p-group, metabelian, nilpotent (class 3), monomial
Aliases: C42.288D4, C42.418C23, C4.602- (1+4), C8⋊Q8⋊14C2, Q8.Q8⋊23C2, C4⋊C8.70C22, (C2×C8).70C23, C4⋊C4.175C23, (C2×C4).434C24, (C22×C4).516D4, C23.703(C2×D4), C4⋊Q8.317C22, C4.Q8.38C22, C8⋊C4.27C22, C22⋊C8.61C22, C42.6C4.2C2, (C4×Q8).113C22, (C2×Q8).166C23, C2.D8.104C22, (C2×C42).895C22, Q8⋊C4.48C22, C23.47D4.2C2, C23.48D4.3C2, C22.694(C22×D4), C22⋊Q8.206C22, C2.65(D8⋊C22), (C22×C4).1099C23, C42.30C22⋊5C2, C22.41(C8.C22), C42.C2.135C22, C23.37C23.41C2, C2.82(C23.38C23), (C2×C4).558(C2×D4), C2.62(C2×C8.C22), (C2×C4⋊C4).651C22, (C2×C42.C2).36C2, SmallGroup(128,1968)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Subgroups: 268 in 162 conjugacy classes, 86 normal (28 characteristic)
C1, C2 [×3], C2 [×2], C4 [×2], C4 [×13], C22, C22 [×2], C22 [×2], C8 [×4], C2×C4 [×6], C2×C4 [×17], Q8 [×6], C23, C42 [×4], C42 [×2], C22⋊C4 [×2], C4⋊C4 [×6], C4⋊C4 [×17], C2×C8 [×4], C22×C4 [×3], C22×C4 [×2], C2×Q8 [×2], C2×Q8, C8⋊C4 [×2], C22⋊C8 [×2], Q8⋊C4 [×8], C4⋊C8 [×2], C4.Q8 [×4], C2.D8 [×4], C2×C42, C2×C4⋊C4 [×2], C2×C4⋊C4 [×2], C42⋊C2, C4×Q8 [×2], C4×Q8, C22⋊Q8 [×2], C22⋊Q8, C42.C2 [×4], C42.C2 [×3], C4⋊Q8 [×2], C42.6C4, Q8.Q8 [×4], C23.47D4 [×2], C23.48D4 [×2], C42.30C22 [×2], C8⋊Q8 [×2], C2×C42.C2, C23.37C23, C42.288D4
Quotients:
C1, C2 [×15], C22 [×35], D4 [×4], C23 [×15], C2×D4 [×6], C24, C8.C22 [×2], C22×D4, 2- (1+4) [×2], C23.38C23, C2×C8.C22, D8⋊C22, C42.288D4
Generators and relations
G = < a,b,c,d | a4=b4=1, c4=d2=b2, ab=ba, cac-1=ab2, dad-1=a-1, cbc-1=a2b-1, dbd-1=a2b, dcd-1=a2b2c3 >
►On 64 points
(1 26 37 41)(2 31 38 46)(3 28 39 43)(4 25 40 48)(5 30 33 45)(6 27 34 42)(7 32 35 47)(8 29 36 44)(9 64 50 23)(10 61 51 20)(11 58 52 17)(12 63 53 22)(13 60 54 19)(14 57 55 24)(15 62 56 21)(16 59 49 18)
(1 35 5 39)(2 4 6 8)(3 37 7 33)(9 11 13 15)(10 49 14 53)(12 51 16 55)(17 19 21 23)(18 57 22 61)(20 59 24 63)(25 27 29 31)(26 47 30 43)(28 41 32 45)(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 37 13 33)(11 35 15 39)(17 47 21 43)(18 31 22 27)(19 45 23 41)(20 29 24 25)(26 60 30 64)(28 58 32 62)(34 49 38 53)(36 55 40 51)(42 59 46 63)(44 57 48 61)
G:=sub<Sym(64)| (1,26,37,41)(2,31,38,46)(3,28,39,43)(4,25,40,48)(5,30,33,45)(6,27,34,42)(7,32,35,47)(8,29,36,44)(9,64,50,23)(10,61,51,20)(11,58,52,17)(12,63,53,22)(13,60,54,19)(14,57,55,24)(15,62,56,21)(16,59,49,18), (1,35,5,39)(2,4,6,8)(3,37,7,33)(9,11,13,15)(10,49,14,53)(12,51,16,55)(17,19,21,23)(18,57,22,61)(20,59,24,63)(25,27,29,31)(26,47,30,43)(28,41,32,45)(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)(9,37,13,33)(11,35,15,39)(17,47,21,43)(18,31,22,27)(19,45,23,41)(20,29,24,25)(26,60,30,64)(28,58,32,62)(34,49,38,53)(36,55,40,51)(42,59,46,63)(44,57,48,61)>;
G:=Group( (1,26,37,41)(2,31,38,46)(3,28,39,43)(4,25,40,48)(5,30,33,45)(6,27,34,42)(7,32,35,47)(8,29,36,44)(9,64,50,23)(10,61,51,20)(11,58,52,17)(12,63,53,22)(13,60,54,19)(14,57,55,24)(15,62,56,21)(16,59,49,18), (1,35,5,39)(2,4,6,8)(3,37,7,33)(9,11,13,15)(10,49,14,53)(12,51,16,55)(17,19,21,23)(18,57,22,61)(20,59,24,63)(25,27,29,31)(26,47,30,43)(28,41,32,45)(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)(9,37,13,33)(11,35,15,39)(17,47,21,43)(18,31,22,27)(19,45,23,41)(20,29,24,25)(26,60,30,64)(28,58,32,62)(34,49,38,53)(36,55,40,51)(42,59,46,63)(44,57,48,61) );
G=PermutationGroup([(1,26,37,41),(2,31,38,46),(3,28,39,43),(4,25,40,48),(5,30,33,45),(6,27,34,42),(7,32,35,47),(8,29,36,44),(9,64,50,23),(10,61,51,20),(11,58,52,17),(12,63,53,22),(13,60,54,19),(14,57,55,24),(15,62,56,21),(16,59,49,18)], [(1,35,5,39),(2,4,6,8),(3,37,7,33),(9,11,13,15),(10,49,14,53),(12,51,16,55),(17,19,21,23),(18,57,22,61),(20,59,24,63),(25,27,29,31),(26,47,30,43),(28,41,32,45),(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),(9,37,13,33),(11,35,15,39),(17,47,21,43),(18,31,22,27),(19,45,23,41),(20,29,24,25),(26,60,30,64),(28,58,32,62),(34,49,38,53),(36,55,40,51),(42,59,46,63),(44,57,48,61)])
Matrix representation ►G ⊆ GL8(𝔽17)
| 0 | 0 | 10 | 10 | 0 | 0 | 0 | 0 |
| 12 | 0 | 7 | 0 | 0 | 0 | 0 | 0 |
| 0 | 12 | 12 | 12 | 0 | 0 | 0 | 0 |
| 5 | 5 | 5 | 5 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 9 | 3 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 8 | 0 | 0 |
| 0 | 0 | 0 | 0 | 16 | 10 | 6 | 6 |
| 0 | 0 | 0 | 0 | 15 | 14 | 8 | 11 |
| 16 | 15 | 0 | 0 | 0 | 0 | 0 | 0 |
| 1 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 1 | 1 | 0 | 16 | 0 | 0 | 0 | 0 |
| 0 | 16 | 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 15 | 0 | 16 | 0 |
| 0 | 0 | 0 | 0 | 5 | 7 | 0 | 16 |
| 12 | 16 | 7 | 0 | 0 | 0 | 0 | 0 |
| 14 | 1 | 5 | 12 | 0 | 0 | 0 | 0 |
| 8 | 4 | 13 | 9 | 0 | 0 | 0 | 0 |
| 0 | 8 | 3 | 8 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 14 | 8 | 13 |
| 0 | 0 | 0 | 0 | 16 | 10 | 4 | 2 |
| 0 | 0 | 0 | 0 | 0 | 5 | 9 | 4 |
| 0 | 0 | 0 | 0 | 14 | 6 | 0 | 14 |
| 6 | 10 | 1 | 16 | 0 | 0 | 0 | 0 |
| 8 | 12 | 16 | 0 | 0 | 0 | 0 | 0 |
| 8 | 4 | 13 | 9 | 0 | 0 | 0 | 0 |
| 6 | 14 | 9 | 3 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 16 | 0 | 16 | 0 |
| 0 | 0 | 0 | 0 | 0 | 10 | 5 | 2 |
| 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 10 | 15 | 7 |
G:=sub<GL(8,GF(17))| [0,12,0,5,0,0,0,0,0,0,12,5,0,0,0,0,10,7,12,5,0,0,0,0,10,0,12,5,0,0,0,0,0,0,0,0,9,1,16,15,0,0,0,0,3,8,10,14,0,0,0,0,0,0,6,8,0,0,0,0,0,0,6,11],[16,1,1,0,0,0,0,0,15,1,1,16,0,0,0,0,0,0,0,1,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,0,1,0,15,5,0,0,0,0,0,1,0,7,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,16],[12,14,8,0,0,0,0,0,16,1,4,8,0,0,0,0,7,5,13,3,0,0,0,0,0,12,9,8,0,0,0,0,0,0,0,0,1,16,0,14,0,0,0,0,14,10,5,6,0,0,0,0,8,4,9,0,0,0,0,0,13,2,4,14],[6,8,8,6,0,0,0,0,10,12,4,14,0,0,0,0,1,16,13,9,0,0,0,0,16,0,9,3,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,10,0,10,0,0,0,0,16,5,1,15,0,0,0,0,0,2,0,7] >;
Character table of C42.288D4
| class | 1 | 2A | 2B | 2C | 2D | 2E | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 4K | 4L | 4M | 4N | 4O | 4P | 8A | 8B | 8C | 8D | |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | |
| ρ1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | trivial |
| ρ2 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | -1 | 1 | -1 | 1 | -1 | -1 | -1 | 1 | 1 | -1 | -1 | 1 | -1 | -1 | 1 | -1 | -1 | 1 | 1 | linear of order 2 |
| ρ3 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | 1 | 1 | 1 | 1 | linear of order 2 |
| ρ4 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | -1 | 1 | -1 | 1 | -1 | -1 | -1 | -1 | -1 | 1 | 1 | -1 | 1 | 1 | -1 | -1 | -1 | 1 | 1 | linear of order 2 |
| ρ5 | 1 | 1 | 1 | 1 | -1 | -1 | 1 | 1 | 1 | 1 | -1 | 1 | -1 | -1 | 1 | -1 | -1 | 1 | -1 | 1 | -1 | 1 | 1 | -1 | -1 | 1 | linear of order 2 |
| ρ6 | 1 | 1 | 1 | 1 | -1 | -1 | 1 | -1 | 1 | -1 | -1 | -1 | 1 | 1 | 1 | -1 | 1 | -1 | -1 | -1 | 1 | 1 | -1 | 1 | -1 | 1 | linear of order 2 |
| ρ7 | 1 | 1 | 1 | 1 | -1 | -1 | 1 | 1 | 1 | 1 | -1 | 1 | -1 | -1 | -1 | 1 | 1 | -1 | 1 | -1 | 1 | -1 | 1 | -1 | -1 | 1 | linear of order 2 |
| ρ8 | 1 | 1 | 1 | 1 | -1 | -1 | 1 | -1 | 1 | -1 | -1 | -1 | 1 | 1 | -1 | 1 | -1 | 1 | 1 | 1 | -1 | -1 | -1 | 1 | -1 | 1 | linear of order 2 |
| ρ9 | 1 | 1 | 1 | 1 | -1 | -1 | 1 | 1 | 1 | 1 | -1 | 1 | -1 | -1 | -1 | -1 | -1 | 1 | 1 | -1 | 1 | 1 | -1 | 1 | 1 | -1 | linear of order 2 |
| ρ10 | 1 | 1 | 1 | 1 | -1 | -1 | 1 | -1 | 1 | -1 | -1 | -1 | 1 | 1 | -1 | -1 | 1 | -1 | 1 | 1 | -1 | 1 | 1 | -1 | 1 | -1 | linear of order 2 |
| ρ11 | 1 | 1 | 1 | 1 | -1 | -1 | 1 | 1 | 1 | 1 | -1 | 1 | -1 | -1 | 1 | 1 | 1 | -1 | -1 | 1 | -1 | -1 | -1 | 1 | 1 | -1 | linear of order 2 |
| ρ12 | 1 | 1 | 1 | 1 | -1 | -1 | 1 | -1 | 1 | -1 | -1 | -1 | 1 | 1 | 1 | 1 | -1 | 1 | -1 | -1 | 1 | -1 | 1 | -1 | 1 | -1 | linear of order 2 |
| ρ13 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | -1 | 1 | 1 | 1 | -1 | -1 | -1 | 1 | -1 | -1 | -1 | -1 | linear of order 2 |
| ρ14 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | -1 | 1 | -1 | 1 | -1 | -1 | -1 | -1 | 1 | -1 | -1 | -1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | linear of order 2 |
| ρ15 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | -1 | linear of order 2 |
| ρ16 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | -1 | 1 | -1 | 1 | -1 | -1 | -1 | 1 | -1 | 1 | 1 | 1 | -1 | -1 | -1 | 1 | 1 | -1 | -1 | linear of order 2 |
| ρ17 | 2 | 2 | 2 | 2 | 2 | 2 | -2 | 2 | -2 | 2 | -2 | -2 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ18 | 2 | 2 | 2 | 2 | 2 | 2 | -2 | -2 | -2 | -2 | -2 | 2 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ19 | 2 | 2 | 2 | 2 | -2 | -2 | -2 | -2 | -2 | -2 | 2 | 2 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ20 | 2 | 2 | 2 | 2 | -2 | -2 | -2 | 2 | -2 | 2 | 2 | -2 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ21 | 4 | -4 | 4 | -4 | 0 | 0 | 4 | 0 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | symplectic lifted from 2- (1+4), Schur index 2 |
| ρ22 | 4 | -4 | -4 | 4 | 4 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | symplectic lifted from C8.C22, Schur index 2 |
| ρ23 | 4 | -4 | 4 | -4 | 0 | 0 | -4 | 0 | 4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | symplectic lifted from 2- (1+4), Schur index 2 |
| ρ24 | 4 | -4 | -4 | 4 | -4 | 4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | symplectic lifted from C8.C22, Schur index 2 |
| ρ25 | 4 | 4 | -4 | -4 | 0 | 0 | 0 | 4i | 0 | 4i | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from D8⋊C22 |
| ρ26 | 4 | 4 | -4 | -4 | 0 | 0 | 0 | 4i | 0 | 4i | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from D8⋊C22 |
In GAP, Magma, Sage, TeX
C_4^2._{288}D_4 % in TeX
G:=Group("C4^2.288D4"); // GroupNames label
G:=SmallGroup(128,1968);
// by ID
G=gap.SmallGroup(128,1968);
# by ID
G:=PCGroup([7,-2,2,2,2,-2,2,-2,448,253,120,758,891,100,675,4037,1027,124]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^4=1,c^4=d^2=b^2,a*b=b*a,c*a*c^-1=a*b^2,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