p-group, metabelian, nilpotent (class 3), monomial
Aliases: C42.416D4, C42.167C23, C4⋊Q8.24C4, (C2×C4).13Q16, C4.37(C2×Q16), C4.55(C2×SD16), (C2×C4).30SD16, C4.10D8⋊34C2, C4⋊C8.205C22, C42.108(C2×C4), (C22×C4).239D4, C4⋊Q8.240C22, C4.109(C8⋊C22), C4.18(Q8⋊C4), C4.7(C4.10D4), C4⋊M4(2).16C2, (C2×C42).211C22, C23.184(C22⋊C4), C42.12C4.26C2, C22.23(Q8⋊C4), C2.12(C23.37D4), (C2×C4⋊Q8).5C2, (C2×C4⋊C4).21C4, C4⋊C4.38(C2×C4), (C2×C4).1238(C2×D4), C2.15(C2×Q8⋊C4), (C22×C4).233(C2×C4), (C2×C4).161(C22×C4), C2.18(C2×C4.10D4), (C2×C4).248(C22⋊C4), C22.225(C2×C22⋊C4), SmallGroup(128,281)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C42.416D4
G = < a,b,c,d | a4=b4=1, c4=a2b2, d2=b, ab=ba, cac-1=a-1, ad=da, cbc-1=b-1, bd=db, dcd-1=a2bc3 >
Subgroups: 236 in 120 conjugacy classes, 56 normal (26 characteristic)
C1, C2, C2, C4, C4, C4, C22, C22, C22, C8, C2×C4, C2×C4, C2×C4, Q8, C23, C42, C4⋊C4, C4⋊C4, C2×C8, M4(2), C22×C4, C22×C4, C2×Q8, C4×C8, C22⋊C8, C4⋊C8, C4⋊C8, C2×C42, C2×C4⋊C4, C2×C4⋊C4, C4⋊Q8, C4⋊Q8, C2×M4(2), C22×Q8, C4.10D8, C4⋊M4(2), C42.12C4, C2×C4⋊Q8, C42.416D4
Quotients: C1, C2, C4, C22, C2×C4, D4, C23, C22⋊C4, SD16, Q16, C22×C4, C2×D4, C4.10D4, Q8⋊C4, C2×C22⋊C4, C2×SD16, C2×Q16, C8⋊C22, C2×C4.10D4, C2×Q8⋊C4, C23.37D4, C42.416D4
►On 64 points
(1 20 64 16)(2 9 57 21)(3 22 58 10)(4 11 59 23)(5 24 60 12)(6 13 61 17)(7 18 62 14)(8 15 63 19)(25 41 52 40)(26 33 53 42)(27 43 54 34)(28 35 55 44)(29 45 56 36)(30 37 49 46)(31 47 50 38)(32 39 51 48)
(1 14 60 22)(2 23 61 15)(3 16 62 24)(4 17 63 9)(5 10 64 18)(6 19 57 11)(7 12 58 20)(8 21 59 13)(25 38 56 43)(26 44 49 39)(27 40 50 45)(28 46 51 33)(29 34 52 47)(30 48 53 35)(31 36 54 41)(32 42 55 37)
(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 14 53 60 35 22 30)(2 52 23 47 61 29 15 34)(3 46 16 51 62 33 24 28)(4 50 17 45 63 27 9 40)(5 44 10 49 64 39 18 26)(6 56 19 43 57 25 11 38)(7 42 12 55 58 37 20 32)(8 54 21 41 59 31 13 36)
G:=sub<Sym(64)| (1,20,64,16)(2,9,57,21)(3,22,58,10)(4,11,59,23)(5,24,60,12)(6,13,61,17)(7,18,62,14)(8,15,63,19)(25,41,52,40)(26,33,53,42)(27,43,54,34)(28,35,55,44)(29,45,56,36)(30,37,49,46)(31,47,50,38)(32,39,51,48), (1,14,60,22)(2,23,61,15)(3,16,62,24)(4,17,63,9)(5,10,64,18)(6,19,57,11)(7,12,58,20)(8,21,59,13)(25,38,56,43)(26,44,49,39)(27,40,50,45)(28,46,51,33)(29,34,52,47)(30,48,53,35)(31,36,54,41)(32,42,55,37), (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), (1,48,14,53,60,35,22,30)(2,52,23,47,61,29,15,34)(3,46,16,51,62,33,24,28)(4,50,17,45,63,27,9,40)(5,44,10,49,64,39,18,26)(6,56,19,43,57,25,11,38)(7,42,12,55,58,37,20,32)(8,54,21,41,59,31,13,36)>;
G:=Group( (1,20,64,16)(2,9,57,21)(3,22,58,10)(4,11,59,23)(5,24,60,12)(6,13,61,17)(7,18,62,14)(8,15,63,19)(25,41,52,40)(26,33,53,42)(27,43,54,34)(28,35,55,44)(29,45,56,36)(30,37,49,46)(31,47,50,38)(32,39,51,48), (1,14,60,22)(2,23,61,15)(3,16,62,24)(4,17,63,9)(5,10,64,18)(6,19,57,11)(7,12,58,20)(8,21,59,13)(25,38,56,43)(26,44,49,39)(27,40,50,45)(28,46,51,33)(29,34,52,47)(30,48,53,35)(31,36,54,41)(32,42,55,37), (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), (1,48,14,53,60,35,22,30)(2,52,23,47,61,29,15,34)(3,46,16,51,62,33,24,28)(4,50,17,45,63,27,9,40)(5,44,10,49,64,39,18,26)(6,56,19,43,57,25,11,38)(7,42,12,55,58,37,20,32)(8,54,21,41,59,31,13,36) );
G=PermutationGroup([[(1,20,64,16),(2,9,57,21),(3,22,58,10),(4,11,59,23),(5,24,60,12),(6,13,61,17),(7,18,62,14),(8,15,63,19),(25,41,52,40),(26,33,53,42),(27,43,54,34),(28,35,55,44),(29,45,56,36),(30,37,49,46),(31,47,50,38),(32,39,51,48)], [(1,14,60,22),(2,23,61,15),(3,16,62,24),(4,17,63,9),(5,10,64,18),(6,19,57,11),(7,12,58,20),(8,21,59,13),(25,38,56,43),(26,44,49,39),(27,40,50,45),(28,46,51,33),(29,34,52,47),(30,48,53,35),(31,36,54,41),(32,42,55,37)], [(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)], [(1,48,14,53,60,35,22,30),(2,52,23,47,61,29,15,34),(3,46,16,51,62,33,24,28),(4,50,17,45,63,27,9,40),(5,44,10,49,64,39,18,26),(6,56,19,43,57,25,11,38),(7,42,12,55,58,37,20,32),(8,54,21,41,59,31,13,36)]])
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 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | - | - | + | |||
| image | C1 | C2 | C2 | C2 | C2 | C4 | C4 | D4 | D4 | SD16 | Q16 | C4.10D4 | C8⋊C22 |
| kernel | C42.416D4 | C4.10D8 | C4⋊M4(2) | C42.12C4 | C2×C4⋊Q8 | C2×C4⋊C4 | C4⋊Q8 | C42 | C22×C4 | C2×C4 | C2×C4 | C4 | C4 |
| # reps | 1 | 4 | 1 | 1 | 1 | 4 | 4 | 2 | 2 | 4 | 4 | 2 | 2 |
Matrix representation of C42.416D4 ►in GL6(𝔽17)
| 0 | 16 | 0 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 16 | 0 | 0 | 0 |
| 0 | 0 | 0 | 16 | 0 | 0 |
| 0 | 0 | 0 | 0 | 16 | 0 |
| 0 | 0 | 0 | 0 | 0 | 16 |
| 0 | 16 | 0 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 16 | 16 | 0 | 0 |
| 0 | 0 | 2 | 1 | 0 | 0 |
| 0 | 0 | 5 | 16 | 0 | 1 |
| 0 | 0 | 7 | 6 | 16 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 12 | 0 | 9 |
| 0 | 0 | 0 | 2 | 8 | 8 |
| 0 | 0 | 0 | 12 | 3 | 4 |
| 0 | 0 | 13 | 11 | 12 | 11 |
| 5 | 5 | 0 | 0 | 0 | 0 |
| 12 | 5 | 0 | 0 | 0 | 0 |
| 0 | 0 | 4 | 5 | 15 | 0 |
| 0 | 0 | 0 | 9 | 2 | 2 |
| 0 | 0 | 0 | 16 | 1 | 5 |
| 0 | 0 | 1 | 12 | 7 | 3 |
G:=sub<GL(6,GF(17))| [0,1,0,0,0,0,16,0,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,16],[0,1,0,0,0,0,16,0,0,0,0,0,0,0,16,2,5,7,0,0,16,1,16,6,0,0,0,0,0,16,0,0,0,0,1,0],[0,1,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,13,0,0,12,2,12,11,0,0,0,8,3,12,0,0,9,8,4,11],[5,12,0,0,0,0,5,5,0,0,0,0,0,0,4,0,0,1,0,0,5,9,16,12,0,0,15,2,1,7,0,0,0,2,5,3] >;
C42.416D4 in GAP, Magma, Sage, TeX
C_4^2._{416}D_4 % in TeX
G:=Group("C4^2.416D4"); // GroupNames label
G:=SmallGroup(128,281);
// by ID
G=gap.SmallGroup(128,281);
# by ID
G:=PCGroup([7,-2,2,2,-2,2,-2,2,112,141,680,758,1123,1018,248,1971,242]);
// 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=a^-1,a*d=d*a,c*b*c^-1=b^-1,b*d=d*b,d*c*d^-1=a^2*b*c^3>;
// generators/relations