p-group, metabelian, nilpotent (class 3), monomial
Aliases: C42.414D4, C42.164C23, (C2×C4).18D8, C4.56(C2×D8), C4⋊Q8.22C4, (C2×C4).27SD16, C4.76(C2×SD16), C4.10D8⋊32C2, C4⋊C8.203C22, C42.105(C2×C4), (C22×C4).237D4, C4⋊Q8.237C22, C4.21(D4⋊C4), C4.6(C4.10D4), C4.104(C8.C22), C4⋊M4(2).14C2, (C2×C42).208C22, C22.27(D4⋊C4), C23.182(C22⋊C4), C42.12C4.24C2, C2.11(C23.38D4), (C2×C4⋊Q8).3C2, (C2×C4⋊C4).20C4, C4⋊C4.36(C2×C4), (C2×C4).1235(C2×D4), C2.15(C2×D4⋊C4), (C2×C4).158(C22×C4), (C22×C4).230(C2×C4), C2.17(C2×C4.10D4), (C2×C4).246(C22⋊C4), C22.222(C2×C22⋊C4), SmallGroup(128,278)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C42.414D4
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=bc3 >
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, C42.414D4
Quotients: C1, C2, C4, C22, C2×C4, D4, C23, C22⋊C4, D8, SD16, C22×C4, C2×D4, C4.10D4, D4⋊C4, C2×C22⋊C4, C2×D8, C2×SD16, C8.C22, C2×C4.10D4, C2×D4⋊C4, C23.38D4, C42.414D4
►On 64 points
(1 11 22 63)(2 64 23 12)(3 13 24 57)(4 58 17 14)(5 15 18 59)(6 60 19 16)(7 9 20 61)(8 62 21 10)(25 51 42 33)(26 34 43 52)(27 53 44 35)(28 36 45 54)(29 55 46 37)(30 38 47 56)(31 49 48 39)(32 40 41 50)
(1 9 18 57)(2 58 19 10)(3 11 20 59)(4 60 21 12)(5 13 22 61)(6 62 23 14)(7 15 24 63)(8 64 17 16)(25 53 46 39)(26 40 47 54)(27 55 48 33)(28 34 41 56)(29 49 42 35)(30 36 43 50)(31 51 44 37)(32 38 45 52)
(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 9 33 18 27 57 55)(2 50 58 30 19 36 10 43)(3 46 11 39 20 25 59 53)(4 56 60 28 21 34 12 41)(5 44 13 37 22 31 61 51)(6 54 62 26 23 40 14 47)(7 42 15 35 24 29 63 49)(8 52 64 32 17 38 16 45)
G:=sub<Sym(64)| (1,11,22,63)(2,64,23,12)(3,13,24,57)(4,58,17,14)(5,15,18,59)(6,60,19,16)(7,9,20,61)(8,62,21,10)(25,51,42,33)(26,34,43,52)(27,53,44,35)(28,36,45,54)(29,55,46,37)(30,38,47,56)(31,49,48,39)(32,40,41,50), (1,9,18,57)(2,58,19,10)(3,11,20,59)(4,60,21,12)(5,13,22,61)(6,62,23,14)(7,15,24,63)(8,64,17,16)(25,53,46,39)(26,40,47,54)(27,55,48,33)(28,34,41,56)(29,49,42,35)(30,36,43,50)(31,51,44,37)(32,38,45,52), (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), (1,48,9,33,18,27,57,55)(2,50,58,30,19,36,10,43)(3,46,11,39,20,25,59,53)(4,56,60,28,21,34,12,41)(5,44,13,37,22,31,61,51)(6,54,62,26,23,40,14,47)(7,42,15,35,24,29,63,49)(8,52,64,32,17,38,16,45)>;
G:=Group( (1,11,22,63)(2,64,23,12)(3,13,24,57)(4,58,17,14)(5,15,18,59)(6,60,19,16)(7,9,20,61)(8,62,21,10)(25,51,42,33)(26,34,43,52)(27,53,44,35)(28,36,45,54)(29,55,46,37)(30,38,47,56)(31,49,48,39)(32,40,41,50), (1,9,18,57)(2,58,19,10)(3,11,20,59)(4,60,21,12)(5,13,22,61)(6,62,23,14)(7,15,24,63)(8,64,17,16)(25,53,46,39)(26,40,47,54)(27,55,48,33)(28,34,41,56)(29,49,42,35)(30,36,43,50)(31,51,44,37)(32,38,45,52), (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), (1,48,9,33,18,27,57,55)(2,50,58,30,19,36,10,43)(3,46,11,39,20,25,59,53)(4,56,60,28,21,34,12,41)(5,44,13,37,22,31,61,51)(6,54,62,26,23,40,14,47)(7,42,15,35,24,29,63,49)(8,52,64,32,17,38,16,45) );
G=PermutationGroup([[(1,11,22,63),(2,64,23,12),(3,13,24,57),(4,58,17,14),(5,15,18,59),(6,60,19,16),(7,9,20,61),(8,62,21,10),(25,51,42,33),(26,34,43,52),(27,53,44,35),(28,36,45,54),(29,55,46,37),(30,38,47,56),(31,49,48,39),(32,40,41,50)], [(1,9,18,57),(2,58,19,10),(3,11,20,59),(4,60,21,12),(5,13,22,61),(6,62,23,14),(7,15,24,63),(8,64,17,16),(25,53,46,39),(26,40,47,54),(27,55,48,33),(28,34,41,56),(29,49,42,35),(30,36,43,50),(31,51,44,37),(32,38,45,52)], [(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)], [(1,48,9,33,18,27,57,55),(2,50,58,30,19,36,10,43),(3,46,11,39,20,25,59,53),(4,56,60,28,21,34,12,41),(5,44,13,37,22,31,61,51),(6,54,62,26,23,40,14,47),(7,42,15,35,24,29,63,49),(8,52,64,32,17,38,16,45)]])
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 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | - | - | |||
| image | C1 | C2 | C2 | C2 | C2 | C4 | C4 | D4 | D4 | D8 | SD16 | C4.10D4 | C8.C22 |
| kernel | C42.414D4 | 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 C42.414D4 ►in GL6(𝔽17)
| 0 | 16 | 0 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 16 | 0 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 16 | 2 | 0 | 0 |
| 0 | 0 | 16 | 1 | 0 | 0 |
| 0 | 0 | 16 | 16 | 16 | 2 |
| 0 | 0 | 7 | 1 | 16 | 1 |
| 5 | 12 | 0 | 0 | 0 | 0 |
| 12 | 12 | 0 | 0 | 0 | 0 |
| 0 | 0 | 14 | 11 | 16 | 13 |
| 0 | 0 | 16 | 13 | 14 | 1 |
| 0 | 0 | 16 | 0 | 3 | 7 |
| 0 | 0 | 8 | 11 | 13 | 4 |
| 12 | 12 | 0 | 0 | 0 | 0 |
| 5 | 12 | 0 | 0 | 0 | 0 |
| 0 | 0 | 12 | 11 | 15 | 0 |
| 0 | 0 | 4 | 7 | 0 | 15 |
| 0 | 0 | 1 | 10 | 5 | 6 |
| 0 | 0 | 13 | 12 | 13 | 10 |
G:=sub<GL(6,GF(17))| [0,1,0,0,0,0,16,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[0,1,0,0,0,0,16,0,0,0,0,0,0,0,16,16,16,7,0,0,2,1,16,1,0,0,0,0,16,16,0,0,0,0,2,1],[5,12,0,0,0,0,12,12,0,0,0,0,0,0,14,16,16,8,0,0,11,13,0,11,0,0,16,14,3,13,0,0,13,1,7,4],[12,5,0,0,0,0,12,12,0,0,0,0,0,0,12,4,1,13,0,0,11,7,10,12,0,0,15,0,5,13,0,0,0,15,6,10] >;
C42.414D4 in GAP, Magma, Sage, TeX
C_4^2._{414}D_4 % in TeX
G:=Group("C4^2.414D4"); // GroupNames label
G:=SmallGroup(128,278);
// by ID
G=gap.SmallGroup(128,278);
# by ID
G:=PCGroup([7,-2,2,2,-2,2,-2,2,112,141,232,758,1123,1018,248,1971,242]);
// 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=b*c^3>;
// generators/relations