p-group, metabelian, nilpotent (class 3), monomial
Aliases: C42.80D4, C42.169C23, C4.99(C4○D8), C4.D8⋊22C2, C4⋊C8.206C22, C4.84(C8⋊C22), C42.110(C2×C4), C4.6Q16⋊22C2, C4.4D4.11C4, (C22×C4).745D4, C4⋊Q8.242C22, C42.6C4⋊40C2, C4.86(C8.C22), C4⋊1D4.128C22, (C2×C42).213C22, C22.3(C4.D4), C23.111(C22⋊C4), C2.14(C23.36D4), C2.16(C23.24D4), C22.26C24.15C2, (C2×C4⋊C8)⋊8C2, (C2×C4○D4).7C4, (C2×D4).32(C2×C4), (C2×Q8).32(C2×C4), (C2×C4).1240(C2×D4), C2.20(C2×C4.D4), (C22×C4).235(C2×C4), (C2×C4).163(C22×C4), (C2×C4).184(C22⋊C4), C22.227(C2×C22⋊C4), SmallGroup(128,283)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C42.80D4
G = < a,b,c,d | a4=b4=1, c4=a2b2, d2=b, ab=ba, cac-1=dad-1=a-1b2, cbc-1=b-1, bd=db, dcd-1=a2b-1c3 >
Subgroups: 284 in 126 conjugacy classes, 48 normal (28 characteristic)
C1, C2, C2, C4, C4, C4, C22, C22, C22, C8, C2×C4, C2×C4, D4, Q8, C23, C23, C42, C22⋊C4, C4⋊C4, C2×C8, C22×C4, C22×C4, C2×D4, C2×D4, C2×Q8, C4○D4, C8⋊C4, C22⋊C8, C4⋊C8, C4⋊C8, C2×C42, C4×D4, C4⋊D4, C4.4D4, C4⋊1D4, C4⋊Q8, C22×C8, C2×C4○D4, C4.D8, C4.6Q16, C2×C4⋊C8, C42.6C4, C22.26C24, C42.80D4
Quotients: C1, C2, C4, C22, C2×C4, D4, C23, C22⋊C4, C22×C4, C2×D4, C4.D4, C2×C22⋊C4, C4○D8, C8⋊C22, C8.C22, C2×C4.D4, C23.24D4, C23.36D4, C42.80D4
►On 64 points
(1 53 20 41)(2 50 21 46)(3 55 22 43)(4 52 23 48)(5 49 24 45)(6 54 17 42)(7 51 18 47)(8 56 19 44)(9 57 40 29)(10 62 33 26)(11 59 34 31)(12 64 35 28)(13 61 36 25)(14 58 37 30)(15 63 38 27)(16 60 39 32)
(1 25 24 57)(2 58 17 26)(3 27 18 59)(4 60 19 28)(5 29 20 61)(6 62 21 30)(7 31 22 63)(8 64 23 32)(9 41 36 49)(10 50 37 42)(11 43 38 51)(12 52 39 44)(13 45 40 53)(14 54 33 46)(15 47 34 55)(16 56 35 48)
(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 64 25 23 24 32 57 8)(2 7 58 31 17 22 26 63)(3 62 27 21 18 30 59 6)(4 5 60 29 19 20 28 61)(9 48 41 16 36 56 49 35)(10 34 50 55 37 15 42 47)(11 46 43 14 38 54 51 33)(12 40 52 53 39 13 44 45)
G:=sub<Sym(64)| (1,53,20,41)(2,50,21,46)(3,55,22,43)(4,52,23,48)(5,49,24,45)(6,54,17,42)(7,51,18,47)(8,56,19,44)(9,57,40,29)(10,62,33,26)(11,59,34,31)(12,64,35,28)(13,61,36,25)(14,58,37,30)(15,63,38,27)(16,60,39,32), (1,25,24,57)(2,58,17,26)(3,27,18,59)(4,60,19,28)(5,29,20,61)(6,62,21,30)(7,31,22,63)(8,64,23,32)(9,41,36,49)(10,50,37,42)(11,43,38,51)(12,52,39,44)(13,45,40,53)(14,54,33,46)(15,47,34,55)(16,56,35,48), (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), (1,64,25,23,24,32,57,8)(2,7,58,31,17,22,26,63)(3,62,27,21,18,30,59,6)(4,5,60,29,19,20,28,61)(9,48,41,16,36,56,49,35)(10,34,50,55,37,15,42,47)(11,46,43,14,38,54,51,33)(12,40,52,53,39,13,44,45)>;
G:=Group( (1,53,20,41)(2,50,21,46)(3,55,22,43)(4,52,23,48)(5,49,24,45)(6,54,17,42)(7,51,18,47)(8,56,19,44)(9,57,40,29)(10,62,33,26)(11,59,34,31)(12,64,35,28)(13,61,36,25)(14,58,37,30)(15,63,38,27)(16,60,39,32), (1,25,24,57)(2,58,17,26)(3,27,18,59)(4,60,19,28)(5,29,20,61)(6,62,21,30)(7,31,22,63)(8,64,23,32)(9,41,36,49)(10,50,37,42)(11,43,38,51)(12,52,39,44)(13,45,40,53)(14,54,33,46)(15,47,34,55)(16,56,35,48), (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), (1,64,25,23,24,32,57,8)(2,7,58,31,17,22,26,63)(3,62,27,21,18,30,59,6)(4,5,60,29,19,20,28,61)(9,48,41,16,36,56,49,35)(10,34,50,55,37,15,42,47)(11,46,43,14,38,54,51,33)(12,40,52,53,39,13,44,45) );
G=PermutationGroup([[(1,53,20,41),(2,50,21,46),(3,55,22,43),(4,52,23,48),(5,49,24,45),(6,54,17,42),(7,51,18,47),(8,56,19,44),(9,57,40,29),(10,62,33,26),(11,59,34,31),(12,64,35,28),(13,61,36,25),(14,58,37,30),(15,63,38,27),(16,60,39,32)], [(1,25,24,57),(2,58,17,26),(3,27,18,59),(4,60,19,28),(5,29,20,61),(6,62,21,30),(7,31,22,63),(8,64,23,32),(9,41,36,49),(10,50,37,42),(11,43,38,51),(12,52,39,44),(13,45,40,53),(14,54,33,46),(15,47,34,55),(16,56,35,48)], [(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)], [(1,64,25,23,24,32,57,8),(2,7,58,31,17,22,26,63),(3,62,27,21,18,30,59,6),(4,5,60,29,19,20,28,61),(9,48,41,16,36,56,49,35),(10,34,50,55,37,15,42,47),(11,46,43,14,38,54,51,33),(12,40,52,53,39,13,44,45)]])
32 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 4A | ··· | 4H | 4I | 4J | 4K | 4L | 8A | ··· | 8H | 8I | 8J | 8K | 8L |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 4 | 4 | 4 | 4 | 8 | ··· | 8 | 8 | 8 | 8 | 8 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 8 | 8 | 2 | ··· | 2 | 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 | 4 |
| type | + | + | + | + | + | + | + | + | + | - | + | |||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C4 | C4 | D4 | D4 | C4○D8 | C8⋊C22 | C8.C22 | C4.D4 |
| kernel | C42.80D4 | C4.D8 | C4.6Q16 | C2×C4⋊C8 | C42.6C4 | C22.26C24 | C4.4D4 | C2×C4○D4 | C42 | C22×C4 | C4 | C4 | C4 | C22 |
| # reps | 1 | 2 | 2 | 1 | 1 | 1 | 4 | 4 | 2 | 2 | 8 | 1 | 1 | 2 |
Matrix representation of C42.80D4 ►in GL6(𝔽17)
| 4 | 0 | 0 | 0 | 0 | 0 |
| 0 | 4 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 16 | 0 |
| 0 | 0 | 0 | 1 | 0 | 15 |
| 0 | 0 | 16 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 16 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 16 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 16 | 16 | 0 | 0 |
| 0 | 0 | 2 | 1 | 0 | 0 |
| 0 | 0 | 0 | 1 | 16 | 15 |
| 0 | 0 | 1 | 0 | 1 | 1 |
| 14 | 14 | 0 | 0 | 0 | 0 |
| 14 | 3 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 1 |
| 0 | 0 | 1 | 1 | 16 | 15 |
| 0 | 0 | 16 | 0 | 0 | 16 |
| 0 | 0 | 1 | 1 | 16 | 16 |
| 3 | 3 | 0 | 0 | 0 | 0 |
| 14 | 3 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 1 |
| 0 | 0 | 16 | 16 | 16 | 0 |
| 0 | 0 | 16 | 0 | 0 | 16 |
| 0 | 0 | 0 | 16 | 0 | 1 |
G:=sub<GL(6,GF(17))| [4,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,16,0,0,0,0,1,0,0,0,0,16,0,0,0,0,0,0,15,0,16],[0,16,0,0,0,0,1,0,0,0,0,0,0,0,16,2,0,1,0,0,16,1,1,0,0,0,0,0,16,1,0,0,0,0,15,1],[14,14,0,0,0,0,14,3,0,0,0,0,0,0,0,1,16,1,0,0,0,1,0,1,0,0,1,16,0,16,0,0,1,15,16,16],[3,14,0,0,0,0,3,3,0,0,0,0,0,0,0,16,16,0,0,0,0,16,0,16,0,0,1,16,0,0,0,0,1,0,16,1] >;
C42.80D4 in GAP, Magma, Sage, TeX
C_4^2._{80}D_4 % in TeX
G:=Group("C4^2.80D4"); // GroupNames label
G:=SmallGroup(128,283);
// by ID
G=gap.SmallGroup(128,283);
# by ID
G:=PCGroup([7,-2,2,2,-2,2,-2,2,112,141,758,352,1123,1018,248,1971,242]);
// 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=d*a*d^-1=a^-1*b^2,c*b*c^-1=b^-1,b*d=d*b,d*c*d^-1=a^2*b^-1*c^3>;
// generators/relations