p-group, metabelian, nilpotent (class 3), monomial
Aliases: C42.53D4, C42.617C23, D4⋊C8⋊33C2, C4⋊Q8.12C4, C4.5(C8○D4), C4⋊D4.8C4, C4⋊1D4.9C4, (C4×C8).9C22, C42.67(C2×C4), (C4×D4).8C22, C4⋊C8.201C22, (C22×C4).207D4, C4.136(C8⋊C22), C42.6C4⋊28C2, C42.12C4⋊14C2, C23.49(C22⋊C4), (C2×C42).170C22, C2.6(C23.37D4), C22.26C24.4C2, C2.13(C42⋊C22), C4⋊C4.57(C2×C4), (C2×D4).55(C2×C4), (C2×C4).1453(C2×D4), (C2×C4).83(C22⋊C4), (C2×C4).322(C22×C4), (C22×C4).192(C2×C4), C22.172(C2×C22⋊C4), C2.22((C22×C8)⋊C2), SmallGroup(128,228)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C42.53D4
G = < a,b,c,d | a4=b4=1, c4=a2b2, d2=b, ab=ba, cac-1=a-1, dad-1=ab2, bc=cb, bd=db, dcd-1=a2b-1c3 >
Subgroups: 276 in 123 conjugacy classes, 46 normal (26 characteristic)
C1, C2, C2, C4, C4, C22, C22, C8, C2×C4, C2×C4, D4, Q8, C23, C23, C42, C22⋊C4, C4⋊C4, C4⋊C4, C2×C8, C22×C4, C22×C4, C2×D4, C2×D4, C2×Q8, C4○D4, C4×C8, C8⋊C4, C22⋊C8, C4⋊C8, C4⋊C8, C2×C42, C4×D4, C4×D4, C4⋊D4, C4⋊D4, C4.4D4, C4⋊1D4, C4⋊Q8, C2×C4○D4, D4⋊C8, C42.12C4, C42.6C4, C22.26C24, C42.53D4
Quotients: C1, C2, C4, C22, C2×C4, D4, C23, C22⋊C4, C22×C4, C2×D4, C2×C22⋊C4, C8○D4, C8⋊C22, (C22×C8)⋊C2, C23.37D4, C42⋊C22, C42.53D4
►On 64 points
(1 35 51 43)(2 44 52 36)(3 37 53 45)(4 46 54 38)(5 39 55 47)(6 48 56 40)(7 33 49 41)(8 42 50 34)(9 63 18 26)(10 27 19 64)(11 57 20 28)(12 29 21 58)(13 59 22 30)(14 31 23 60)(15 61 24 32)(16 25 17 62)
(1 28 55 61)(2 29 56 62)(3 30 49 63)(4 31 50 64)(5 32 51 57)(6 25 52 58)(7 26 53 59)(8 27 54 60)(9 45 22 33)(10 46 23 34)(11 47 24 35)(12 48 17 36)(13 41 18 37)(14 42 19 38)(15 43 20 39)(16 44 21 40)
(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 27 28 54 55 60 61 8)(2 53 29 59 56 7 62 26)(3 58 30 6 49 25 63 52)(4 5 31 32 50 51 64 57)(9 40 45 16 22 44 33 21)(10 15 46 43 23 20 34 39)(11 42 47 19 24 38 35 14)(12 18 48 37 17 13 36 41)
G:=sub<Sym(64)| (1,35,51,43)(2,44,52,36)(3,37,53,45)(4,46,54,38)(5,39,55,47)(6,48,56,40)(7,33,49,41)(8,42,50,34)(9,63,18,26)(10,27,19,64)(11,57,20,28)(12,29,21,58)(13,59,22,30)(14,31,23,60)(15,61,24,32)(16,25,17,62), (1,28,55,61)(2,29,56,62)(3,30,49,63)(4,31,50,64)(5,32,51,57)(6,25,52,58)(7,26,53,59)(8,27,54,60)(9,45,22,33)(10,46,23,34)(11,47,24,35)(12,48,17,36)(13,41,18,37)(14,42,19,38)(15,43,20,39)(16,44,21,40), (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), (1,27,28,54,55,60,61,8)(2,53,29,59,56,7,62,26)(3,58,30,6,49,25,63,52)(4,5,31,32,50,51,64,57)(9,40,45,16,22,44,33,21)(10,15,46,43,23,20,34,39)(11,42,47,19,24,38,35,14)(12,18,48,37,17,13,36,41)>;
G:=Group( (1,35,51,43)(2,44,52,36)(3,37,53,45)(4,46,54,38)(5,39,55,47)(6,48,56,40)(7,33,49,41)(8,42,50,34)(9,63,18,26)(10,27,19,64)(11,57,20,28)(12,29,21,58)(13,59,22,30)(14,31,23,60)(15,61,24,32)(16,25,17,62), (1,28,55,61)(2,29,56,62)(3,30,49,63)(4,31,50,64)(5,32,51,57)(6,25,52,58)(7,26,53,59)(8,27,54,60)(9,45,22,33)(10,46,23,34)(11,47,24,35)(12,48,17,36)(13,41,18,37)(14,42,19,38)(15,43,20,39)(16,44,21,40), (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), (1,27,28,54,55,60,61,8)(2,53,29,59,56,7,62,26)(3,58,30,6,49,25,63,52)(4,5,31,32,50,51,64,57)(9,40,45,16,22,44,33,21)(10,15,46,43,23,20,34,39)(11,42,47,19,24,38,35,14)(12,18,48,37,17,13,36,41) );
G=PermutationGroup([[(1,35,51,43),(2,44,52,36),(3,37,53,45),(4,46,54,38),(5,39,55,47),(6,48,56,40),(7,33,49,41),(8,42,50,34),(9,63,18,26),(10,27,19,64),(11,57,20,28),(12,29,21,58),(13,59,22,30),(14,31,23,60),(15,61,24,32),(16,25,17,62)], [(1,28,55,61),(2,29,56,62),(3,30,49,63),(4,31,50,64),(5,32,51,57),(6,25,52,58),(7,26,53,59),(8,27,54,60),(9,45,22,33),(10,46,23,34),(11,47,24,35),(12,48,17,36),(13,41,18,37),(14,42,19,38),(15,43,20,39),(16,44,21,40)], [(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)], [(1,27,28,54,55,60,61,8),(2,53,29,59,56,7,62,26),(3,58,30,6,49,25,63,52),(4,5,31,32,50,51,64,57),(9,40,45,16,22,44,33,21),(10,15,46,43,23,20,34,39),(11,42,47,19,24,38,35,14),(12,18,48,37,17,13,36,41)]])
32 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 4K | 4L | 4M | 8A | ··· | 8H | 8I | 8J | 8K | 8L |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 8 | ··· | 8 | 8 | 8 | 8 | 8 |
| size | 1 | 1 | 1 | 1 | 4 | 8 | 8 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 8 | 8 | 4 | ··· | 4 | 8 | 8 | 8 | 8 |
32 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | |||||
| image | C1 | C2 | C2 | C2 | C2 | C4 | C4 | C4 | D4 | D4 | C8○D4 | C8⋊C22 | C42⋊C22 |
| kernel | C42.53D4 | D4⋊C8 | C42.12C4 | C42.6C4 | C22.26C24 | C4⋊D4 | C4⋊1D4 | C4⋊Q8 | C42 | C22×C4 | C4 | C4 | C2 |
| # reps | 1 | 4 | 1 | 1 | 1 | 4 | 2 | 2 | 2 | 2 | 8 | 2 | 2 |
Matrix representation of C42.53D4 ►in GL6(𝔽17)
| 10 | 15 | 0 | 0 | 0 | 0 |
| 7 | 7 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 16 | 0 |
| 0 | 0 | 1 | 0 | 16 | 15 |
| 0 | 0 | 2 | 0 | 16 | 15 |
| 0 | 0 | 16 | 1 | 0 | 1 |
| 4 | 0 | 0 | 0 | 0 | 0 |
| 0 | 4 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 12 | 1 | 0 | 0 | 0 | 0 |
| 5 | 5 | 0 | 0 | 0 | 0 |
| 0 | 0 | 6 | 7 | 4 | 4 |
| 0 | 0 | 6 | 10 | 0 | 4 |
| 0 | 0 | 0 | 3 | 0 | 3 |
| 0 | 0 | 6 | 7 | 11 | 1 |
| 12 | 1 | 0 | 0 | 0 | 0 |
| 13 | 5 | 0 | 0 | 0 | 0 |
| 0 | 0 | 6 | 7 | 4 | 4 |
| 0 | 0 | 11 | 7 | 0 | 13 |
| 0 | 0 | 0 | 14 | 0 | 14 |
| 0 | 0 | 6 | 10 | 11 | 4 |
G:=sub<GL(6,GF(17))| [10,7,0,0,0,0,15,7,0,0,0,0,0,0,0,1,2,16,0,0,1,0,0,1,0,0,16,16,16,0,0,0,0,15,15,1],[4,0,0,0,0,0,0,4,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[12,5,0,0,0,0,1,5,0,0,0,0,0,0,6,6,0,6,0,0,7,10,3,7,0,0,4,0,0,11,0,0,4,4,3,1],[12,13,0,0,0,0,1,5,0,0,0,0,0,0,6,11,0,6,0,0,7,7,14,10,0,0,4,0,0,11,0,0,4,13,14,4] >;
C42.53D4 in GAP, Magma, Sage, TeX
C_4^2._{53}D_4 % in TeX
G:=Group("C4^2.53D4"); // GroupNames label
G:=SmallGroup(128,228);
// by ID
G=gap.SmallGroup(128,228);
# by ID
G:=PCGroup([7,-2,2,2,-2,2,-2,2,112,141,1430,1059,184,1123,570,136,172]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^4=1,c^4=a^2*b^2,d^2=b,a*b=b*a,c*a*c^-1=a^-1,d*a*d^-1=a*b^2,b*c=c*b,b*d=d*b,d*c*d^-1=a^2*b^-1*c^3>;
// generators/relations