p-group, metabelian, nilpotent (class 3), monomial
Aliases: M5(2)⋊5C4, (C2×C8).2C8, C4.9(C4⋊C8), C8.34(C4⋊C4), (C2×C8).27Q8, (C2×C8).374D4, (C22×C4).4C8, C22.3(C4×C8), C4.3(C8⋊C4), C22.4(C4⋊C8), (C22×C8).15C4, C23.28(C2×C8), (C2×C4).54C42, (C2×C42).12C4, C4.31(C22⋊C8), C8.50(C22⋊C4), C2.3(C23.C8), (C2×M5(2)).8C2, (C2×C4).70M4(2), C22.38(C22⋊C8), (C22×C8).368C22, C4.27(C2.C42), C2.15(C22.7C42), (C2×C4).75(C2×C8), (C2×C8).116(C2×C4), (C2×C8⋊C4).18C2, (C2×C4).105(C4⋊C4), (C22×C4).468(C2×C4), (C2×C4).385(C22⋊C4), SmallGroup(128,109)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for M5(2)⋊C4
G = < a,b,c | a16=b2=c4=1, bab=a9, cac-1=ab, bc=cb >
Subgroups: 104 in 70 conjugacy classes, 44 normal (24 characteristic)
C1, C2, C2, C2, C4, C4, C22, C22, C8, C8, C2×C4, C2×C4, C23, C16, C42, C2×C8, C2×C8, C2×C8, C22×C4, C22×C4, C8⋊C4, C2×C16, M5(2), M5(2), C2×C42, C22×C8, C2×C8⋊C4, C2×M5(2), M5(2)⋊C4
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, C23.C8, M5(2)⋊C4
►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 50)(2 59)(3 52)(4 61)(5 54)(6 63)(7 56)(8 49)(9 58)(10 51)(11 60)(12 53)(13 62)(14 55)(15 64)(16 57)(17 42)(18 35)(19 44)(20 37)(21 46)(22 39)(23 48)(24 41)(25 34)(26 43)(27 36)(28 45)(29 38)(30 47)(31 40)(32 33)
(1 36 50 27)(2 20 51 45)(3 46 52 21)(4 30 53 39)(5 40 54 31)(6 24 55 33)(7 34 56 25)(8 18 57 43)(9 44 58 19)(10 28 59 37)(11 38 60 29)(12 22 61 47)(13 48 62 23)(14 32 63 41)(15 42 64 17)(16 26 49 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), (1,50)(2,59)(3,52)(4,61)(5,54)(6,63)(7,56)(8,49)(9,58)(10,51)(11,60)(12,53)(13,62)(14,55)(15,64)(16,57)(17,42)(18,35)(19,44)(20,37)(21,46)(22,39)(23,48)(24,41)(25,34)(26,43)(27,36)(28,45)(29,38)(30,47)(31,40)(32,33), (1,36,50,27)(2,20,51,45)(3,46,52,21)(4,30,53,39)(5,40,54,31)(6,24,55,33)(7,34,56,25)(8,18,57,43)(9,44,58,19)(10,28,59,37)(11,38,60,29)(12,22,61,47)(13,48,62,23)(14,32,63,41)(15,42,64,17)(16,26,49,35)>;
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), (1,50)(2,59)(3,52)(4,61)(5,54)(6,63)(7,56)(8,49)(9,58)(10,51)(11,60)(12,53)(13,62)(14,55)(15,64)(16,57)(17,42)(18,35)(19,44)(20,37)(21,46)(22,39)(23,48)(24,41)(25,34)(26,43)(27,36)(28,45)(29,38)(30,47)(31,40)(32,33), (1,36,50,27)(2,20,51,45)(3,46,52,21)(4,30,53,39)(5,40,54,31)(6,24,55,33)(7,34,56,25)(8,18,57,43)(9,44,58,19)(10,28,59,37)(11,38,60,29)(12,22,61,47)(13,48,62,23)(14,32,63,41)(15,42,64,17)(16,26,49,35) );
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)], [(1,50),(2,59),(3,52),(4,61),(5,54),(6,63),(7,56),(8,49),(9,58),(10,51),(11,60),(12,53),(13,62),(14,55),(15,64),(16,57),(17,42),(18,35),(19,44),(20,37),(21,46),(22,39),(23,48),(24,41),(25,34),(26,43),(27,36),(28,45),(29,38),(30,47),(31,40),(32,33)], [(1,36,50,27),(2,20,51,45),(3,46,52,21),(4,30,53,39),(5,40,54,31),(6,24,55,33),(7,34,56,25),(8,18,57,43),(9,44,58,19),(10,28,59,37),(11,38,60,29),(12,22,61,47),(13,48,62,23),(14,32,63,41),(15,42,64,17),(16,26,49,35)]])
44 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 | 16A | ··· | 16P |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 8 | ··· | 8 | 8 | 8 | 8 | 8 | 16 | ··· | 16 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 1 | 1 | 1 | 1 | 2 | 2 | 4 | 4 | 4 | 4 | 2 | ··· | 2 | 4 | 4 | 4 | 4 | 4 | ··· | 4 |
44 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 4 |
| type | + | + | + | + | - | |||||||
| image | C1 | C2 | C2 | C4 | C4 | C4 | C8 | C8 | D4 | Q8 | M4(2) | C23.C8 |
| kernel | M5(2)⋊C4 | C2×C8⋊C4 | C2×M5(2) | M5(2) | C2×C42 | C22×C8 | C2×C8 | C22×C4 | C2×C8 | C2×C8 | C2×C4 | C2 |
| # reps | 1 | 1 | 2 | 8 | 2 | 2 | 8 | 8 | 3 | 1 | 4 | 4 |
Matrix representation of M5(2)⋊C4 ►in GL6(𝔽17)
| 13 | 11 | 0 | 0 | 0 | 0 |
| 11 | 4 | 0 | 0 | 0 | 0 |
| 0 | 0 | 9 | 1 | 0 | 12 |
| 0 | 0 | 15 | 8 | 3 | 2 |
| 0 | 0 | 4 | 3 | 8 | 12 |
| 0 | 0 | 9 | 7 | 14 | 9 |
| 16 | 0 | 0 | 0 | 0 | 0 |
| 0 | 16 | 0 | 0 | 0 | 0 |
| 0 | 0 | 16 | 0 | 6 | 5 |
| 0 | 0 | 0 | 16 | 15 | 8 |
| 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 | 16 | 3 | 15 | 4 |
| 0 | 0 | 5 | 1 | 7 | 10 |
| 0 | 0 | 0 | 0 | 10 | 10 |
| 0 | 0 | 0 | 0 | 12 | 7 |
G:=sub<GL(6,GF(17))| [13,11,0,0,0,0,11,4,0,0,0,0,0,0,9,15,4,9,0,0,1,8,3,7,0,0,0,3,8,14,0,0,12,2,12,9],[16,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,6,15,1,0,0,0,5,8,0,1],[0,16,0,0,0,0,1,0,0,0,0,0,0,0,16,5,0,0,0,0,3,1,0,0,0,0,15,7,10,12,0,0,4,10,10,7] >;
M5(2)⋊C4 in GAP, Magma, Sage, TeX
M_5(2)\rtimes C_4
% in TeX
G:=Group("M5(2):C4"); // GroupNames label
G:=SmallGroup(128,109);
// by ID
G=gap.SmallGroup(128,109);
# by ID
G:=PCGroup([7,-2,2,-2,2,2,-2,-2,56,85,120,1430,1018,136,124]);
// Polycyclic
G:=Group<a,b,c|a^16=b^2=c^4=1,b*a*b=a^9,c*a*c^-1=a*b,b*c=c*b>;
// generators/relations