p-group, metabelian, nilpotent (class 2), monomial
Aliases: C42.286C23, C8○2(C4⋊Q8), (C2×C8)⋊16Q8, (C8×Q8)⋊27C2, C8○(C8⋊4Q8), C4⋊Q8.39C4, C8.45(C2×Q8), C4.21(C4×Q8), C8⋊4Q8⋊50C2, C8○2(C22⋊Q8), C8○(C42.C2), C4.19(C8○D4), C22.6(C4×Q8), C22⋊Q8.35C4, C4.64(C22×Q8), C4⋊C8.358C22, (C2×C8).640C23, C42.286(C2×C4), (C2×C4).657C24, (C4×C8).329C22, C42.C2.27C4, C8○2(C4⋊M4(2)), (C4×Q8).277C22, C8⋊C4.158C22, C8○2M4(2).22C2, C4⋊M4(2).39C2, C23.142(C22×C4), (C22×C8).514C22, C22.183(C23×C4), C8○2(C42.6C22), C8○(C23.37C23), (C22×C4).1519C23, (C2×C42).1117C22, C42⋊C2.303C22, (C2×M4(2)).360C22, C42.6C22.15C2, C23.37C23.47C2, (C2×C4×C8).69C2, (C2×C8)○(C4⋊Q8), C2.24(C2×C4×Q8), C2.21(C2×C8○D4), C4⋊C4.162(C2×C4), C4.308(C2×C4○D4), (C2×C4).242(C2×Q8), C22⋊C4.38(C2×C4), (C2×C4).74(C22×C4), (C2×Q8).163(C2×C4), (C2×C4).695(C4○D4), (C22×C4).419(C2×C4), (C2×C8)○(C23.37C23), SmallGroup(128,1692)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Subgroups: 220 in 182 conjugacy classes, 144 normal (20 characteristic)
C1, C2, C2 [×2], C2 [×2], C4 [×2], C4 [×6], C4 [×10], C22, C22 [×2], C22 [×2], C8 [×4], C8 [×6], C2×C4 [×2], C2×C4 [×16], C2×C4 [×4], Q8 [×8], C23, C42 [×2], C42 [×6], C22⋊C4 [×4], C4⋊C4 [×16], C2×C8 [×2], C2×C8 [×10], C2×C8 [×2], M4(2) [×4], C22×C4, C22×C4 [×2], C2×Q8 [×4], C4×C8 [×8], C8⋊C4 [×4], 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×C4×C8, C8○2M4(2) [×2], C4⋊M4(2), C42.6C22 [×2], C8×Q8 [×4], C8⋊4Q8 [×4], C23.37C23, C42.286C23
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], C8○D4 [×4], C23×C4, C22×Q8, C2×C4○D4, C2×C4×Q8, C2×C8○D4 [×2], C42.286C23
Generators and relations
G = < a,b,c,d,e | a4=b4=e2=1, c2=a2, d2=b, ab=ba, cac-1=a-1b2, ad=da, eae=ab2, bc=cb, bd=db, be=eb, dcd-1=a2b2c, ce=ec, ede=b2d >
►On 64 points
(1 50 61 21)(2 51 62 22)(3 52 63 23)(4 53 64 24)(5 54 57 17)(6 55 58 18)(7 56 59 19)(8 49 60 20)(9 36 26 42)(10 37 27 43)(11 38 28 44)(12 39 29 45)(13 40 30 46)(14 33 31 47)(15 34 32 48)(16 35 25 41)
(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 39 61 45)(2 42 62 36)(3 33 63 47)(4 44 64 38)(5 35 57 41)(6 46 58 40)(7 37 59 43)(8 48 60 34)(9 55 26 18)(10 23 27 52)(11 49 28 20)(12 17 29 54)(13 51 30 22)(14 19 31 56)(15 53 32 24)(16 21 25 50)
(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)
(2 6)(4 8)(10 14)(12 16)(17 21)(19 23)(25 29)(27 31)(34 38)(36 40)(42 46)(44 48)(50 54)(52 56)(58 62)(60 64)
G:=sub<Sym(64)| (1,50,61,21)(2,51,62,22)(3,52,63,23)(4,53,64,24)(5,54,57,17)(6,55,58,18)(7,56,59,19)(8,49,60,20)(9,36,26,42)(10,37,27,43)(11,38,28,44)(12,39,29,45)(13,40,30,46)(14,33,31,47)(15,34,32,48)(16,35,25,41), (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), (1,39,61,45)(2,42,62,36)(3,33,63,47)(4,44,64,38)(5,35,57,41)(6,46,58,40)(7,37,59,43)(8,48,60,34)(9,55,26,18)(10,23,27,52)(11,49,28,20)(12,17,29,54)(13,51,30,22)(14,19,31,56)(15,53,32,24)(16,21,25,50), (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), (2,6)(4,8)(10,14)(12,16)(17,21)(19,23)(25,29)(27,31)(34,38)(36,40)(42,46)(44,48)(50,54)(52,56)(58,62)(60,64)>;
G:=Group( (1,50,61,21)(2,51,62,22)(3,52,63,23)(4,53,64,24)(5,54,57,17)(6,55,58,18)(7,56,59,19)(8,49,60,20)(9,36,26,42)(10,37,27,43)(11,38,28,44)(12,39,29,45)(13,40,30,46)(14,33,31,47)(15,34,32,48)(16,35,25,41), (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), (1,39,61,45)(2,42,62,36)(3,33,63,47)(4,44,64,38)(5,35,57,41)(6,46,58,40)(7,37,59,43)(8,48,60,34)(9,55,26,18)(10,23,27,52)(11,49,28,20)(12,17,29,54)(13,51,30,22)(14,19,31,56)(15,53,32,24)(16,21,25,50), (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), (2,6)(4,8)(10,14)(12,16)(17,21)(19,23)(25,29)(27,31)(34,38)(36,40)(42,46)(44,48)(50,54)(52,56)(58,62)(60,64) );
G=PermutationGroup([(1,50,61,21),(2,51,62,22),(3,52,63,23),(4,53,64,24),(5,54,57,17),(6,55,58,18),(7,56,59,19),(8,49,60,20),(9,36,26,42),(10,37,27,43),(11,38,28,44),(12,39,29,45),(13,40,30,46),(14,33,31,47),(15,34,32,48),(16,35,25,41)], [(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)], [(1,39,61,45),(2,42,62,36),(3,33,63,47),(4,44,64,38),(5,35,57,41),(6,46,58,40),(7,37,59,43),(8,48,60,34),(9,55,26,18),(10,23,27,52),(11,49,28,20),(12,17,29,54),(13,51,30,22),(14,19,31,56),(15,53,32,24),(16,21,25,50)], [(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)], [(2,6),(4,8),(10,14),(12,16),(17,21),(19,23),(25,29),(27,31),(34,38),(36,40),(42,46),(44,48),(50,54),(52,56),(58,62),(60,64)])
Matrix representation ►G ⊆ GL4(𝔽17) generated by
| 0 | 15 | 0 | 0 |
| 8 | 0 | 0 | 0 |
| 0 | 0 | 9 | 16 |
| 0 | 0 | 14 | 8 |
| 4 | 0 | 0 | 0 |
| 0 | 4 | 0 | 0 |
| 0 | 0 | 13 | 0 |
| 0 | 0 | 0 | 13 |
| 1 | 0 | 0 | 0 |
| 0 | 16 | 0 | 0 |
| 0 | 0 | 4 | 0 |
| 0 | 0 | 0 | 4 |
| 0 | 13 | 0 | 0 |
| 16 | 0 | 0 | 0 |
| 0 | 0 | 16 | 2 |
| 0 | 0 | 6 | 1 |
| 1 | 0 | 0 | 0 |
| 0 | 16 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 1 | 16 |
G:=sub<GL(4,GF(17))| [0,8,0,0,15,0,0,0,0,0,9,14,0,0,16,8],[4,0,0,0,0,4,0,0,0,0,13,0,0,0,0,13],[1,0,0,0,0,16,0,0,0,0,4,0,0,0,0,4],[0,16,0,0,13,0,0,0,0,0,16,6,0,0,2,1],[1,0,0,0,0,16,0,0,0,0,1,1,0,0,0,16] >;
56 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 4A | 4B | 4C | 4D | 4E | ··· | 4N | 4O | ··· | 4V | 8A | ··· | 8H | 8I | ··· | 8T | 8U | ··· | 8AB |
| 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 | 1 | 1 | 1 | 1 | 2 | ··· | 2 | 4 | ··· | 4 | 1 | ··· | 1 | 2 | ··· | 2 | 4 | ··· | 4 |
56 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | + | - | |||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C4 | C4 | C4 | Q8 | C4○D4 | C8○D4 |
| kernel | C42.286C23 | C2×C4×C8 | C8○2M4(2) | C4⋊M4(2) | C42.6C22 | C8×Q8 | C8⋊4Q8 | C23.37C23 | C22⋊Q8 | C42.C2 | C4⋊Q8 | C2×C8 | C2×C4 | C4 |
| # reps | 1 | 1 | 2 | 1 | 2 | 4 | 4 | 1 | 8 | 4 | 4 | 4 | 4 | 16 |
In GAP, Magma, Sage, TeX
C_4^2._{286}C_2^3 % in TeX
G:=Group("C4^2.286C2^3"); // GroupNames label
G:=SmallGroup(128,1692);
// by ID
G=gap.SmallGroup(128,1692);
# by ID
G:=PCGroup([7,-2,2,2,2,-2,2,-2,224,253,120,268,2019,124]);
// Polycyclic
G:=Group<a,b,c,d,e|a^4=b^4=e^2=1,c^2=a^2,d^2=b,a*b=b*a,c*a*c^-1=a^-1*b^2,a*d=d*a,e*a*e=a*b^2,b*c=c*b,b*d=d*b,b*e=e*b,d*c*d^-1=a^2*b^2*c,c*e=e*c,e*d*e=b^2*d>;
// generators/relations