p-group, metabelian, nilpotent (class 3), monomial
Aliases: C42.267D4, C42.398C23, C4.1012+ 1+4, (C2×C4)⋊4Q16, C4⋊Q16⋊8C2, C4⋊2Q16⋊8C2, C4.47(C2×Q16), (C4×C8).74C22, C4⋊C8.286C22, C4⋊C4.148C23, (C2×C8).160C23, (C2×C4).407C24, C4.SD16⋊12C2, (C22×C4).497D4, C23.691(C2×D4), C4⋊Q8.301C22, C22⋊Q16.2C2, C2.15(C22×Q16), C22.18(C2×Q16), C4.26(C8.C22), (C2×Q8).144C23, Q8⋊C4.3C22, (C2×Q16).25C22, (C4×Q8).100C22, C22⋊C8.178C22, (C2×C42).874C22, C22.667(C22×D4), C22⋊Q8.192C22, C42.12C4.35C2, (C22×C4).1078C23, (C22×Q8).320C22, C2.78(C22.29C24), C23.37C23.37C2, (C2×C4⋊Q8).53C2, (C2×C4).867(C2×D4), C2.54(C2×C8.C22), SmallGroup(128,1941)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C42.267D4
G = < a,b,c,d | a4=b4=1, c4=d2=b2, ab=ba, ac=ca, dad-1=a-1, cbc-1=a2b-1, dbd-1=a2b, dcd-1=b2c3 >
Subgroups: 348 in 192 conjugacy classes, 96 normal (26 characteristic)
C1, C2, C2, C4, C4, C22, C22, C22, C8, C2×C4, C2×C4, C2×C4, Q8, C23, C42, C42, C22⋊C4, C4⋊C4, C4⋊C4, C2×C8, Q16, C22×C4, C22×C4, C2×Q8, C2×Q8, C4×C8, C22⋊C8, Q8⋊C4, C4⋊C8, C2×C42, C2×C4⋊C4, C42⋊C2, C4×Q8, C4×Q8, C22⋊Q8, C22⋊Q8, C42.C2, C4⋊Q8, C4⋊Q8, C4⋊Q8, C2×Q16, C22×Q8, C42.12C4, C22⋊Q16, C4⋊2Q16, C4.SD16, C4⋊Q16, C2×C4⋊Q8, C23.37C23, C42.267D4
Quotients: C1, C2, C22, D4, C23, Q16, C2×D4, C24, C2×Q16, C8.C22, C22×D4, 2+ 1+4, C22.29C24, C22×Q16, C2×C8.C22, C42.267D4
►On 64 points
(1 40 19 48)(2 33 20 41)(3 34 21 42)(4 35 22 43)(5 36 23 44)(6 37 24 45)(7 38 17 46)(8 39 18 47)(9 30 54 60)(10 31 55 61)(11 32 56 62)(12 25 49 63)(13 26 50 64)(14 27 51 57)(15 28 52 58)(16 29 53 59)
(1 7 5 3)(2 22 6 18)(4 24 8 20)(9 56 13 52)(10 16 14 12)(11 50 15 54)(17 23 21 19)(25 31 29 27)(26 58 30 62)(28 60 32 64)(33 43 37 47)(34 40 38 36)(35 45 39 41)(42 48 46 44)(49 55 53 51)(57 63 61 59)
(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 9 5 13)(2 16 6 12)(3 15 7 11)(4 14 8 10)(17 56 21 52)(18 55 22 51)(19 54 23 50)(20 53 24 49)(25 41 29 45)(26 48 30 44)(27 47 31 43)(28 46 32 42)(33 59 37 63)(34 58 38 62)(35 57 39 61)(36 64 40 60)
G:=sub<Sym(64)| (1,40,19,48)(2,33,20,41)(3,34,21,42)(4,35,22,43)(5,36,23,44)(6,37,24,45)(7,38,17,46)(8,39,18,47)(9,30,54,60)(10,31,55,61)(11,32,56,62)(12,25,49,63)(13,26,50,64)(14,27,51,57)(15,28,52,58)(16,29,53,59), (1,7,5,3)(2,22,6,18)(4,24,8,20)(9,56,13,52)(10,16,14,12)(11,50,15,54)(17,23,21,19)(25,31,29,27)(26,58,30,62)(28,60,32,64)(33,43,37,47)(34,40,38,36)(35,45,39,41)(42,48,46,44)(49,55,53,51)(57,63,61,59), (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), (1,9,5,13)(2,16,6,12)(3,15,7,11)(4,14,8,10)(17,56,21,52)(18,55,22,51)(19,54,23,50)(20,53,24,49)(25,41,29,45)(26,48,30,44)(27,47,31,43)(28,46,32,42)(33,59,37,63)(34,58,38,62)(35,57,39,61)(36,64,40,60)>;
G:=Group( (1,40,19,48)(2,33,20,41)(3,34,21,42)(4,35,22,43)(5,36,23,44)(6,37,24,45)(7,38,17,46)(8,39,18,47)(9,30,54,60)(10,31,55,61)(11,32,56,62)(12,25,49,63)(13,26,50,64)(14,27,51,57)(15,28,52,58)(16,29,53,59), (1,7,5,3)(2,22,6,18)(4,24,8,20)(9,56,13,52)(10,16,14,12)(11,50,15,54)(17,23,21,19)(25,31,29,27)(26,58,30,62)(28,60,32,64)(33,43,37,47)(34,40,38,36)(35,45,39,41)(42,48,46,44)(49,55,53,51)(57,63,61,59), (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), (1,9,5,13)(2,16,6,12)(3,15,7,11)(4,14,8,10)(17,56,21,52)(18,55,22,51)(19,54,23,50)(20,53,24,49)(25,41,29,45)(26,48,30,44)(27,47,31,43)(28,46,32,42)(33,59,37,63)(34,58,38,62)(35,57,39,61)(36,64,40,60) );
G=PermutationGroup([[(1,40,19,48),(2,33,20,41),(3,34,21,42),(4,35,22,43),(5,36,23,44),(6,37,24,45),(7,38,17,46),(8,39,18,47),(9,30,54,60),(10,31,55,61),(11,32,56,62),(12,25,49,63),(13,26,50,64),(14,27,51,57),(15,28,52,58),(16,29,53,59)], [(1,7,5,3),(2,22,6,18),(4,24,8,20),(9,56,13,52),(10,16,14,12),(11,50,15,54),(17,23,21,19),(25,31,29,27),(26,58,30,62),(28,60,32,64),(33,43,37,47),(34,40,38,36),(35,45,39,41),(42,48,46,44),(49,55,53,51),(57,63,61,59)], [(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)], [(1,9,5,13),(2,16,6,12),(3,15,7,11),(4,14,8,10),(17,56,21,52),(18,55,22,51),(19,54,23,50),(20,53,24,49),(25,41,29,45),(26,48,30,44),(27,47,31,43),(28,46,32,42),(33,59,37,63),(34,58,38,62),(35,57,39,61),(36,64,40,60)]])
32 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 4A | ··· | 4H | 4I | 4J | 4K | ··· | 4R | 8A | ··· | 8H |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 4 | 4 | 4 | ··· | 4 | 8 | ··· | 8 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 2 | ··· | 2 | 4 | 4 | 8 | ··· | 8 | 4 | ··· | 4 |
32 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | - | - | + |
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | D4 | D4 | Q16 | C8.C22 | 2+ 1+4 |
| kernel | C42.267D4 | C42.12C4 | C22⋊Q16 | C4⋊2Q16 | C4.SD16 | C4⋊Q16 | C2×C4⋊Q8 | C23.37C23 | C42 | C22×C4 | C2×C4 | C4 | C4 |
| # reps | 1 | 1 | 4 | 4 | 2 | 2 | 1 | 1 | 2 | 2 | 8 | 2 | 2 |
Matrix representation of C42.267D4 ►in GL6(𝔽17)
| 0 | 1 | 0 | 0 | 0 | 0 |
| 16 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 16 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 14 | 0 | 0 | 1 |
| 0 | 0 | 0 | 14 | 16 | 0 |
| 0 | 16 | 0 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 16 | 0 | 0 | 0 |
| 0 | 0 | 0 | 16 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 11 | 0 | 0 | 1 |
| 14 | 3 | 0 | 0 | 0 | 0 |
| 14 | 14 | 0 | 0 | 0 | 0 |
| 0 | 0 | 11 | 0 | 0 | 2 |
| 0 | 0 | 0 | 0 | 2 | 0 |
| 0 | 0 | 0 | 8 | 0 | 0 |
| 0 | 0 | 7 | 0 | 0 | 6 |
| 5 | 5 | 0 | 0 | 0 | 0 |
| 5 | 12 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 2 | 0 |
| 0 | 0 | 11 | 0 | 0 | 2 |
| 0 | 0 | 9 | 0 | 0 | 0 |
| 0 | 0 | 0 | 9 | 6 | 0 |
G:=sub<GL(6,GF(17))| [0,16,0,0,0,0,1,0,0,0,0,0,0,0,0,1,14,0,0,0,16,0,0,14,0,0,0,0,0,16,0,0,0,0,1,0],[0,1,0,0,0,0,16,0,0,0,0,0,0,0,16,0,0,11,0,0,0,16,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[14,14,0,0,0,0,3,14,0,0,0,0,0,0,11,0,0,7,0,0,0,0,8,0,0,0,0,2,0,0,0,0,2,0,0,6],[5,5,0,0,0,0,5,12,0,0,0,0,0,0,0,11,9,0,0,0,0,0,0,9,0,0,2,0,0,6,0,0,0,2,0,0] >;
C42.267D4 in GAP, Magma, Sage, TeX
C_4^2._{267}D_4 % in TeX
G:=Group("C4^2.267D4"); // GroupNames label
G:=SmallGroup(128,1941);
// by ID
G=gap.SmallGroup(128,1941);
# by ID
G:=PCGroup([7,-2,2,2,2,-2,2,-2,448,253,120,758,219,352,675,4037,1027,124]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^4=1,c^4=d^2=b^2,a*b=b*a,a*c=c*a,d*a*d^-1=a^-1,c*b*c^-1=a^2*b^-1,d*b*d^-1=a^2*b,d*c*d^-1=b^2*c^3>;
// generators/relations