p-group, metabelian, nilpotent (class 3), monomial
Aliases: C82⋊5C2, C8.13D8, C42.656C23, C4.2(C2×D8), C4⋊Q16⋊4C2, C4.4(C4○D8), (C2×C8).223D4, C4.4D8⋊8C2, C8⋊4D4.6C2, C2.5(C8⋊4D4), C4⋊Q8.81C22, (C4×C8).369C22, C2.9(C8.12D4), C4⋊1D4.43C22, C22.57(C4⋊1D4), (C2×C4).713(C2×D4), SmallGroup(128,441)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C82⋊5C2
G = < a,b,c | a8=b8=c2=1, ab=ba, cac=a3b4, cbc=a4b3 >
Subgroups: 304 in 106 conjugacy classes, 40 normal (10 characteristic)
C1, C2, C2, C2, C4, C4, C22, C22, C8, C8, C2×C4, C2×C4, C2×C4, D4, Q8, C23, C42, C4⋊C4, C2×C8, D8, Q16, C2×D4, C2×Q8, C4×C8, C4×C8, D4⋊C4, C4⋊1D4, C4⋊Q8, C2×D8, C2×Q16, C82, C4.4D8, C8⋊4D4, C4⋊Q16, C82⋊5C2
Quotients: C1, C2, C22, D4, C23, D8, C2×D4, C4⋊1D4, C2×D8, C4○D8, C8⋊4D4, C8.12D4, C82⋊5C2
►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 13 57 25 24 47 38 49)(2 14 58 26 17 48 39 50)(3 15 59 27 18 41 40 51)(4 16 60 28 19 42 33 52)(5 9 61 29 20 43 34 53)(6 10 62 30 21 44 35 54)(7 11 63 31 22 45 36 55)(8 12 64 32 23 46 37 56)
(2 19)(3 7)(4 17)(6 23)(8 21)(9 25)(10 52)(11 31)(12 50)(13 29)(14 56)(15 27)(16 54)(18 22)(26 46)(28 44)(30 42)(32 48)(33 39)(34 61)(35 37)(36 59)(38 57)(40 63)(41 51)(43 49)(45 55)(47 53)(58 60)(62 64)
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), (1,13,57,25,24,47,38,49)(2,14,58,26,17,48,39,50)(3,15,59,27,18,41,40,51)(4,16,60,28,19,42,33,52)(5,9,61,29,20,43,34,53)(6,10,62,30,21,44,35,54)(7,11,63,31,22,45,36,55)(8,12,64,32,23,46,37,56), (2,19)(3,7)(4,17)(6,23)(8,21)(9,25)(10,52)(11,31)(12,50)(13,29)(14,56)(15,27)(16,54)(18,22)(26,46)(28,44)(30,42)(32,48)(33,39)(34,61)(35,37)(36,59)(38,57)(40,63)(41,51)(43,49)(45,55)(47,53)(58,60)(62,64)>;
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), (1,13,57,25,24,47,38,49)(2,14,58,26,17,48,39,50)(3,15,59,27,18,41,40,51)(4,16,60,28,19,42,33,52)(5,9,61,29,20,43,34,53)(6,10,62,30,21,44,35,54)(7,11,63,31,22,45,36,55)(8,12,64,32,23,46,37,56), (2,19)(3,7)(4,17)(6,23)(8,21)(9,25)(10,52)(11,31)(12,50)(13,29)(14,56)(15,27)(16,54)(18,22)(26,46)(28,44)(30,42)(32,48)(33,39)(34,61)(35,37)(36,59)(38,57)(40,63)(41,51)(43,49)(45,55)(47,53)(58,60)(62,64) );
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)], [(1,13,57,25,24,47,38,49),(2,14,58,26,17,48,39,50),(3,15,59,27,18,41,40,51),(4,16,60,28,19,42,33,52),(5,9,61,29,20,43,34,53),(6,10,62,30,21,44,35,54),(7,11,63,31,22,45,36,55),(8,12,64,32,23,46,37,56)], [(2,19),(3,7),(4,17),(6,23),(8,21),(9,25),(10,52),(11,31),(12,50),(13,29),(14,56),(15,27),(16,54),(18,22),(26,46),(28,44),(30,42),(32,48),(33,39),(34,61),(35,37),(36,59),(38,57),(40,63),(41,51),(43,49),(45,55),(47,53),(58,60),(62,64)]])
38 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 4A | ··· | 4F | 4G | 4H | 8A | ··· | 8X |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 4 | 4 | 8 | ··· | 8 |
| size | 1 | 1 | 1 | 1 | 16 | 16 | 2 | ··· | 2 | 16 | 16 | 2 | ··· | 2 |
38 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | |
| image | C1 | C2 | C2 | C2 | C2 | D4 | D8 | C4○D8 |
| kernel | C82⋊5C2 | C82 | C4.4D8 | C8⋊4D4 | C4⋊Q16 | C2×C8 | C8 | C4 |
| # reps | 1 | 1 | 4 | 1 | 1 | 6 | 8 | 16 |
Matrix representation of C82⋊5C2 ►in GL4(𝔽17) generated by
| 0 | 13 | 0 | 0 |
| 4 | 0 | 0 | 0 |
| 0 | 0 | 3 | 3 |
| 0 | 0 | 14 | 3 |
| 5 | 12 | 0 | 0 |
| 5 | 5 | 0 | 0 |
| 0 | 0 | 3 | 14 |
| 0 | 0 | 3 | 3 |
| 1 | 0 | 0 | 0 |
| 0 | 16 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 16 |
G:=sub<GL(4,GF(17))| [0,4,0,0,13,0,0,0,0,0,3,14,0,0,3,3],[5,5,0,0,12,5,0,0,0,0,3,3,0,0,14,3],[1,0,0,0,0,16,0,0,0,0,1,0,0,0,0,16] >;
C82⋊5C2 in GAP, Magma, Sage, TeX
C_8^2\rtimes_5C_2
% in TeX
G:=Group("C8^2:5C2"); // GroupNames label
G:=SmallGroup(128,441);
// by ID
G=gap.SmallGroup(128,441);
# by ID
G:=PCGroup([7,-2,2,2,-2,2,-2,2,141,512,422,268,1123,136,2804,172]);
// Polycyclic
G:=Group<a,b,c|a^8=b^8=c^2=1,a*b=b*a,c*a*c=a^3*b^4,c*b*c=a^4*b^3>;
// generators/relations