p-group, metabelian, nilpotent (class 3), monomial
Aliases: C42.81D4, C42.170C23, C4.100(C4○D8), C4.10D8⋊36C2, C4⋊C8.207C22, C4.85(C8⋊C22), C42⋊C2.7C4, C42.111(C2×C4), (C22×C4).746D4, C4⋊Q8.243C22, C42.C2.13C4, C4.87(C8.C22), C42.6C4.23C2, (C2×C42).214C22, C23.112(C22⋊C4), C22.3(C4.10D4), C2.17(C23.24D4), C2.15(C23.36D4), C23.37C23.15C2, (C2×C4⋊C8).15C2, C4⋊C4.40(C2×C4), (C2×C4).1241(C2×D4), (C22×C4).236(C2×C4), (C2×C4).164(C22×C4), C2.19(C2×C4.10D4), (C2×C4).185(C22⋊C4), C22.228(C2×C22⋊C4), SmallGroup(128,284)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C42.81D4
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=bc3 >
Subgroups: 188 in 104 conjugacy classes, 48 normal (28 characteristic)
C1, C2, C2, C4, C4, C4, C22, C22, C22, C8, C2×C4, C2×C4, Q8, C23, C42, C42, C22⋊C4, C4⋊C4, C4⋊C4, C2×C8, C22×C4, C2×Q8, C8⋊C4, C22⋊C8, C4⋊C8, C4⋊C8, C2×C42, C42⋊C2, C4×Q8, C22⋊Q8, C42.C2, C4⋊Q8, C22×C8, C4.10D8, C2×C4⋊C8, C42.6C4, C23.37C23, C42.81D4
Quotients: C1, C2, C4, C22, C2×C4, D4, C23, C22⋊C4, C22×C4, C2×D4, C4.10D4, C2×C22⋊C4, C4○D8, C8⋊C22, C8.C22, C2×C4.10D4, C23.24D4, C23.36D4, C42.81D4
►On 64 points
(1 59 23 9)(2 64 24 14)(3 61 17 11)(4 58 18 16)(5 63 19 13)(6 60 20 10)(7 57 21 15)(8 62 22 12)(25 38 46 53)(26 35 47 50)(27 40 48 55)(28 37 41 52)(29 34 42 49)(30 39 43 54)(31 36 44 51)(32 33 45 56)
(1 15 19 61)(2 62 20 16)(3 9 21 63)(4 64 22 10)(5 11 23 57)(6 58 24 12)(7 13 17 59)(8 60 18 14)(25 51 42 40)(26 33 43 52)(27 53 44 34)(28 35 45 54)(29 55 46 36)(30 37 47 56)(31 49 48 38)(32 39 41 50)
(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 48 15 38 19 31 61 49)(2 52 62 26 20 33 16 43)(3 46 9 36 21 29 63 55)(4 50 64 32 22 39 10 41)(5 44 11 34 23 27 57 53)(6 56 58 30 24 37 12 47)(7 42 13 40 17 25 59 51)(8 54 60 28 18 35 14 45)
G:=sub<Sym(64)| (1,59,23,9)(2,64,24,14)(3,61,17,11)(4,58,18,16)(5,63,19,13)(6,60,20,10)(7,57,21,15)(8,62,22,12)(25,38,46,53)(26,35,47,50)(27,40,48,55)(28,37,41,52)(29,34,42,49)(30,39,43,54)(31,36,44,51)(32,33,45,56), (1,15,19,61)(2,62,20,16)(3,9,21,63)(4,64,22,10)(5,11,23,57)(6,58,24,12)(7,13,17,59)(8,60,18,14)(25,51,42,40)(26,33,43,52)(27,53,44,34)(28,35,45,54)(29,55,46,36)(30,37,47,56)(31,49,48,38)(32,39,41,50), (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), (1,48,15,38,19,31,61,49)(2,52,62,26,20,33,16,43)(3,46,9,36,21,29,63,55)(4,50,64,32,22,39,10,41)(5,44,11,34,23,27,57,53)(6,56,58,30,24,37,12,47)(7,42,13,40,17,25,59,51)(8,54,60,28,18,35,14,45)>;
G:=Group( (1,59,23,9)(2,64,24,14)(3,61,17,11)(4,58,18,16)(5,63,19,13)(6,60,20,10)(7,57,21,15)(8,62,22,12)(25,38,46,53)(26,35,47,50)(27,40,48,55)(28,37,41,52)(29,34,42,49)(30,39,43,54)(31,36,44,51)(32,33,45,56), (1,15,19,61)(2,62,20,16)(3,9,21,63)(4,64,22,10)(5,11,23,57)(6,58,24,12)(7,13,17,59)(8,60,18,14)(25,51,42,40)(26,33,43,52)(27,53,44,34)(28,35,45,54)(29,55,46,36)(30,37,47,56)(31,49,48,38)(32,39,41,50), (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), (1,48,15,38,19,31,61,49)(2,52,62,26,20,33,16,43)(3,46,9,36,21,29,63,55)(4,50,64,32,22,39,10,41)(5,44,11,34,23,27,57,53)(6,56,58,30,24,37,12,47)(7,42,13,40,17,25,59,51)(8,54,60,28,18,35,14,45) );
G=PermutationGroup([[(1,59,23,9),(2,64,24,14),(3,61,17,11),(4,58,18,16),(5,63,19,13),(6,60,20,10),(7,57,21,15),(8,62,22,12),(25,38,46,53),(26,35,47,50),(27,40,48,55),(28,37,41,52),(29,34,42,49),(30,39,43,54),(31,36,44,51),(32,33,45,56)], [(1,15,19,61),(2,62,20,16),(3,9,21,63),(4,64,22,10),(5,11,23,57),(6,58,24,12),(7,13,17,59),(8,60,18,14),(25,51,42,40),(26,33,43,52),(27,53,44,34),(28,35,45,54),(29,55,46,36),(30,37,47,56),(31,49,48,38),(32,39,41,50)], [(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)], [(1,48,15,38,19,31,61,49),(2,52,62,26,20,33,16,43),(3,46,9,36,21,29,63,55),(4,50,64,32,22,39,10,41),(5,44,11,34,23,27,57,53),(6,56,58,30,24,37,12,47),(7,42,13,40,17,25,59,51),(8,54,60,28,18,35,14,45)]])
32 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 4A | ··· | 4H | 4I | 4J | 4K | 4L | 4M | 4N | 8A | ··· | 8H | 8I | 8J | 8K | 8L |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 8 | ··· | 8 | 8 | 8 | 8 | 8 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 2 | ··· | 2 | 4 | 4 | 8 | 8 | 8 | 8 | 4 | ··· | 4 | 8 | 8 | 8 | 8 |
32 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | - | - | |||
| image | C1 | C2 | C2 | C2 | C2 | C4 | C4 | D4 | D4 | C4○D8 | C8⋊C22 | C8.C22 | C4.10D4 |
| kernel | C42.81D4 | C4.10D8 | C2×C4⋊C8 | C42.6C4 | C23.37C23 | C42⋊C2 | C42.C2 | C42 | C22×C4 | C4 | C4 | C4 | C22 |
| # reps | 1 | 4 | 1 | 1 | 1 | 4 | 4 | 2 | 2 | 8 | 1 | 1 | 2 |
Matrix representation of C42.81D4 ►in GL6(𝔽17)
| 4 | 0 | 0 | 0 | 0 | 0 |
| 0 | 4 | 0 | 0 | 0 | 0 |
| 0 | 0 | 13 | 0 | 2 | 0 |
| 0 | 0 | 16 | 1 | 2 | 15 |
| 0 | 0 | 1 | 0 | 4 | 0 |
| 0 | 0 | 11 | 0 | 4 | 16 |
| 13 | 0 | 0 | 0 | 0 | 0 |
| 2 | 4 | 0 | 0 | 0 | 0 |
| 0 | 0 | 16 | 2 | 0 | 0 |
| 0 | 0 | 16 | 1 | 0 | 0 |
| 0 | 0 | 16 | 5 | 1 | 15 |
| 0 | 0 | 6 | 4 | 1 | 16 |
| 16 | 13 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 15 | 5 | 14 | 15 |
| 0 | 0 | 5 | 5 | 1 | 12 |
| 0 | 0 | 7 | 16 | 1 | 0 |
| 0 | 0 | 5 | 10 | 5 | 13 |
| 9 | 0 | 0 | 0 | 0 | 0 |
| 11 | 2 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 1 | 7 | 16 |
| 0 | 0 | 10 | 9 | 15 | 10 |
| 0 | 0 | 15 | 10 | 0 | 7 |
| 0 | 0 | 15 | 10 | 5 | 7 |
G:=sub<GL(6,GF(17))| [4,0,0,0,0,0,0,4,0,0,0,0,0,0,13,16,1,11,0,0,0,1,0,0,0,0,2,2,4,4,0,0,0,15,0,16],[13,2,0,0,0,0,0,4,0,0,0,0,0,0,16,16,16,6,0,0,2,1,5,4,0,0,0,0,1,1,0,0,0,0,15,16],[16,0,0,0,0,0,13,1,0,0,0,0,0,0,15,5,7,5,0,0,5,5,16,10,0,0,14,1,1,5,0,0,15,12,0,13],[9,11,0,0,0,0,0,2,0,0,0,0,0,0,1,10,15,15,0,0,1,9,10,10,0,0,7,15,0,5,0,0,16,10,7,7] >;
C42.81D4 in GAP, Magma, Sage, TeX
C_4^2._{81}D_4 % in TeX
G:=Group("C4^2.81D4"); // GroupNames label
G:=SmallGroup(128,284);
// by ID
G=gap.SmallGroup(128,284);
# by ID
G:=PCGroup([7,-2,2,2,-2,2,-2,2,112,141,232,758,352,1123,1018,248,1971,242]);
// 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=b*c^3>;
// generators/relations