p-group, metabelian, nilpotent (class 2), monomial
Aliases: C42.287C23, (C2×C8)⋊9Q8, C4⋊Q8.30C4, C4.22(C4×Q8), C8.28(C2×Q8), C8⋊4Q8⋊32C2, C22.7(C4×Q8), C22⋊Q8.19C4, C4.65(C22×Q8), C4⋊C8.233C22, C42.212(C2×C4), (C2×C4).658C24, (C2×C8).424C23, (C4×C8).330C22, C42.C2.15C4, (C4×Q8).57C22, C8⋊C4.159C22, C8○2M4(2).23C2, C2.19(Q8○M4(2)), C4⋊M4(2).37C2, C22.184(C23×C4), (C22×C8).444C22, C23.143(C22×C4), (C2×C42).771C22, (C22×C4).1520C23, C42⋊C2.304C22, (C2×M4(2)).361C22, C42.6C22.13C2, C23.37C23.22C2, C2.25(C2×C4×Q8), C4⋊C4.117(C2×C4), C4.309(C2×C4○D4), (C2×C4).243(C2×Q8), (C2×C8⋊C4).39C2, C22⋊C4.39(C2×C4), (C2×C4).75(C22×C4), (C2×Q8).118(C2×C4), (C2×C4).696(C4○D4), (C22×C4).345(C2×C4), SmallGroup(128,1693)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Subgroups: 220 in 174 conjugacy classes, 140 normal (18 characteristic)
C1, C2, C2 [×2], C2 [×2], C4 [×2], C4 [×2], C4 [×12], C22, C22 [×2], C22 [×2], C8 [×4], C8 [×6], C2×C4 [×2], C2×C4 [×16], C2×C4 [×2], Q8 [×4], C23, C42 [×2], C42 [×6], C22⋊C4 [×4], C4⋊C4 [×16], C2×C8 [×12], C2×C8 [×2], M4(2) [×4], C22×C4, C22×C4 [×2], C2×Q8 [×4], C4×C8 [×4], C8⋊C4 [×8], C4⋊C8 [×12], C2×C42, C42⋊C2 [×2], C4×Q8 [×4], C22⋊Q8 [×4], C42.C2 [×2], C4⋊Q8 [×2], C22×C8 [×2], C2×M4(2) [×2], C2×C8⋊C4, C8○2M4(2) [×2], C4⋊M4(2), C42.6C22 [×2], C8⋊4Q8 [×8], C23.37C23, C42.287C23
Quotients:
C1, C2 [×15], C4 [×8], C22 [×35], C2×C4 [×28], Q8 [×4], C23 [×15], C22×C4 [×14], C2×Q8 [×6], C4○D4 [×2], C24, C4×Q8 [×4], C23×C4, C22×Q8, C2×C4○D4, C2×C4×Q8, Q8○M4(2) [×2], C42.287C23
Generators and relations
G = < a,b,c,d,e | a4=b4=1, c2=b, d2=a2b, e2=a2, ab=ba, cac-1=a-1b2, dad-1=ab2, ae=ea, bc=cb, bd=db, be=eb, cd=dc, ece-1=a2c, ede-1=b2d >
►On 64 points
(1 63 55 47)(2 44 56 60)(3 57 49 41)(4 46 50 62)(5 59 51 43)(6 48 52 64)(7 61 53 45)(8 42 54 58)(9 30 38 22)(10 19 39 27)(11 32 40 24)(12 21 33 29)(13 26 34 18)(14 23 35 31)(15 28 36 20)(16 17 37 25)
(1 3 5 7)(2 4 6 8)(9 11 13 15)(10 12 14 16)(17 19 21 23)(18 20 22 24)(25 27 29 31)(26 28 30 32)(33 35 37 39)(34 36 38 40)(41 43 45 47)(42 44 46 48)(49 51 53 55)(50 52 54 56)(57 59 61 63)(58 60 62 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 25 49 19 5 29 53 23)(2 26 50 20 6 30 54 24)(3 27 51 21 7 31 55 17)(4 28 52 22 8 32 56 18)(9 46 40 64 13 42 36 60)(10 47 33 57 14 43 37 61)(11 48 34 58 15 44 38 62)(12 41 35 59 16 45 39 63)
(1 63 55 47)(2 48 56 64)(3 57 49 41)(4 42 50 58)(5 59 51 43)(6 44 52 60)(7 61 53 45)(8 46 54 62)(9 26 38 18)(10 19 39 27)(11 28 40 20)(12 21 33 29)(13 30 34 22)(14 23 35 31)(15 32 36 24)(16 17 37 25)
G:=sub<Sym(64)| (1,63,55,47)(2,44,56,60)(3,57,49,41)(4,46,50,62)(5,59,51,43)(6,48,52,64)(7,61,53,45)(8,42,54,58)(9,30,38,22)(10,19,39,27)(11,32,40,24)(12,21,33,29)(13,26,34,18)(14,23,35,31)(15,28,36,20)(16,17,37,25), (1,3,5,7)(2,4,6,8)(9,11,13,15)(10,12,14,16)(17,19,21,23)(18,20,22,24)(25,27,29,31)(26,28,30,32)(33,35,37,39)(34,36,38,40)(41,43,45,47)(42,44,46,48)(49,51,53,55)(50,52,54,56)(57,59,61,63)(58,60,62,64), (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), (1,25,49,19,5,29,53,23)(2,26,50,20,6,30,54,24)(3,27,51,21,7,31,55,17)(4,28,52,22,8,32,56,18)(9,46,40,64,13,42,36,60)(10,47,33,57,14,43,37,61)(11,48,34,58,15,44,38,62)(12,41,35,59,16,45,39,63), (1,63,55,47)(2,48,56,64)(3,57,49,41)(4,42,50,58)(5,59,51,43)(6,44,52,60)(7,61,53,45)(8,46,54,62)(9,26,38,18)(10,19,39,27)(11,28,40,20)(12,21,33,29)(13,30,34,22)(14,23,35,31)(15,32,36,24)(16,17,37,25)>;
G:=Group( (1,63,55,47)(2,44,56,60)(3,57,49,41)(4,46,50,62)(5,59,51,43)(6,48,52,64)(7,61,53,45)(8,42,54,58)(9,30,38,22)(10,19,39,27)(11,32,40,24)(12,21,33,29)(13,26,34,18)(14,23,35,31)(15,28,36,20)(16,17,37,25), (1,3,5,7)(2,4,6,8)(9,11,13,15)(10,12,14,16)(17,19,21,23)(18,20,22,24)(25,27,29,31)(26,28,30,32)(33,35,37,39)(34,36,38,40)(41,43,45,47)(42,44,46,48)(49,51,53,55)(50,52,54,56)(57,59,61,63)(58,60,62,64), (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), (1,25,49,19,5,29,53,23)(2,26,50,20,6,30,54,24)(3,27,51,21,7,31,55,17)(4,28,52,22,8,32,56,18)(9,46,40,64,13,42,36,60)(10,47,33,57,14,43,37,61)(11,48,34,58,15,44,38,62)(12,41,35,59,16,45,39,63), (1,63,55,47)(2,48,56,64)(3,57,49,41)(4,42,50,58)(5,59,51,43)(6,44,52,60)(7,61,53,45)(8,46,54,62)(9,26,38,18)(10,19,39,27)(11,28,40,20)(12,21,33,29)(13,30,34,22)(14,23,35,31)(15,32,36,24)(16,17,37,25) );
G=PermutationGroup([(1,63,55,47),(2,44,56,60),(3,57,49,41),(4,46,50,62),(5,59,51,43),(6,48,52,64),(7,61,53,45),(8,42,54,58),(9,30,38,22),(10,19,39,27),(11,32,40,24),(12,21,33,29),(13,26,34,18),(14,23,35,31),(15,28,36,20),(16,17,37,25)], [(1,3,5,7),(2,4,6,8),(9,11,13,15),(10,12,14,16),(17,19,21,23),(18,20,22,24),(25,27,29,31),(26,28,30,32),(33,35,37,39),(34,36,38,40),(41,43,45,47),(42,44,46,48),(49,51,53,55),(50,52,54,56),(57,59,61,63),(58,60,62,64)], [(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)], [(1,25,49,19,5,29,53,23),(2,26,50,20,6,30,54,24),(3,27,51,21,7,31,55,17),(4,28,52,22,8,32,56,18),(9,46,40,64,13,42,36,60),(10,47,33,57,14,43,37,61),(11,48,34,58,15,44,38,62),(12,41,35,59,16,45,39,63)], [(1,63,55,47),(2,48,56,64),(3,57,49,41),(4,42,50,58),(5,59,51,43),(6,44,52,60),(7,61,53,45),(8,46,54,62),(9,26,38,18),(10,19,39,27),(11,28,40,20),(12,21,33,29),(13,30,34,22),(14,23,35,31),(15,32,36,24),(16,17,37,25)])
Matrix representation ►G ⊆ GL6(𝔽17)
| 14 | 15 | 0 | 0 | 0 | 0 |
| 5 | 3 | 0 | 0 | 0 | 0 |
| 0 | 0 | 11 | 15 | 0 | 0 |
| 0 | 0 | 9 | 6 | 0 | 0 |
| 0 | 0 | 0 | 0 | 6 | 2 |
| 0 | 0 | 0 | 0 | 8 | 11 |
| 16 | 0 | 0 | 0 | 0 | 0 |
| 0 | 16 | 0 | 0 | 0 | 0 |
| 0 | 0 | 13 | 0 | 0 | 0 |
| 0 | 0 | 0 | 13 | 0 | 0 |
| 0 | 0 | 0 | 0 | 13 | 0 |
| 0 | 0 | 0 | 0 | 0 | 13 |
| 14 | 7 | 0 | 0 | 0 | 0 |
| 1 | 3 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 13 | 0 | 0 | 0 |
| 0 | 0 | 0 | 13 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 13 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 13 | 0 |
| 3 | 2 | 0 | 0 | 0 | 0 |
| 12 | 14 | 0 | 0 | 0 | 0 |
| 0 | 0 | 11 | 15 | 0 | 0 |
| 0 | 0 | 9 | 6 | 0 | 0 |
| 0 | 0 | 0 | 0 | 11 | 15 |
| 0 | 0 | 0 | 0 | 9 | 6 |
G:=sub<GL(6,GF(17))| [14,5,0,0,0,0,15,3,0,0,0,0,0,0,11,9,0,0,0,0,15,6,0,0,0,0,0,0,6,8,0,0,0,0,2,11],[16,0,0,0,0,0,0,16,0,0,0,0,0,0,13,0,0,0,0,0,0,13,0,0,0,0,0,0,13,0,0,0,0,0,0,13],[14,1,0,0,0,0,7,3,0,0,0,0,0,0,0,0,13,0,0,0,0,0,0,13,0,0,1,0,0,0,0,0,0,1,0,0],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,13,0,0,0,0,1,0,0,0,0,0,0,0,0,13,0,0,0,0,1,0],[3,12,0,0,0,0,2,14,0,0,0,0,0,0,11,9,0,0,0,0,15,6,0,0,0,0,0,0,11,9,0,0,0,0,15,6] >;
44 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 4A | 4B | 4C | 4D | 4E | 4F | 4G | ··· | 4R | 8A | ··· | 8H | 8I | ··· | 8T |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 8 | ··· | 8 | 8 | ··· | 8 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 1 | 1 | 1 | 1 | 2 | 2 | 4 | ··· | 4 | 2 | ··· | 2 | 4 | ··· | 4 |
44 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 4 |
| type | + | + | + | + | + | + | + | - | |||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C4 | C4 | C4 | Q8 | C4○D4 | Q8○M4(2) |
| kernel | C42.287C23 | C2×C8⋊C4 | C8○2M4(2) | C4⋊M4(2) | C42.6C22 | C8⋊4Q8 | C23.37C23 | C22⋊Q8 | C42.C2 | C4⋊Q8 | C2×C8 | C2×C4 | C2 |
| # reps | 1 | 1 | 2 | 1 | 2 | 8 | 1 | 8 | 4 | 4 | 4 | 4 | 4 |
In GAP, Magma, Sage, TeX
C_4^2._{287}C_2^3 % in TeX
G:=Group("C4^2.287C2^3"); // GroupNames label
G:=SmallGroup(128,1693);
// by ID
G=gap.SmallGroup(128,1693);
# by ID
G:=PCGroup([7,-2,2,2,2,-2,2,-2,224,253,120,723,268,2019,124]);
// Polycyclic
G:=Group<a,b,c,d,e|a^4=b^4=1,c^2=b,d^2=a^2*b,e^2=a^2,a*b=b*a,c*a*c^-1=a^-1*b^2,d*a*d^-1=a*b^2,a*e=e*a,b*c=c*b,b*d=d*b,b*e=e*b,c*d=d*c,e*c*e^-1=a^2*c,e*d*e^-1=b^2*d>;
// generators/relations