p-group, metabelian, nilpotent (class 3), monomial
Aliases: C42.355D4, C42.706C23, C4○(C4.4D8), C4.23(C4○D8), C4.4D8⋊45C2, C4⋊C4.84C23, C4○(C4.SD16), (C2×C4).329C24, (C4×C8).383C22, (C2×C8).491C23, C4.SD16⋊46C2, (C2×D4).98C23, (C22×C4).610D4, C23.387(C2×D4), C4⋊Q8.272C22, (C2×Q8).86C23, C4.97(C4.4D4), C23.24D4⋊5C2, C4⋊1D4.144C22, (C22×C8).520C22, C22.4(C4.4D4), C22.589(C22×D4), D4⋊C4.144C22, C4○(C42.78C22), C23.37C23⋊8C2, (C22×C4).1551C23, (C2×C42).1124C22, Q8⋊C4.136C22, C4.4D4.133C22, C42.C2.109C22, C42.78C22⋊32C2, C42⋊C2.137C22, C22.26C24.33C2, (C2×C4×C8)⋊22C2, C2.29(C2×C4○D8), C4.38(C2×C4○D4), (C2×C4)○(C4.4D8), (C2×C4).694(C2×D4), (C2×C4)○(C4.SD16), C2.40(C2×C4.4D4), (C2×C4).708(C4○D4), (C2×C4○D4).147C22, (C2×C4)○(C42.78C22), SmallGroup(128,1863)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Subgroups: 372 in 200 conjugacy classes, 96 normal (16 characteristic)
C1, C2, C2 [×2], C2 [×4], C4 [×8], C4 [×8], C22, C22 [×2], C22 [×8], C8 [×4], C2×C4 [×2], C2×C4 [×8], C2×C4 [×14], D4 [×12], Q8 [×8], C23, C23 [×2], C42 [×2], C42 [×2], C42 [×2], C22⋊C4 [×6], C4⋊C4 [×4], C4⋊C4 [×8], C2×C8 [×4], C2×C8 [×4], C22×C4, C22×C4 [×2], C22×C4 [×2], C2×D4 [×2], C2×D4 [×4], C2×Q8 [×2], C2×Q8 [×2], C4○D4 [×8], C4×C8 [×4], D4⋊C4 [×8], Q8⋊C4 [×8], C2×C42, C42⋊C2 [×2], C4×D4 [×2], C4×Q8 [×2], C4⋊D4 [×2], C22⋊Q8 [×2], C4.4D4 [×2], C42.C2 [×2], C4⋊1D4, C4⋊Q8, C4⋊Q8 [×2], C22×C8 [×2], C2×C4○D4 [×2], C2×C4×C8, C23.24D4 [×4], C4.4D8 [×2], C4.SD16 [×2], C42.78C22 [×4], C22.26C24, C23.37C23, C42.355D4
Quotients:
C1, C2 [×15], C22 [×35], D4 [×4], C23 [×15], C2×D4 [×6], C4○D4 [×4], C24, C4.4D4 [×4], C4○D8 [×4], C22×D4, C2×C4○D4 [×2], C2×C4.4D4, C2×C4○D8 [×2], C42.355D4
Generators and relations
G = < a,b,c,d | a4=b4=1, c4=b2, d2=a2, ab=ba, ac=ca, dad-1=a-1, bc=cb, bd=db, dcd-1=a2b2c3 >
►On 64 points
(1 60 55 20)(2 61 56 21)(3 62 49 22)(4 63 50 23)(5 64 51 24)(6 57 52 17)(7 58 53 18)(8 59 54 19)(9 46 35 26)(10 47 36 27)(11 48 37 28)(12 41 38 29)(13 42 39 30)(14 43 40 31)(15 44 33 32)(16 45 34 25)
(1 45 5 41)(2 46 6 42)(3 47 7 43)(4 48 8 44)(9 17 13 21)(10 18 14 22)(11 19 15 23)(12 20 16 24)(25 51 29 55)(26 52 30 56)(27 53 31 49)(28 54 32 50)(33 63 37 59)(34 64 38 60)(35 57 39 61)(36 58 40 62)
(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 50 55 4)(2 3 56 49)(5 54 51 8)(6 7 52 53)(9 36 35 10)(11 34 37 16)(12 15 38 33)(13 40 39 14)(17 58 57 18)(19 64 59 24)(20 23 60 63)(21 62 61 22)(25 48 45 28)(26 27 46 47)(29 44 41 32)(30 31 42 43)
G:=sub<Sym(64)| (1,60,55,20)(2,61,56,21)(3,62,49,22)(4,63,50,23)(5,64,51,24)(6,57,52,17)(7,58,53,18)(8,59,54,19)(9,46,35,26)(10,47,36,27)(11,48,37,28)(12,41,38,29)(13,42,39,30)(14,43,40,31)(15,44,33,32)(16,45,34,25), (1,45,5,41)(2,46,6,42)(3,47,7,43)(4,48,8,44)(9,17,13,21)(10,18,14,22)(11,19,15,23)(12,20,16,24)(25,51,29,55)(26,52,30,56)(27,53,31,49)(28,54,32,50)(33,63,37,59)(34,64,38,60)(35,57,39,61)(36,58,40,62), (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), (1,50,55,4)(2,3,56,49)(5,54,51,8)(6,7,52,53)(9,36,35,10)(11,34,37,16)(12,15,38,33)(13,40,39,14)(17,58,57,18)(19,64,59,24)(20,23,60,63)(21,62,61,22)(25,48,45,28)(26,27,46,47)(29,44,41,32)(30,31,42,43)>;
G:=Group( (1,60,55,20)(2,61,56,21)(3,62,49,22)(4,63,50,23)(5,64,51,24)(6,57,52,17)(7,58,53,18)(8,59,54,19)(9,46,35,26)(10,47,36,27)(11,48,37,28)(12,41,38,29)(13,42,39,30)(14,43,40,31)(15,44,33,32)(16,45,34,25), (1,45,5,41)(2,46,6,42)(3,47,7,43)(4,48,8,44)(9,17,13,21)(10,18,14,22)(11,19,15,23)(12,20,16,24)(25,51,29,55)(26,52,30,56)(27,53,31,49)(28,54,32,50)(33,63,37,59)(34,64,38,60)(35,57,39,61)(36,58,40,62), (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), (1,50,55,4)(2,3,56,49)(5,54,51,8)(6,7,52,53)(9,36,35,10)(11,34,37,16)(12,15,38,33)(13,40,39,14)(17,58,57,18)(19,64,59,24)(20,23,60,63)(21,62,61,22)(25,48,45,28)(26,27,46,47)(29,44,41,32)(30,31,42,43) );
G=PermutationGroup([(1,60,55,20),(2,61,56,21),(3,62,49,22),(4,63,50,23),(5,64,51,24),(6,57,52,17),(7,58,53,18),(8,59,54,19),(9,46,35,26),(10,47,36,27),(11,48,37,28),(12,41,38,29),(13,42,39,30),(14,43,40,31),(15,44,33,32),(16,45,34,25)], [(1,45,5,41),(2,46,6,42),(3,47,7,43),(4,48,8,44),(9,17,13,21),(10,18,14,22),(11,19,15,23),(12,20,16,24),(25,51,29,55),(26,52,30,56),(27,53,31,49),(28,54,32,50),(33,63,37,59),(34,64,38,60),(35,57,39,61),(36,58,40,62)], [(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)], [(1,50,55,4),(2,3,56,49),(5,54,51,8),(6,7,52,53),(9,36,35,10),(11,34,37,16),(12,15,38,33),(13,40,39,14),(17,58,57,18),(19,64,59,24),(20,23,60,63),(21,62,61,22),(25,48,45,28),(26,27,46,47),(29,44,41,32),(30,31,42,43)])
Matrix representation ►G ⊆ GL4(𝔽17) generated by
| 4 | 8 | 0 | 0 |
| 0 | 13 | 0 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 0 | 16 | 0 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 13 | 0 |
| 0 | 0 | 0 | 13 |
| 4 | 0 | 0 | 0 |
| 0 | 4 | 0 | 0 |
| 0 | 0 | 5 | 5 |
| 0 | 0 | 12 | 5 |
| 4 | 0 | 0 | 0 |
| 13 | 13 | 0 | 0 |
| 0 | 0 | 5 | 12 |
| 0 | 0 | 12 | 12 |
G:=sub<GL(4,GF(17))| [4,0,0,0,8,13,0,0,0,0,0,16,0,0,1,0],[1,0,0,0,0,1,0,0,0,0,13,0,0,0,0,13],[4,0,0,0,0,4,0,0,0,0,5,12,0,0,5,5],[4,13,0,0,0,13,0,0,0,0,5,12,0,0,12,12] >;
44 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 4A | 4B | 4C | 4D | 4E | ··· | 4N | 4O | ··· | 4T | 8A | ··· | 8P |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 4 | ··· | 4 | 8 | ··· | 8 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 8 | 8 | 1 | 1 | 1 | 1 | 2 | ··· | 2 | 8 | ··· | 8 | 2 | ··· | 2 |
44 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | + | + | + | ||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | D4 | D4 | C4○D4 | C4○D8 |
| kernel | C42.355D4 | C2×C4×C8 | C23.24D4 | C4.4D8 | C4.SD16 | C42.78C22 | C22.26C24 | C23.37C23 | C42 | C22×C4 | C2×C4 | C4 |
| # reps | 1 | 1 | 4 | 2 | 2 | 4 | 1 | 1 | 2 | 2 | 8 | 16 |
In GAP, Magma, Sage, TeX
C_4^2._{355}D_4 % in TeX
G:=Group("C4^2.355D4"); // GroupNames label
G:=SmallGroup(128,1863);
// by ID
G=gap.SmallGroup(128,1863);
# by ID
G:=PCGroup([7,-2,2,2,2,-2,2,-2,253,232,758,100,248,2804,172,4037,124]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^4=1,c^4=b^2,d^2=a^2,a*b=b*a,a*c=c*a,d*a*d^-1=a^-1,b*c=c*b,b*d=d*b,d*c*d^-1=a^2*b^2*c^3>;
// generators/relations