p-group, metabelian, nilpotent (class 3), monomial
Aliases: C42.239D4, C42.365C23, C4⋊C4.88C23, (C2×C8).454C23, (C2×C4).333C24, C23.388(C2×D4), (C22×C4).459D4, C4⋊Q8.274C22, (C2×Q8).89C23, C4.98(C4.4D4), (C2×D4).101C23, C8⋊C4.167C22, C23.24D4⋊42C2, C4⋊1D4.146C22, (C2×C42).846C22, (C22×C8).460C22, C22.5(C4.4D4), C22.593(C22×D4), D4⋊C4.201C22, C2.35(D8⋊C22), C4○(C42.28C22), C4○(C42.29C22), C4○(C42.30C22), C23.37C23⋊9C2, (C22×C4).1555C23, Q8⋊C4.202C22, C4.4D4.135C22, C42.C2.111C22, C42.29C22⋊23C2, C42⋊C2.138C22, C42.28C22⋊37C2, C42.30C22⋊23C2, C22.26C24.34C2, (C2×C8⋊C4)⋊39C2, C4.42(C2×C4○D4), (C2×C4).513(C2×D4), C2.44(C2×C4.4D4), (C2×C4).712(C4○D4), (C2×C4○D4).148C22, (C2×C4)○(C42.30C22), (C2×C4)○(C42.28C22), (C2×C4)○(C42.29C22), SmallGroup(128,1867)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Subgroups: 372 in 196 conjugacy classes, 92 normal (16 characteristic)
C1, C2, C2 [×2], C2 [×4], C4 [×4], C4 [×10], C22, C22 [×2], C22 [×8], C8 [×4], C2×C4 [×2], C2×C4 [×8], C2×C4 [×12], 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], C8⋊C4 [×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×C8⋊C4, C23.24D4 [×4], C42.28C22 [×4], C42.29C22 [×2], C42.30C22 [×2], C22.26C24, C23.37C23, C42.239D4
Quotients:
C1, C2 [×15], C22 [×35], D4 [×4], C23 [×15], C2×D4 [×6], C4○D4 [×4], C24, C4.4D4 [×4], C22×D4, C2×C4○D4 [×2], C2×C4.4D4, D8⋊C22 [×2], C42.239D4
Generators and relations
G = < a,b,c,d | a4=b4=1, c4=b2, d2=a2b2, ab=ba, cac-1=ab2, dad-1=a-1, bc=cb, bd=db, dcd-1=a2c3 >
►On 64 points
(1 64 51 20)(2 61 52 17)(3 58 53 22)(4 63 54 19)(5 60 55 24)(6 57 56 21)(7 62 49 18)(8 59 50 23)(9 44 33 28)(10 41 34 25)(11 46 35 30)(12 43 36 27)(13 48 37 32)(14 45 38 29)(15 42 39 26)(16 47 40 31)
(1 45 5 41)(2 46 6 42)(3 47 7 43)(4 48 8 44)(9 19 13 23)(10 20 14 24)(11 21 15 17)(12 22 16 18)(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 34 37 14)(10 13 38 33)(11 40 39 12)(15 36 35 16)(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)| (1,64,51,20)(2,61,52,17)(3,58,53,22)(4,63,54,19)(5,60,55,24)(6,57,56,21)(7,62,49,18)(8,59,50,23)(9,44,33,28)(10,41,34,25)(11,46,35,30)(12,43,36,27)(13,48,37,32)(14,45,38,29)(15,42,39,26)(16,47,40,31), (1,45,5,41)(2,46,6,42)(3,47,7,43)(4,48,8,44)(9,19,13,23)(10,20,14,24)(11,21,15,17)(12,22,16,18)(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)(9,34,37,14)(10,13,38,33)(11,40,39,12)(15,36,35,16)(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( (1,64,51,20)(2,61,52,17)(3,58,53,22)(4,63,54,19)(5,60,55,24)(6,57,56,21)(7,62,49,18)(8,59,50,23)(9,44,33,28)(10,41,34,25)(11,46,35,30)(12,43,36,27)(13,48,37,32)(14,45,38,29)(15,42,39,26)(16,47,40,31), (1,45,5,41)(2,46,6,42)(3,47,7,43)(4,48,8,44)(9,19,13,23)(10,20,14,24)(11,21,15,17)(12,22,16,18)(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)(9,34,37,14)(10,13,38,33)(11,40,39,12)(15,36,35,16)(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([(1,64,51,20),(2,61,52,17),(3,58,53,22),(4,63,54,19),(5,60,55,24),(6,57,56,21),(7,62,49,18),(8,59,50,23),(9,44,33,28),(10,41,34,25),(11,46,35,30),(12,43,36,27),(13,48,37,32),(14,45,38,29),(15,42,39,26),(16,47,40,31)], [(1,45,5,41),(2,46,6,42),(3,47,7,43),(4,48,8,44),(9,19,13,23),(10,20,14,24),(11,21,15,17),(12,22,16,18),(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),(9,34,37,14),(10,13,38,33),(11,40,39,12),(15,36,35,16),(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 ⊆ GL6(𝔽17)
| 16 | 9 | 0 | 0 | 0 | 0 |
| 13 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 10 | 0 | 5 | 5 |
| 0 | 0 | 0 | 10 | 12 | 5 |
| 0 | 0 | 12 | 5 | 7 | 0 |
| 0 | 0 | 12 | 12 | 0 | 7 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 4 | 0 | 0 | 0 |
| 0 | 0 | 0 | 4 | 0 | 0 |
| 0 | 0 | 0 | 0 | 4 | 0 |
| 0 | 0 | 0 | 0 | 0 | 4 |
| 13 | 2 | 0 | 0 | 0 | 0 |
| 1 | 4 | 0 | 0 | 0 | 0 |
| 0 | 0 | 5 | 12 | 10 | 0 |
| 0 | 0 | 5 | 5 | 0 | 10 |
| 0 | 0 | 0 | 10 | 12 | 5 |
| 0 | 0 | 7 | 0 | 12 | 12 |
| 13 | 2 | 0 | 0 | 0 | 0 |
| 0 | 4 | 0 | 0 | 0 | 0 |
| 0 | 0 | 12 | 5 | 7 | 0 |
| 0 | 0 | 5 | 5 | 0 | 10 |
| 0 | 0 | 10 | 0 | 5 | 5 |
| 0 | 0 | 0 | 7 | 5 | 12 |
G:=sub<GL(6,GF(17))| [16,13,0,0,0,0,9,1,0,0,0,0,0,0,10,0,12,12,0,0,0,10,5,12,0,0,5,12,7,0,0,0,5,5,0,7],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,4],[13,1,0,0,0,0,2,4,0,0,0,0,0,0,5,5,0,7,0,0,12,5,10,0,0,0,10,0,12,12,0,0,0,10,5,12],[13,0,0,0,0,0,2,4,0,0,0,0,0,0,12,5,10,0,0,0,5,5,0,7,0,0,7,0,5,5,0,0,0,10,5,12] >;
32 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 4K | ··· | 4P | 8A | ··· | 8H |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 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 | 4 | 4 | 4 | 4 | 8 | ··· | 8 | 4 | ··· | 4 |
32 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | ||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | D4 | D4 | C4○D4 | D8⋊C22 |
| kernel | C42.239D4 | C2×C8⋊C4 | C23.24D4 | C42.28C22 | C42.29C22 | C42.30C22 | C22.26C24 | C23.37C23 | C42 | C22×C4 | C2×C4 | C2 |
| # reps | 1 | 1 | 4 | 4 | 2 | 2 | 1 | 1 | 2 | 2 | 8 | 4 |
In GAP, Magma, Sage, TeX
C_4^2._{239}D_4 % in TeX
G:=Group("C4^2.239D4"); // GroupNames label
G:=SmallGroup(128,1867);
// by ID
G=gap.SmallGroup(128,1867);
# by ID
G:=PCGroup([7,-2,2,2,2,-2,2,-2,253,232,758,723,100,248,2804,172,4037,124]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^4=1,c^4=b^2,d^2=a^2*b^2,a*b=b*a,c*a*c^-1=a*b^2,d*a*d^-1=a^-1,b*c=c*b,b*d=d*b,d*c*d^-1=a^2*c^3>;
// generators/relations