p-group, metabelian, nilpotent (class 3), monomial
Aliases: M4(2).33D4, C4.93(C4×D4), C4⋊C4.224D4, (C2×Q16)⋊13C4, (C2×C8).116D4, C22.64(C4×D4), C4.89(C4⋊D4), C4.22(C4⋊1D4), C8.23(C22⋊C4), C2.5(D4.5D4), C8○2M4(2).5C2, (C22×Q16).12C2, C23.271(C4○D4), (C22×C8).228C22, (C22×C4).705C23, C23.38D4.6C2, (C22×Q8).43C22, C22.151(C4⋊D4), C22.20(C4.4D4), C42⋊C2.280C22, (C2×M4(2)).215C22, C2.31(C24.3C22), (C2×C8).78(C2×C4), (C2×C4).35(C2×D4), C4.47(C2×C22⋊C4), (C2×C4).69(C4○D4), (C2×Q8).102(C2×C4), (C2×C8.C4).19C2, (C2×C4).203(C22×C4), (C2×C4.10D4).10C2, SmallGroup(128,711)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for M4(2).33D4
G = < a,b,c,d | a8=b2=1, c4=a4, d2=a2b, bab=a5, ac=ca, dad-1=ab, bc=cb, dbd-1=a4b, dcd-1=a4c3 >
Subgroups: 252 in 138 conjugacy classes, 56 normal (26 characteristic)
C1, C2, C2, C2, C4, C4, C22, C22, C8, C8, C2×C4, C2×C4, Q8, C23, C42, C22⋊C4, C4⋊C4, C2×C8, C2×C8, C2×C8, M4(2), M4(2), Q16, C22×C4, C22×C4, C2×Q8, C2×Q8, C4×C8, C8⋊C4, C4.10D4, Q8⋊C4, C8.C4, C42⋊C2, C22×C8, C2×M4(2), C2×M4(2), C2×Q16, C2×Q16, C22×Q8, C8○2M4(2), C2×C4.10D4, C23.38D4, C2×C8.C4, C22×Q16, M4(2).33D4
Quotients: C1, C2, C4, C22, C2×C4, D4, C23, C22⋊C4, C22×C4, C2×D4, C4○D4, C2×C22⋊C4, C4×D4, C4⋊D4, C4.4D4, C4⋊1D4, C24.3C22, D4.5D4, M4(2).33D4
►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 31)(2 28)(3 25)(4 30)(5 27)(6 32)(7 29)(8 26)(9 33)(10 38)(11 35)(12 40)(13 37)(14 34)(15 39)(16 36)(17 47)(18 44)(19 41)(20 46)(21 43)(22 48)(23 45)(24 42)(49 59)(50 64)(51 61)(52 58)(53 63)(54 60)(55 57)(56 62)
(1 40 25 14 5 36 29 10)(2 33 26 15 6 37 30 11)(3 34 27 16 7 38 31 12)(4 35 28 9 8 39 32 13)(17 49 45 57 21 53 41 61)(18 50 46 58 22 54 42 62)(19 51 47 59 23 55 43 63)(20 52 48 60 24 56 44 64)
(1 23 25 47 5 19 29 43)(2 46 30 22 6 42 26 18)(3 21 27 45 7 17 31 41)(4 44 32 20 8 48 28 24)(9 60 35 56 13 64 39 52)(10 55 40 59 14 51 36 63)(11 58 37 54 15 62 33 50)(12 53 34 57 16 49 38 61)
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), (1,31)(2,28)(3,25)(4,30)(5,27)(6,32)(7,29)(8,26)(9,33)(10,38)(11,35)(12,40)(13,37)(14,34)(15,39)(16,36)(17,47)(18,44)(19,41)(20,46)(21,43)(22,48)(23,45)(24,42)(49,59)(50,64)(51,61)(52,58)(53,63)(54,60)(55,57)(56,62), (1,40,25,14,5,36,29,10)(2,33,26,15,6,37,30,11)(3,34,27,16,7,38,31,12)(4,35,28,9,8,39,32,13)(17,49,45,57,21,53,41,61)(18,50,46,58,22,54,42,62)(19,51,47,59,23,55,43,63)(20,52,48,60,24,56,44,64), (1,23,25,47,5,19,29,43)(2,46,30,22,6,42,26,18)(3,21,27,45,7,17,31,41)(4,44,32,20,8,48,28,24)(9,60,35,56,13,64,39,52)(10,55,40,59,14,51,36,63)(11,58,37,54,15,62,33,50)(12,53,34,57,16,49,38,61)>;
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), (1,31)(2,28)(3,25)(4,30)(5,27)(6,32)(7,29)(8,26)(9,33)(10,38)(11,35)(12,40)(13,37)(14,34)(15,39)(16,36)(17,47)(18,44)(19,41)(20,46)(21,43)(22,48)(23,45)(24,42)(49,59)(50,64)(51,61)(52,58)(53,63)(54,60)(55,57)(56,62), (1,40,25,14,5,36,29,10)(2,33,26,15,6,37,30,11)(3,34,27,16,7,38,31,12)(4,35,28,9,8,39,32,13)(17,49,45,57,21,53,41,61)(18,50,46,58,22,54,42,62)(19,51,47,59,23,55,43,63)(20,52,48,60,24,56,44,64), (1,23,25,47,5,19,29,43)(2,46,30,22,6,42,26,18)(3,21,27,45,7,17,31,41)(4,44,32,20,8,48,28,24)(9,60,35,56,13,64,39,52)(10,55,40,59,14,51,36,63)(11,58,37,54,15,62,33,50)(12,53,34,57,16,49,38,61) );
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)], [(1,31),(2,28),(3,25),(4,30),(5,27),(6,32),(7,29),(8,26),(9,33),(10,38),(11,35),(12,40),(13,37),(14,34),(15,39),(16,36),(17,47),(18,44),(19,41),(20,46),(21,43),(22,48),(23,45),(24,42),(49,59),(50,64),(51,61),(52,58),(53,63),(54,60),(55,57),(56,62)], [(1,40,25,14,5,36,29,10),(2,33,26,15,6,37,30,11),(3,34,27,16,7,38,31,12),(4,35,28,9,8,39,32,13),(17,49,45,57,21,53,41,61),(18,50,46,58,22,54,42,62),(19,51,47,59,23,55,43,63),(20,52,48,60,24,56,44,64)], [(1,23,25,47,5,19,29,43),(2,46,30,22,6,42,26,18),(3,21,27,45,7,17,31,41),(4,44,32,20,8,48,28,24),(9,60,35,56,13,64,39,52),(10,55,40,59,14,51,36,63),(11,58,37,54,15,62,33,50),(12,53,34,57,16,49,38,61)]])
32 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 4K | 4L | 8A | 8B | 8C | 8D | 8E | ··· | 8J | 8K | 8L | 8M | 8N |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 8 | 8 | 8 | 8 | 8 | ··· | 8 | 8 | 8 | 8 | 8 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 8 | 8 | 8 | 8 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 8 | 8 | 8 | 8 |
32 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 4 |
| type | + | + | + | + | + | + | + | + | + | - | |||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C4 | D4 | D4 | D4 | C4○D4 | C4○D4 | D4.5D4 |
| kernel | M4(2).33D4 | C8○2M4(2) | C2×C4.10D4 | C23.38D4 | C2×C8.C4 | C22×Q16 | C2×Q16 | C4⋊C4 | C2×C8 | M4(2) | C2×C4 | C23 | C2 |
| # reps | 1 | 1 | 2 | 2 | 1 | 1 | 8 | 2 | 4 | 2 | 2 | 2 | 4 |
Matrix representation of M4(2).33D4 ►in GL6(𝔽17)
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 13 | 2 | 0 | 0 |
| 0 | 0 | 11 | 4 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 16 |
| 0 | 0 | 11 | 3 | 4 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 16 | 0 | 0 | 0 |
| 0 | 0 | 13 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 16 |
| 10 | 4 | 0 | 0 | 0 | 0 |
| 13 | 7 | 0 | 0 | 0 | 0 |
| 0 | 0 | 8 | 0 | 0 | 0 |
| 0 | 0 | 0 | 8 | 0 | 0 |
| 0 | 0 | 16 | 9 | 15 | 0 |
| 0 | 0 | 16 | 0 | 0 | 15 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 12 | 16 | 0 | 0 | 0 | 0 |
| 0 | 0 | 7 | 5 | 2 | 0 |
| 0 | 0 | 2 | 10 | 4 | 16 |
| 0 | 0 | 11 | 0 | 0 | 11 |
| 0 | 0 | 9 | 4 | 10 | 0 |
G:=sub<GL(6,GF(17))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,13,11,0,11,0,0,2,4,0,3,0,0,0,0,0,4,0,0,0,0,16,0],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,16,13,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,16],[10,13,0,0,0,0,4,7,0,0,0,0,0,0,8,0,16,16,0,0,0,8,9,0,0,0,0,0,15,0,0,0,0,0,0,15],[1,12,0,0,0,0,0,16,0,0,0,0,0,0,7,2,11,9,0,0,5,10,0,4,0,0,2,4,0,10,0,0,0,16,11,0] >;
M4(2).33D4 in GAP, Magma, Sage, TeX
M_4(2)._{33}D_4 % in TeX
G:=Group("M4(2).33D4"); // GroupNames label
G:=SmallGroup(128,711);
// by ID
G=gap.SmallGroup(128,711);
# by ID
G:=PCGroup([7,-2,2,2,-2,2,2,-2,224,141,736,422,436,2019,1018,2804,172,2028,124]);
// Polycyclic
G:=Group<a,b,c,d|a^8=b^2=1,c^4=a^4,d^2=a^2*b,b*a*b=a^5,a*c=c*a,d*a*d^-1=a*b,b*c=c*b,d*b*d^-1=a^4*b,d*c*d^-1=a^4*c^3>;
// generators/relations