p-group, metabelian, nilpotent (class 3), monomial
Aliases: C42.405D4, C42.143C23, C4.26C4≀C2, C4⋊Q8.14C4, C4⋊1D4.10C4, C42.84(C2×C4), (C4×M4(2))⋊17C2, (C22×C4).223D4, C4.23(C4.D4), C8⋊C4.143C22, C23.56(C22⋊C4), (C2×C42).187C22, C4○2(C42.C22), C42.C22⋊17C2, C4.4D4.112C22, C22.26C24.9C2, C2.30(C2×C4≀C2), (C2×C4○D4).3C4, (C2×D4).19(C2×C4), (C2×Q8).19(C2×C4), (C2×C4).1171(C2×D4), C2.10(C2×C4.D4), (C2×C4).137(C22×C4), (C22×C4).209(C2×C4), (C2×C4).243(C22⋊C4), C22.201(C2×C22⋊C4), (C2×C4)○(C42.C22), SmallGroup(128,257)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C42.405D4
G = < a,b,c,d | a4=b4=1, c4=a2b2, d2=b, ab=ba, ac=ca, ad=da, cbc-1=a2b-1, bd=db, dcd-1=a2b-1c3 >
Subgroups: 292 in 135 conjugacy classes, 48 normal (14 characteristic)
C1, C2, C2, C2, C4, C4, C22, C22, C8, C2×C4, C2×C4, C2×C4, D4, Q8, C23, C23, C42, C42, C22⋊C4, C4⋊C4, C2×C8, M4(2), C22×C4, C22×C4, C22×C4, C2×D4, C2×D4, C2×Q8, C4○D4, C4×C8, C8⋊C4, C2×C42, C4×D4, C4⋊D4, C4.4D4, C4⋊1D4, C4⋊Q8, C2×M4(2), C2×C4○D4, C42.C22, C4×M4(2), C22.26C24, C42.405D4
Quotients: C1, C2, C4, C22, C2×C4, D4, C23, C22⋊C4, C22×C4, C2×D4, C4.D4, C4≀C2, C2×C22⋊C4, C2×C4.D4, C2×C4≀C2, C42.405D4
►On 64 points
(1 45 39 30)(2 46 40 31)(3 47 33 32)(4 48 34 25)(5 41 35 26)(6 42 36 27)(7 43 37 28)(8 44 38 29)(9 55 19 61)(10 56 20 62)(11 49 21 63)(12 50 22 64)(13 51 23 57)(14 52 24 58)(15 53 17 59)(16 54 18 60)
(1 58 35 56)(2 63 36 53)(3 60 37 50)(4 57 38 55)(5 62 39 52)(6 59 40 49)(7 64 33 54)(8 61 34 51)(9 25 23 44)(10 30 24 41)(11 27 17 46)(12 32 18 43)(13 29 19 48)(14 26 20 45)(15 31 21 42)(16 28 22 47)
(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 25 58 23 35 44 56 9)(2 18 63 43 36 12 53 32)(3 46 60 11 37 27 50 17)(4 14 57 26 38 20 55 45)(5 29 62 19 39 48 52 13)(6 22 59 47 40 16 49 28)(7 42 64 15 33 31 54 21)(8 10 61 30 34 24 51 41)
G:=sub<Sym(64)| (1,45,39,30)(2,46,40,31)(3,47,33,32)(4,48,34,25)(5,41,35,26)(6,42,36,27)(7,43,37,28)(8,44,38,29)(9,55,19,61)(10,56,20,62)(11,49,21,63)(12,50,22,64)(13,51,23,57)(14,52,24,58)(15,53,17,59)(16,54,18,60), (1,58,35,56)(2,63,36,53)(3,60,37,50)(4,57,38,55)(5,62,39,52)(6,59,40,49)(7,64,33,54)(8,61,34,51)(9,25,23,44)(10,30,24,41)(11,27,17,46)(12,32,18,43)(13,29,19,48)(14,26,20,45)(15,31,21,42)(16,28,22,47), (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), (1,25,58,23,35,44,56,9)(2,18,63,43,36,12,53,32)(3,46,60,11,37,27,50,17)(4,14,57,26,38,20,55,45)(5,29,62,19,39,48,52,13)(6,22,59,47,40,16,49,28)(7,42,64,15,33,31,54,21)(8,10,61,30,34,24,51,41)>;
G:=Group( (1,45,39,30)(2,46,40,31)(3,47,33,32)(4,48,34,25)(5,41,35,26)(6,42,36,27)(7,43,37,28)(8,44,38,29)(9,55,19,61)(10,56,20,62)(11,49,21,63)(12,50,22,64)(13,51,23,57)(14,52,24,58)(15,53,17,59)(16,54,18,60), (1,58,35,56)(2,63,36,53)(3,60,37,50)(4,57,38,55)(5,62,39,52)(6,59,40,49)(7,64,33,54)(8,61,34,51)(9,25,23,44)(10,30,24,41)(11,27,17,46)(12,32,18,43)(13,29,19,48)(14,26,20,45)(15,31,21,42)(16,28,22,47), (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), (1,25,58,23,35,44,56,9)(2,18,63,43,36,12,53,32)(3,46,60,11,37,27,50,17)(4,14,57,26,38,20,55,45)(5,29,62,19,39,48,52,13)(6,22,59,47,40,16,49,28)(7,42,64,15,33,31,54,21)(8,10,61,30,34,24,51,41) );
G=PermutationGroup([[(1,45,39,30),(2,46,40,31),(3,47,33,32),(4,48,34,25),(5,41,35,26),(6,42,36,27),(7,43,37,28),(8,44,38,29),(9,55,19,61),(10,56,20,62),(11,49,21,63),(12,50,22,64),(13,51,23,57),(14,52,24,58),(15,53,17,59),(16,54,18,60)], [(1,58,35,56),(2,63,36,53),(3,60,37,50),(4,57,38,55),(5,62,39,52),(6,59,40,49),(7,64,33,54),(8,61,34,51),(9,25,23,44),(10,30,24,41),(11,27,17,46),(12,32,18,43),(13,29,19,48),(14,26,20,45),(15,31,21,42),(16,28,22,47)], [(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)], [(1,25,58,23,35,44,56,9),(2,18,63,43,36,12,53,32),(3,46,60,11,37,27,50,17),(4,14,57,26,38,20,55,45),(5,29,62,19,39,48,52,13),(6,22,59,47,40,16,49,28),(7,42,64,15,33,31,54,21),(8,10,61,30,34,24,51,41)]])
38 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 4A | 4B | 4C | 4D | 4E | ··· | 4L | 4M | 4N | 4O | 8A | ··· | 8P |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 4 | 4 | 4 | 8 | ··· | 8 |
| size | 1 | 1 | 1 | 1 | 4 | 8 | 8 | 1 | 1 | 1 | 1 | 2 | ··· | 2 | 4 | 8 | 8 | 4 | ··· | 4 |
38 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 4 |
| type | + | + | + | + | + | + | + | ||||
| image | C1 | C2 | C2 | C2 | C4 | C4 | C4 | D4 | D4 | C4≀C2 | C4.D4 |
| kernel | C42.405D4 | C42.C22 | C4×M4(2) | C22.26C24 | C4⋊1D4 | C4⋊Q8 | C2×C4○D4 | C42 | C22×C4 | C4 | C4 |
| # reps | 1 | 4 | 2 | 1 | 2 | 2 | 4 | 2 | 2 | 16 | 2 |
Matrix representation of C42.405D4 ►in GL4(𝔽17) generated by
| 13 | 0 | 0 | 0 |
| 0 | 13 | 0 | 0 |
| 0 | 0 | 4 | 0 |
| 0 | 0 | 0 | 4 |
| 0 | 1 | 0 | 0 |
| 1 | 0 | 0 | 0 |
| 0 | 0 | 4 | 0 |
| 0 | 0 | 0 | 4 |
| 6 | 7 | 0 | 0 |
| 10 | 11 | 0 | 0 |
| 0 | 0 | 0 | 12 |
| 0 | 0 | 11 | 12 |
| 11 | 7 | 0 | 0 |
| 7 | 11 | 0 | 0 |
| 0 | 0 | 0 | 5 |
| 0 | 0 | 11 | 0 |
G:=sub<GL(4,GF(17))| [13,0,0,0,0,13,0,0,0,0,4,0,0,0,0,4],[0,1,0,0,1,0,0,0,0,0,4,0,0,0,0,4],[6,10,0,0,7,11,0,0,0,0,0,11,0,0,12,12],[11,7,0,0,7,11,0,0,0,0,0,11,0,0,5,0] >;
C42.405D4 in GAP, Magma, Sage, TeX
C_4^2._{405}D_4 % in TeX
G:=Group("C4^2.405D4"); // GroupNames label
G:=SmallGroup(128,257);
// by ID
G=gap.SmallGroup(128,257);
# by ID
G:=PCGroup([7,-2,2,2,-2,2,-2,2,112,141,352,1123,1018,248,1971,102]);
// 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,a*c=c*a,a*d=d*a,c*b*c^-1=a^2*b^-1,b*d=d*b,d*c*d^-1=a^2*b^-1*c^3>;
// generators/relations