p-group, metabelian, nilpotent (class 3), monomial
Aliases: C42.104D4, C8⋊C4.15C4, (C22×C4).45Q8, C23.82(C2×Q8), C42.143(C2×C4), C4.83(C4.4D4), C2.11(C42⋊8C4), C4.61(C42⋊C2), C4⋊M4(2).28C2, C4.C42.10C2, (C22×C8).386C22, (C2×C42).256C22, C2.9(M4(2).C4), C22.2(C42.C2), (C22×C4).1342C23, (C2×M4(2)).168C22, (C2×C4).46(C4⋊C4), (C2×C8).145(C2×C4), (C2×C8⋊C4).29C2, (C2×C4).1522(C2×D4), C22.100(C2×C4⋊C4), (C2×C4).560(C4○D4), (C2×C4).540(C22×C4), SmallGroup(128,570)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C42.104D4
G = < a,b,c,d | a4=b4=1, c4=b2, d2=a2b-1, ab=ba, cac-1=ab2, dad-1=a-1, bc=cb, bd=db, dcd-1=a2c3 >
Subgroups: 140 in 90 conjugacy classes, 52 normal (10 characteristic)
C1, C2, C2, C2, C4, C4, C22, C22, C22, C8, C2×C4, C2×C4, C2×C4, C23, C42, C2×C8, C2×C8, M4(2), C22×C4, C22×C4, C8⋊C4, C4⋊C8, C2×C42, C22×C8, C2×M4(2), C4.C42, C2×C8⋊C4, C4⋊M4(2), C42.104D4
Quotients: C1, C2, C4, C22, C2×C4, D4, Q8, C23, C4⋊C4, C22×C4, C2×D4, C2×Q8, C4○D4, C2×C4⋊C4, C42⋊C2, C4.4D4, C42.C2, C42⋊8C4, M4(2).C4, C42.104D4
►On 64 points
(1 38 31 15)(2 35 32 12)(3 40 25 9)(4 37 26 14)(5 34 27 11)(6 39 28 16)(7 36 29 13)(8 33 30 10)(17 64 48 55)(18 61 41 52)(19 58 42 49)(20 63 43 54)(21 60 44 51)(22 57 45 56)(23 62 46 53)(24 59 47 50)
(1 25 5 29)(2 26 6 30)(3 27 7 31)(4 28 8 32)(9 34 13 38)(10 35 14 39)(11 36 15 40)(12 37 16 33)(17 46 21 42)(18 47 22 43)(19 48 23 44)(20 41 24 45)(49 64 53 60)(50 57 54 61)(51 58 55 62)(52 59 56 63)
(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 58 7 60 5 62 3 64)(2 52 8 54 6 56 4 50)(9 48 15 42 13 44 11 46)(10 20 16 22 14 24 12 18)(17 38 19 36 21 34 23 40)(25 55 31 49 29 51 27 53)(26 59 32 61 30 63 28 57)(33 43 39 45 37 47 35 41)
G:=sub<Sym(64)| (1,38,31,15)(2,35,32,12)(3,40,25,9)(4,37,26,14)(5,34,27,11)(6,39,28,16)(7,36,29,13)(8,33,30,10)(17,64,48,55)(18,61,41,52)(19,58,42,49)(20,63,43,54)(21,60,44,51)(22,57,45,56)(23,62,46,53)(24,59,47,50), (1,25,5,29)(2,26,6,30)(3,27,7,31)(4,28,8,32)(9,34,13,38)(10,35,14,39)(11,36,15,40)(12,37,16,33)(17,46,21,42)(18,47,22,43)(19,48,23,44)(20,41,24,45)(49,64,53,60)(50,57,54,61)(51,58,55,62)(52,59,56,63), (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), (1,58,7,60,5,62,3,64)(2,52,8,54,6,56,4,50)(9,48,15,42,13,44,11,46)(10,20,16,22,14,24,12,18)(17,38,19,36,21,34,23,40)(25,55,31,49,29,51,27,53)(26,59,32,61,30,63,28,57)(33,43,39,45,37,47,35,41)>;
G:=Group( (1,38,31,15)(2,35,32,12)(3,40,25,9)(4,37,26,14)(5,34,27,11)(6,39,28,16)(7,36,29,13)(8,33,30,10)(17,64,48,55)(18,61,41,52)(19,58,42,49)(20,63,43,54)(21,60,44,51)(22,57,45,56)(23,62,46,53)(24,59,47,50), (1,25,5,29)(2,26,6,30)(3,27,7,31)(4,28,8,32)(9,34,13,38)(10,35,14,39)(11,36,15,40)(12,37,16,33)(17,46,21,42)(18,47,22,43)(19,48,23,44)(20,41,24,45)(49,64,53,60)(50,57,54,61)(51,58,55,62)(52,59,56,63), (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), (1,58,7,60,5,62,3,64)(2,52,8,54,6,56,4,50)(9,48,15,42,13,44,11,46)(10,20,16,22,14,24,12,18)(17,38,19,36,21,34,23,40)(25,55,31,49,29,51,27,53)(26,59,32,61,30,63,28,57)(33,43,39,45,37,47,35,41) );
G=PermutationGroup([[(1,38,31,15),(2,35,32,12),(3,40,25,9),(4,37,26,14),(5,34,27,11),(6,39,28,16),(7,36,29,13),(8,33,30,10),(17,64,48,55),(18,61,41,52),(19,58,42,49),(20,63,43,54),(21,60,44,51),(22,57,45,56),(23,62,46,53),(24,59,47,50)], [(1,25,5,29),(2,26,6,30),(3,27,7,31),(4,28,8,32),(9,34,13,38),(10,35,14,39),(11,36,15,40),(12,37,16,33),(17,46,21,42),(18,47,22,43),(19,48,23,44),(20,41,24,45),(49,64,53,60),(50,57,54,61),(51,58,55,62),(52,59,56,63)], [(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)], [(1,58,7,60,5,62,3,64),(2,52,8,54,6,56,4,50),(9,48,15,42,13,44,11,46),(10,20,16,22,14,24,12,18),(17,38,19,36,21,34,23,40),(25,55,31,49,29,51,27,53),(26,59,32,61,30,63,28,57),(33,43,39,45,37,47,35,41)]])
32 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 8A | ··· | 8H | 8I | ··· | 8P |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 8 | ··· | 8 | 8 | ··· | 8 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 1 | 1 | 1 | 1 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 8 | ··· | 8 |
32 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 4 |
| type | + | + | + | + | + | - | |||
| image | C1 | C2 | C2 | C2 | C4 | D4 | Q8 | C4○D4 | M4(2).C4 |
| kernel | C42.104D4 | C4.C42 | C2×C8⋊C4 | C4⋊M4(2) | C8⋊C4 | C42 | C22×C4 | C2×C4 | C2 |
| # reps | 1 | 4 | 1 | 2 | 8 | 2 | 2 | 8 | 4 |
Matrix representation of C42.104D4 ►in GL6(𝔽17)
| 13 | 4 | 0 | 0 | 0 | 0 |
| 0 | 4 | 0 | 0 | 0 | 0 |
| 0 | 0 | 12 | 14 | 0 | 0 |
| 0 | 0 | 3 | 5 | 0 | 0 |
| 0 | 0 | 4 | 4 | 12 | 10 |
| 0 | 0 | 13 | 8 | 11 | 5 |
| 16 | 0 | 0 | 0 | 0 | 0 |
| 0 | 16 | 0 | 0 | 0 | 0 |
| 0 | 0 | 13 | 0 | 0 | 0 |
| 0 | 0 | 0 | 13 | 0 | 0 |
| 0 | 0 | 0 | 0 | 13 | 0 |
| 0 | 0 | 0 | 0 | 0 | 13 |
| 13 | 0 | 0 | 0 | 0 | 0 |
| 0 | 13 | 0 | 0 | 0 | 0 |
| 0 | 0 | 6 | 10 | 0 | 0 |
| 0 | 0 | 7 | 11 | 0 | 0 |
| 0 | 0 | 12 | 6 | 10 | 3 |
| 0 | 0 | 9 | 16 | 5 | 7 |
| 4 | 13 | 0 | 0 | 0 | 0 |
| 8 | 13 | 0 | 0 | 0 | 0 |
| 0 | 0 | 2 | 13 | 3 | 6 |
| 0 | 0 | 8 | 1 | 12 | 10 |
| 0 | 0 | 3 | 1 | 14 | 15 |
| 0 | 0 | 11 | 10 | 0 | 0 |
G:=sub<GL(6,GF(17))| [13,0,0,0,0,0,4,4,0,0,0,0,0,0,12,3,4,13,0,0,14,5,4,8,0,0,0,0,12,11,0,0,0,0,10,5],[16,0,0,0,0,0,0,16,0,0,0,0,0,0,13,0,0,0,0,0,0,13,0,0,0,0,0,0,13,0,0,0,0,0,0,13],[13,0,0,0,0,0,0,13,0,0,0,0,0,0,6,7,12,9,0,0,10,11,6,16,0,0,0,0,10,5,0,0,0,0,3,7],[4,8,0,0,0,0,13,13,0,0,0,0,0,0,2,8,3,11,0,0,13,1,1,10,0,0,3,12,14,0,0,0,6,10,15,0] >;
C42.104D4 in GAP, Magma, Sage, TeX
C_4^2._{104}D_4 % in TeX
G:=Group("C4^2.104D4"); // GroupNames label
G:=SmallGroup(128,570);
// by ID
G=gap.SmallGroup(128,570);
# by ID
G:=PCGroup([7,-2,2,2,-2,2,2,-2,224,141,120,422,723,58,2019,248,2804,172,124]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^4=1,c^4=b^2,d^2=a^2*b^-1,a*b=b*a,c*a*c^-1=a*b^2,d*a*d^-1=a^-1,b*c=c*b,b*d=d*b,d*c*d^-1=a^2*c^3>;
// generators/relations