p-group, metabelian, nilpotent (class 3), monomial
Aliases: C42.281D4, C42.414C23, C4.532- 1+4, C8⋊3Q8⋊22C2, Q8⋊Q8⋊37C2, (C2×C4).58SD16, C4.87(C2×SD16), C4⋊C8.338C22, C4⋊C4.168C23, (C2×C4).427C24, (C4×C8).268C22, (C2×C8).332C23, C4.SD16⋊27C2, (C22×C4).510D4, C23.699(C2×D4), C4⋊Q8.311C22, C4.Q8.85C22, C4.29(C8.C22), (C2×Q8).161C23, (C4×Q8).108C22, C22.26(C2×SD16), C2.25(C22×SD16), C22⋊C8.221C22, (C2×C42).888C22, C23.47D4.5C2, C22.687(C22×D4), C22⋊Q8.201C22, C42.12C4.40C2, (C22×C4).1092C23, Q8⋊C4.104C22, C23.37C23.38C2, C2.75(C23.38C23), (C2×C4⋊Q8).54C2, (C2×C4).870(C2×D4), C2.60(C2×C8.C22), (C2×C4⋊C4).647C22, SmallGroup(128,1961)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C42.281D4
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=a2b2c3 >
Subgroups: 316 in 180 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, C22×C4, C22×C4, C2×Q8, C2×Q8, C4×C8, C22⋊C8, Q8⋊C4, C4⋊C8, C4.Q8, C2×C42, C2×C4⋊C4, C2×C4⋊C4, C42⋊C2, C4×Q8, C4×Q8, C22⋊Q8, C22⋊Q8, C42.C2, C4⋊Q8, C4⋊Q8, C4⋊Q8, C22×Q8, C42.12C4, Q8⋊Q8, C23.47D4, C4.SD16, C8⋊3Q8, C2×C4⋊Q8, C23.37C23, C42.281D4
Quotients: C1, C2, C22, D4, C23, SD16, C2×D4, C24, C2×SD16, C8.C22, C22×D4, 2- 1+4, C23.38C23, C22×SD16, C2×C8.C22, C42.281D4
►On 64 points
(1 40 17 45)(2 33 18 46)(3 34 19 47)(4 35 20 48)(5 36 21 41)(6 37 22 42)(7 38 23 43)(8 39 24 44)(9 64 50 26)(10 57 51 27)(11 58 52 28)(12 59 53 29)(13 60 54 30)(14 61 55 31)(15 62 56 32)(16 63 49 25)
(1 23 5 19)(2 4 6 8)(3 17 7 21)(9 11 13 15)(10 49 14 53)(12 51 16 55)(18 20 22 24)(25 61 29 57)(26 28 30 32)(27 63 31 59)(33 35 37 39)(34 45 38 41)(36 47 40 43)(42 44 46 48)(50 52 54 56)(58 60 62 64)
(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 54 5 50)(2 12 6 16)(3 52 7 56)(4 10 8 14)(9 17 13 21)(11 23 15 19)(18 53 22 49)(20 51 24 55)(25 33 29 37)(26 45 30 41)(27 39 31 35)(28 43 32 47)(34 58 38 62)(36 64 40 60)(42 63 46 59)(44 61 48 57)
G:=sub<Sym(64)| (1,40,17,45)(2,33,18,46)(3,34,19,47)(4,35,20,48)(5,36,21,41)(6,37,22,42)(7,38,23,43)(8,39,24,44)(9,64,50,26)(10,57,51,27)(11,58,52,28)(12,59,53,29)(13,60,54,30)(14,61,55,31)(15,62,56,32)(16,63,49,25), (1,23,5,19)(2,4,6,8)(3,17,7,21)(9,11,13,15)(10,49,14,53)(12,51,16,55)(18,20,22,24)(25,61,29,57)(26,28,30,32)(27,63,31,59)(33,35,37,39)(34,45,38,41)(36,47,40,43)(42,44,46,48)(50,52,54,56)(58,60,62,64), (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), (1,54,5,50)(2,12,6,16)(3,52,7,56)(4,10,8,14)(9,17,13,21)(11,23,15,19)(18,53,22,49)(20,51,24,55)(25,33,29,37)(26,45,30,41)(27,39,31,35)(28,43,32,47)(34,58,38,62)(36,64,40,60)(42,63,46,59)(44,61,48,57)>;
G:=Group( (1,40,17,45)(2,33,18,46)(3,34,19,47)(4,35,20,48)(5,36,21,41)(6,37,22,42)(7,38,23,43)(8,39,24,44)(9,64,50,26)(10,57,51,27)(11,58,52,28)(12,59,53,29)(13,60,54,30)(14,61,55,31)(15,62,56,32)(16,63,49,25), (1,23,5,19)(2,4,6,8)(3,17,7,21)(9,11,13,15)(10,49,14,53)(12,51,16,55)(18,20,22,24)(25,61,29,57)(26,28,30,32)(27,63,31,59)(33,35,37,39)(34,45,38,41)(36,47,40,43)(42,44,46,48)(50,52,54,56)(58,60,62,64), (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), (1,54,5,50)(2,12,6,16)(3,52,7,56)(4,10,8,14)(9,17,13,21)(11,23,15,19)(18,53,22,49)(20,51,24,55)(25,33,29,37)(26,45,30,41)(27,39,31,35)(28,43,32,47)(34,58,38,62)(36,64,40,60)(42,63,46,59)(44,61,48,57) );
G=PermutationGroup([[(1,40,17,45),(2,33,18,46),(3,34,19,47),(4,35,20,48),(5,36,21,41),(6,37,22,42),(7,38,23,43),(8,39,24,44),(9,64,50,26),(10,57,51,27),(11,58,52,28),(12,59,53,29),(13,60,54,30),(14,61,55,31),(15,62,56,32),(16,63,49,25)], [(1,23,5,19),(2,4,6,8),(3,17,7,21),(9,11,13,15),(10,49,14,53),(12,51,16,55),(18,20,22,24),(25,61,29,57),(26,28,30,32),(27,63,31,59),(33,35,37,39),(34,45,38,41),(36,47,40,43),(42,44,46,48),(50,52,54,56),(58,60,62,64)], [(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)], [(1,54,5,50),(2,12,6,16),(3,52,7,56),(4,10,8,14),(9,17,13,21),(11,23,15,19),(18,53,22,49),(20,51,24,55),(25,33,29,37),(26,45,30,41),(27,39,31,35),(28,43,32,47),(34,58,38,62),(36,64,40,60),(42,63,46,59),(44,61,48,57)]])
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 | SD16 | C8.C22 | 2- 1+4 |
| kernel | C42.281D4 | C42.12C4 | Q8⋊Q8 | C23.47D4 | C4.SD16 | C8⋊3Q8 | 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 C42.281D4 ►in GL6(𝔽17)
| 1 | 15 | 0 | 0 | 0 | 0 |
| 1 | 16 | 0 | 0 | 0 | 0 |
| 0 | 0 | 16 | 0 | 0 | 0 |
| 0 | 0 | 0 | 16 | 0 | 0 |
| 0 | 0 | 0 | 0 | 16 | 0 |
| 0 | 0 | 0 | 0 | 0 | 16 |
| 16 | 2 | 0 | 0 | 0 | 0 |
| 16 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 13 | 0 | 0 | 0 |
| 0 | 0 | 0 | 4 | 0 | 0 |
| 0 | 0 | 0 | 9 | 13 | 0 |
| 0 | 0 | 11 | 0 | 0 | 4 |
| 0 | 7 | 0 | 0 | 0 | 0 |
| 5 | 7 | 0 | 0 | 0 | 0 |
| 0 | 0 | 10 | 1 | 0 | 15 |
| 0 | 0 | 16 | 2 | 2 | 0 |
| 0 | 0 | 6 | 5 | 15 | 16 |
| 0 | 0 | 5 | 6 | 1 | 7 |
| 13 | 0 | 0 | 0 | 0 | 0 |
| 13 | 4 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 15 | 15 | 0 |
| 0 | 0 | 10 | 1 | 0 | 15 |
| 0 | 0 | 8 | 15 | 16 | 2 |
| 0 | 0 | 10 | 8 | 7 | 16 |
G:=sub<GL(6,GF(17))| [1,1,0,0,0,0,15,16,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,16],[16,16,0,0,0,0,2,1,0,0,0,0,0,0,13,0,0,11,0,0,0,4,9,0,0,0,0,0,13,0,0,0,0,0,0,4],[0,5,0,0,0,0,7,7,0,0,0,0,0,0,10,16,6,5,0,0,1,2,5,6,0,0,0,2,15,1,0,0,15,0,16,7],[13,13,0,0,0,0,0,4,0,0,0,0,0,0,1,10,8,10,0,0,15,1,15,8,0,0,15,0,16,7,0,0,0,15,2,16] >;
C42.281D4 in GAP, Magma, Sage, TeX
C_4^2._{281}D_4 % in TeX
G:=Group("C4^2.281D4"); // GroupNames label
G:=SmallGroup(128,1961);
// by ID
G=gap.SmallGroup(128,1961);
# by ID
G:=PCGroup([7,-2,2,2,2,-2,2,-2,448,253,120,758,219,100,675,4037,1027,124]);
// 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=a^2*b^2*c^3>;
// generators/relations