b*a*b=a^9,c*a*c^-1=a^13*b,c*b*c^-1=a^8*b>;
// generators/relations