p-group, metacyclic, nilpotent (class 2), monomial
Aliases: M7(2), C4.C32, C64⋊3C2, C32.2C4, C16.5C8, C8.3C16, C22.C32, C32.8C22, C8.29(C2×C8), C2.3(C2×C32), (C2×C4).5C16, (C2×C8).16C8, (C2×C32).8C2, C4.13(C2×C16), (C2×C16).18C4, C16.25(C2×C4), SmallGroup(128,160)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
| C1 — C2 — C2 — C2 — C2 — C2 — C2 — C2 — C2 — C2 — C2 — C2 — C2 — C2 — C2 — C2 — C2 — C4 — C4 — C4 — C4 — C4 — C4 — C4 — C4 — C8 — C8 — C8 — C8 — C16 — C16 — C32 — M7(2) |
Generators and relations for M7(2)
G = < a,b | a64=b2=1, bab=a33 >
Smallest permutation representation of M7(2)
►On 64 points
Generators in S64
(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 34)(4 36)(6 38)(8 40)(10 42)(12 44)(14 46)(16 48)(18 50)(20 52)(22 54)(24 56)(26 58)(28 60)(30 62)(32 64)
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), (2,34)(4,36)(6,38)(8,40)(10,42)(12,44)(14,46)(16,48)(18,50)(20,52)(22,54)(24,56)(26,58)(28,60)(30,62)(32,64)>;
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), (2,34)(4,36)(6,38)(8,40)(10,42)(12,44)(14,46)(16,48)(18,50)(20,52)(22,54)(24,56)(26,58)(28,60)(30,62)(32,64) );
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)], [(2,34),(4,36),(6,38),(8,40),(10,42),(12,44),(14,46),(16,48),(18,50),(20,52),(22,54),(24,56),(26,58),(28,60),(30,62),(32,64)]])
80 conjugacy classes
| class | 1 | 2A | 2B | 4A | 4B | 4C | 8A | 8B | 8C | 8D | 8E | 8F | 16A | ··· | 16H | 16I | 16J | 16K | 16L | 32A | ··· | 32P | 32Q | ··· | 32X | 64A | ··· | 64AF |
| order | 1 | 2 | 2 | 4 | 4 | 4 | 8 | 8 | 8 | 8 | 8 | 8 | 16 | ··· | 16 | 16 | 16 | 16 | 16 | 32 | ··· | 32 | 32 | ··· | 32 | 64 | ··· | 64 |
| size | 1 | 1 | 2 | 1 | 1 | 2 | 1 | 1 | 1 | 1 | 2 | 2 | 1 | ··· | 1 | 2 | 2 | 2 | 2 | 1 | ··· | 1 | 2 | ··· | 2 | 2 | ··· | 2 |
80 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 |
| type | + | + | + | |||||||||
| image | C1 | C2 | C2 | C4 | C4 | C8 | C8 | C16 | C16 | C32 | C32 | M7(2) |
| kernel | M7(2) | C64 | C2×C32 | C32 | C2×C16 | C16 | C2×C8 | C8 | C2×C4 | C4 | C22 | C1 |
| # reps | 1 | 2 | 1 | 2 | 2 | 4 | 4 | 8 | 8 | 16 | 16 | 16 |
Matrix representation of M7(2) ►in GL2(𝔽193) generated by
| 9 | 132 |
| 46 | 184 |
| 1 | 8 |
| 0 | 192 |
G:=sub<GL(2,GF(193))| [9,46,132,184],[1,0,8,192] >;
M7(2) in GAP, Magma, Sage, TeX
M_7(2)
% in TeX
G:=Group("M7(2)"); // GroupNames label
G:=SmallGroup(128,160);
// by ID
G=gap.SmallGroup(128,160);
# by ID
G:=PCGroup([7,-2,2,-2,-2,-2,-2,-2,28,925,58,80,102,124]);
// Polycyclic
G:=Group<a,b|a^64=b^2=1,b*a*b=a^33>;
// generators/relations
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