p-group, metabelian, nilpotent (class 3), monomial
Aliases: C42.281D4, C42.414C23, C4.532- (1+4), C8⋊3Q8⋊22C2, Q8⋊Q8⋊37C2, C4.87(C2×SD16), (C2×C4).58SD16, C4⋊C4.168C23, C4⋊C8.338C22, (C2×C4).427C24, (C4×C8).268C22, (C2×C8).332C23, C4.SD16⋊27C2, C23.699(C2×D4), (C22×C4).510D4, C4⋊Q8.311C22, C4.Q8.85C22, C4.29(C8.C22), (C4×Q8).108C22, (C2×Q8).161C23, C2.25(C22×SD16), C22.26(C2×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
Subgroups: 316 in 180 conjugacy classes, 96 normal (26 characteristic)
C1, C2 [×3], C2 [×2], C4 [×8], C4 [×10], C22, C22 [×2], C22 [×2], C8 [×4], C2×C4 [×6], C2×C4 [×4], C2×C4 [×16], Q8 [×14], C23, C42 [×4], C42 [×2], C22⋊C4 [×2], C4⋊C4 [×6], C4⋊C4 [×13], C2×C8 [×4], C22×C4 [×3], C22×C4 [×2], C2×Q8 [×2], C2×Q8 [×9], C4×C8 [×2], C22⋊C8 [×2], Q8⋊C4 [×8], C4⋊C8 [×2], C4.Q8 [×8], C2×C42, C2×C4⋊C4 [×2], C2×C4⋊C4, C42⋊C2, C4×Q8 [×2], C4×Q8, C22⋊Q8 [×2], C22⋊Q8, C42.C2, C4⋊Q8 [×2], C4⋊Q8 [×4], C4⋊Q8 [×2], C22×Q8, C42.12C4, Q8⋊Q8 [×4], C23.47D4 [×4], C4.SD16 [×2], C8⋊3Q8 [×2], C2×C4⋊Q8, C23.37C23, C42.281D4
Quotients:
C1, C2 [×15], C22 [×35], D4 [×4], C23 [×15], SD16 [×4], C2×D4 [×6], C24, C2×SD16 [×6], C8.C22 [×2], C22×D4, 2- (1+4) [×2], C23.38C23, C22×SD16, C2×C8.C22, C42.281D4
Generators and relations
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 >
►On 64 points
(1 25 33 45)(2 26 34 46)(3 27 35 47)(4 28 36 48)(5 29 37 41)(6 30 38 42)(7 31 39 43)(8 32 40 44)(9 64 50 18)(10 57 51 19)(11 58 52 20)(12 59 53 21)(13 60 54 22)(14 61 55 23)(15 62 56 24)(16 63 49 17)
(1 39 5 35)(2 4 6 8)(3 33 7 37)(9 11 13 15)(10 49 14 53)(12 51 16 55)(17 61 21 57)(18 20 22 24)(19 63 23 59)(25 43 29 47)(26 28 30 32)(27 45 31 41)(34 36 38 40)(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 33 13 37)(11 39 15 35)(17 26 21 30)(18 45 22 41)(19 32 23 28)(20 43 24 47)(25 60 29 64)(27 58 31 62)(34 53 38 49)(36 51 40 55)(42 63 46 59)(44 61 48 57)
G:=sub<Sym(64)| (1,25,33,45)(2,26,34,46)(3,27,35,47)(4,28,36,48)(5,29,37,41)(6,30,38,42)(7,31,39,43)(8,32,40,44)(9,64,50,18)(10,57,51,19)(11,58,52,20)(12,59,53,21)(13,60,54,22)(14,61,55,23)(15,62,56,24)(16,63,49,17), (1,39,5,35)(2,4,6,8)(3,33,7,37)(9,11,13,15)(10,49,14,53)(12,51,16,55)(17,61,21,57)(18,20,22,24)(19,63,23,59)(25,43,29,47)(26,28,30,32)(27,45,31,41)(34,36,38,40)(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)(9,33,13,37)(11,39,15,35)(17,26,21,30)(18,45,22,41)(19,32,23,28)(20,43,24,47)(25,60,29,64)(27,58,31,62)(34,53,38,49)(36,51,40,55)(42,63,46,59)(44,61,48,57)>;
G:=Group( (1,25,33,45)(2,26,34,46)(3,27,35,47)(4,28,36,48)(5,29,37,41)(6,30,38,42)(7,31,39,43)(8,32,40,44)(9,64,50,18)(10,57,51,19)(11,58,52,20)(12,59,53,21)(13,60,54,22)(14,61,55,23)(15,62,56,24)(16,63,49,17), (1,39,5,35)(2,4,6,8)(3,33,7,37)(9,11,13,15)(10,49,14,53)(12,51,16,55)(17,61,21,57)(18,20,22,24)(19,63,23,59)(25,43,29,47)(26,28,30,32)(27,45,31,41)(34,36,38,40)(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)(9,33,13,37)(11,39,15,35)(17,26,21,30)(18,45,22,41)(19,32,23,28)(20,43,24,47)(25,60,29,64)(27,58,31,62)(34,53,38,49)(36,51,40,55)(42,63,46,59)(44,61,48,57) );
G=PermutationGroup([(1,25,33,45),(2,26,34,46),(3,27,35,47),(4,28,36,48),(5,29,37,41),(6,30,38,42),(7,31,39,43),(8,32,40,44),(9,64,50,18),(10,57,51,19),(11,58,52,20),(12,59,53,21),(13,60,54,22),(14,61,55,23),(15,62,56,24),(16,63,49,17)], [(1,39,5,35),(2,4,6,8),(3,33,7,37),(9,11,13,15),(10,49,14,53),(12,51,16,55),(17,61,21,57),(18,20,22,24),(19,63,23,59),(25,43,29,47),(26,28,30,32),(27,45,31,41),(34,36,38,40),(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),(9,33,13,37),(11,39,15,35),(17,26,21,30),(18,45,22,41),(19,32,23,28),(20,43,24,47),(25,60,29,64),(27,58,31,62),(34,53,38,49),(36,51,40,55),(42,63,46,59),(44,61,48,57)])
Matrix representation ►G ⊆ 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] >;
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 |
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