p-group, metabelian, nilpotent (class 3), monomial
Aliases: C42.259D4, C42.725C23, C8⋊2(C4○D4), C8⋊5D4⋊7C2, C8⋊D4⋊10C2, C8⋊2Q8⋊30C2, C4.4D8⋊43C2, (C4×M4(2))⋊9C2, C4⋊C4.121C23, C4.23(C8⋊C22), (C4×C8).185C22, (C2×C8).462C23, (C2×C4).380C24, C4.SD16⋊44C2, (C22×C4).479D4, C23.267(C2×D4), C4⋊Q8.296C22, SD16⋊C4⋊25C2, (C4×Q8).97C22, C2.D8.99C22, (C4×D4).100C22, (C2×D4).134C23, C4.23(C8.C22), (C2×Q8).122C23, C8⋊C4.137C22, C4⋊D4.177C22, C4⋊1D4.158C22, (C2×C42).866C22, (C2×SD16).28C22, C22.640(C22×D4), C22⋊Q8.182C22, D4⋊C4.138C22, (C22×C4).1058C23, Q8⋊C4.131C22, C23.37C23⋊15C2, (C2×M4(2)).288C22, C22.26C24.39C2, C2.77(C22.26C24), C4.65(C2×C4○D4), C2.47(C2×C8⋊C22), (C2×C4).1224(C2×D4), C2.47(C2×C8.C22), SmallGroup(128,1914)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Subgroups: 388 in 201 conjugacy classes, 92 normal (34 characteristic)
C1, C2 [×3], C2 [×3], C4 [×6], C4 [×9], C22, C22 [×9], C8 [×4], C8 [×2], C2×C4 [×6], C2×C4 [×16], D4 [×12], Q8 [×8], C23, C23 [×2], C42 [×4], C42 [×2], C22⋊C4 [×6], C4⋊C4 [×4], C4⋊C4 [×8], C2×C8 [×4], M4(2) [×4], SD16 [×8], C22×C4 [×3], C22×C4 [×2], C2×D4 [×2], C2×D4 [×4], C2×Q8 [×2], C2×Q8 [×2], C4○D4 [×4], C4×C8 [×2], C8⋊C4 [×2], D4⋊C4 [×4], Q8⋊C4 [×4], C2.D8 [×4], C2×C42, C42⋊C2, C4×D4 [×2], C4×D4, C4×Q8 [×2], C4×Q8, C4⋊D4 [×2], C4⋊D4, C22⋊Q8 [×2], C22⋊Q8, C4.4D4, C42.C2, C4⋊1D4, C4⋊Q8 [×3], C2×M4(2) [×2], C2×SD16 [×4], C2×C4○D4, C4×M4(2), SD16⋊C4 [×4], C8⋊D4 [×4], C4.4D8, C4.SD16, C8⋊5D4, C8⋊2Q8, C22.26C24, C23.37C23, C42.259D4
Quotients:
C1, C2 [×15], C22 [×35], D4 [×4], C23 [×15], C2×D4 [×6], C4○D4 [×4], C24, C8⋊C22 [×2], C8.C22 [×2], C22×D4, C2×C4○D4 [×2], C22.26C24, C2×C8⋊C22, C2×C8.C22, C42.259D4
Generators and relations
G = < a,b,c,d | a4=b4=1, c4=d2=a2, ab=ba, cac-1=a-1, dad-1=a-1b2, cbc-1=a2b, bd=db, dcd-1=c3 >
►On 64 points
(1 48 5 44)(2 45 6 41)(3 42 7 46)(4 47 8 43)(9 55 13 51)(10 52 14 56)(11 49 15 53)(12 54 16 50)(17 34 21 38)(18 39 22 35)(19 36 23 40)(20 33 24 37)(25 63 29 59)(26 60 30 64)(27 57 31 61)(28 62 32 58)
(1 40 27 14)(2 37 28 11)(3 34 29 16)(4 39 30 13)(5 36 31 10)(6 33 32 15)(7 38 25 12)(8 35 26 9)(17 63 54 46)(18 60 55 43)(19 57 56 48)(20 62 49 45)(21 59 50 42)(22 64 51 47)(23 61 52 44)(24 58 53 41)
(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 4 5 8)(2 7 6 3)(9 14 13 10)(11 12 15 16)(17 49 21 53)(18 52 22 56)(19 55 23 51)(20 50 24 54)(25 32 29 28)(26 27 30 31)(33 34 37 38)(35 40 39 36)(41 63 45 59)(42 58 46 62)(43 61 47 57)(44 64 48 60)
G:=sub<Sym(64)| (1,48,5,44)(2,45,6,41)(3,42,7,46)(4,47,8,43)(9,55,13,51)(10,52,14,56)(11,49,15,53)(12,54,16,50)(17,34,21,38)(18,39,22,35)(19,36,23,40)(20,33,24,37)(25,63,29,59)(26,60,30,64)(27,57,31,61)(28,62,32,58), (1,40,27,14)(2,37,28,11)(3,34,29,16)(4,39,30,13)(5,36,31,10)(6,33,32,15)(7,38,25,12)(8,35,26,9)(17,63,54,46)(18,60,55,43)(19,57,56,48)(20,62,49,45)(21,59,50,42)(22,64,51,47)(23,61,52,44)(24,58,53,41), (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), (1,4,5,8)(2,7,6,3)(9,14,13,10)(11,12,15,16)(17,49,21,53)(18,52,22,56)(19,55,23,51)(20,50,24,54)(25,32,29,28)(26,27,30,31)(33,34,37,38)(35,40,39,36)(41,63,45,59)(42,58,46,62)(43,61,47,57)(44,64,48,60)>;
G:=Group( (1,48,5,44)(2,45,6,41)(3,42,7,46)(4,47,8,43)(9,55,13,51)(10,52,14,56)(11,49,15,53)(12,54,16,50)(17,34,21,38)(18,39,22,35)(19,36,23,40)(20,33,24,37)(25,63,29,59)(26,60,30,64)(27,57,31,61)(28,62,32,58), (1,40,27,14)(2,37,28,11)(3,34,29,16)(4,39,30,13)(5,36,31,10)(6,33,32,15)(7,38,25,12)(8,35,26,9)(17,63,54,46)(18,60,55,43)(19,57,56,48)(20,62,49,45)(21,59,50,42)(22,64,51,47)(23,61,52,44)(24,58,53,41), (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), (1,4,5,8)(2,7,6,3)(9,14,13,10)(11,12,15,16)(17,49,21,53)(18,52,22,56)(19,55,23,51)(20,50,24,54)(25,32,29,28)(26,27,30,31)(33,34,37,38)(35,40,39,36)(41,63,45,59)(42,58,46,62)(43,61,47,57)(44,64,48,60) );
G=PermutationGroup([(1,48,5,44),(2,45,6,41),(3,42,7,46),(4,47,8,43),(9,55,13,51),(10,52,14,56),(11,49,15,53),(12,54,16,50),(17,34,21,38),(18,39,22,35),(19,36,23,40),(20,33,24,37),(25,63,29,59),(26,60,30,64),(27,57,31,61),(28,62,32,58)], [(1,40,27,14),(2,37,28,11),(3,34,29,16),(4,39,30,13),(5,36,31,10),(6,33,32,15),(7,38,25,12),(8,35,26,9),(17,63,54,46),(18,60,55,43),(19,57,56,48),(20,62,49,45),(21,59,50,42),(22,64,51,47),(23,61,52,44),(24,58,53,41)], [(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)], [(1,4,5,8),(2,7,6,3),(9,14,13,10),(11,12,15,16),(17,49,21,53),(18,52,22,56),(19,55,23,51),(20,50,24,54),(25,32,29,28),(26,27,30,31),(33,34,37,38),(35,40,39,36),(41,63,45,59),(42,58,46,62),(43,61,47,57),(44,64,48,60)])
Matrix representation ►G ⊆ GL6(𝔽17)
| 4 | 9 | 0 | 0 | 0 | 0 |
| 4 | 13 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 16 | 0 |
| 0 | 0 | 0 | 0 | 0 | 16 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 4 | 0 | 0 | 0 | 0 | 0 |
| 0 | 4 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 16 | 0 |
| 0 | 0 | 0 | 16 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 1 | 15 | 0 | 0 | 0 | 0 |
| 1 | 16 | 0 | 0 | 0 | 0 |
| 0 | 0 | 3 | 14 | 4 | 13 |
| 0 | 0 | 3 | 3 | 4 | 4 |
| 0 | 0 | 4 | 13 | 14 | 3 |
| 0 | 0 | 4 | 4 | 14 | 14 |
| 16 | 2 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 3 | 14 | 4 | 13 |
| 0 | 0 | 14 | 14 | 13 | 13 |
| 0 | 0 | 4 | 13 | 14 | 3 |
| 0 | 0 | 13 | 13 | 3 | 3 |
G:=sub<GL(6,GF(17))| [4,4,0,0,0,0,9,13,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,16,0,0,0,0,0,0,16,0,0],[4,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,1,0,0,0,0,16,0,0,0,0,16,0,0,0,0,1,0,0,0],[1,1,0,0,0,0,15,16,0,0,0,0,0,0,3,3,4,4,0,0,14,3,13,4,0,0,4,4,14,14,0,0,13,4,3,14],[16,0,0,0,0,0,2,1,0,0,0,0,0,0,3,14,4,13,0,0,14,14,13,13,0,0,4,13,14,3,0,0,13,13,3,3] >;
32 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 4A | ··· | 4J | 4K | 4L | ··· | 4Q | 8A | ··· | 8H |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 4 | 4 | ··· | 4 | 8 | ··· | 8 |
| size | 1 | 1 | 1 | 1 | 4 | 8 | 8 | 2 | ··· | 2 | 4 | 8 | ··· | 8 | 4 | ··· | 4 |
32 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | - | |
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | D4 | D4 | C4○D4 | C8⋊C22 | C8.C22 |
| kernel | C42.259D4 | C4×M4(2) | SD16⋊C4 | C8⋊D4 | C4.4D8 | C4.SD16 | C8⋊5D4 | C8⋊2Q8 | C22.26C24 | C23.37C23 | C42 | C22×C4 | C8 | C4 | C4 |
| # reps | 1 | 1 | 4 | 4 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 8 | 2 | 2 |
In GAP, Magma, Sage, TeX
C_4^2._{259}D_4 % in TeX
G:=Group("C4^2.259D4"); // GroupNames label
G:=SmallGroup(128,1914);
// by ID
G=gap.SmallGroup(128,1914);
# by ID
G:=PCGroup([7,-2,2,2,2,-2,2,-2,253,758,723,184,521,80,4037,124]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^4=1,c^4=d^2=a^2,a*b=b*a,c*a*c^-1=a^-1,d*a*d^-1=a^-1*b^2,c*b*c^-1=a^2*b,b*d=d*b,d*c*d^-1=c^3>;
// generators/relations