p-group, metabelian, nilpotent (class 3), monomial
Aliases: C42.56C23, C4.662+ 1+4, C8⋊8D4⋊54C2, C8⋊2D4⋊26C2, C8⋊9D4⋊23C2, C4⋊C4.370D4, D4⋊D4⋊46C2, Q8⋊D4⋊22C2, Q8⋊5D4⋊10C2, Q8⋊Q8⋊18C2, (C2×D4).174D4, D4.2D4⋊43C2, C4⋊C8.107C22, C4⋊C4.413C23, (C2×C8).355C23, (C2×C4).513C24, Q8.27(C4○D4), C22⋊C4.170D4, (C2×D8).86C22, C23.330(C2×D4), C4⋊Q8.154C22, SD16⋊C4⋊38C2, C4.Q8.59C22, C8⋊C4.48C22, C2.79(D4○SD16), (C4×D4).164C22, (C2×D4).239C23, C4⋊D4.88C22, C22⋊C8.85C22, (C4×Q8).162C22, (C2×Q8).224C23, C2.149(D4⋊5D4), C2.D8.122C22, C22⋊Q8.87C22, D4⋊C4.75C22, C23.20D4⋊38C2, C23.38D4⋊15C2, C23.19D4⋊37C2, C23.24D4⋊33C2, (C22×C8).366C22, (C2×SD16).59C22, C4.4D4.69C22, C22.773(C22×D4), C2.89(D8⋊C22), C22.49C24⋊6C2, (C22×C4).1157C23, Q8⋊C4.182C22, (C22×Q8).346C22, C42⋊C2.193C22, C42.28C22⋊19C2, (C2×M4(2)).119C22, C4.238(C2×C4○D4), (C2×C4).610(C2×D4), (C2×C4○D4).215C22, SmallGroup(128,2053)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C42.56C23
G = < a,b,c,d,e | a4=b4=1, c2=a2b2, d2=e2=b2, ab=ba, cac-1=eae-1=a-1, dad-1=ab2, cbc-1=dbd-1=b-1, be=eb, dcd-1=bc, ece-1=a2c, ede-1=b2d >
Subgroups: 400 in 196 conjugacy classes, 86 normal (84 characteristic)
C1, C2, C2, C4, C4, C22, C22, C8, C2×C4, C2×C4, D4, Q8, Q8, C23, C23, C42, C42, C22⋊C4, C22⋊C4, C4⋊C4, C4⋊C4, C2×C8, C2×C8, M4(2), D8, SD16, C22×C4, C22×C4, C2×D4, C2×D4, C2×Q8, C2×Q8, C4○D4, C8⋊C4, C22⋊C8, D4⋊C4, Q8⋊C4, C4⋊C8, C4.Q8, C2.D8, C42⋊C2, C42⋊C2, C4×D4, C4×D4, C4×Q8, C4⋊D4, C4⋊D4, C22⋊Q8, C22⋊Q8, C4.4D4, C4.4D4, C4⋊Q8, C22×C8, C2×M4(2), C2×D8, C2×SD16, C22×Q8, C2×C4○D4, C23.24D4, C23.38D4, C8⋊9D4, SD16⋊C4, Q8⋊D4, D4⋊D4, D4.2D4, C8⋊8D4, C8⋊2D4, Q8⋊Q8, C23.19D4, C23.20D4, C42.28C22, Q8⋊5D4, C22.49C24, C42.56C23
Quotients: C1, C2, C22, D4, C23, C2×D4, C4○D4, C24, C22×D4, C2×C4○D4, 2+ 1+4, D4⋊5D4, D8⋊C22, D4○SD16, C42.56C23
Character table of C42.56C23
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 4K | 4L | 4M | 4N | 4O | 8A | 8B | 8C | 8D | 8E | 8F | |
| size | 1 | 1 | 1 | 1 | 4 | 4 | 8 | 8 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 8 | 8 | 8 | 8 | 4 | 4 | 4 | 4 | 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 | 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 | -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 | -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 | 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 | -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 | 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 | 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 | -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 | 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 | -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 | -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 | 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 | -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 | 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 | 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 | -1 | -1 | 1 | linear of order 2 |
| ρ17 | 2 | 2 | 2 | 2 | 2 | -2 | 0 | 0 | -2 | -2 | -2 | -2 | -2 | 2 | 0 | 0 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ18 | 2 | 2 | 2 | 2 | -2 | -2 | 0 | 0 | -2 | -2 | 2 | 2 | 2 | 2 | 0 | 0 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ19 | 2 | 2 | 2 | 2 | 2 | 2 | 0 | 0 | -2 | -2 | -2 | -2 | 2 | -2 | 0 | 0 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ20 | 2 | 2 | 2 | 2 | -2 | 2 | 0 | 0 | -2 | -2 | 2 | 2 | -2 | -2 | 0 | 0 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ21 | 2 | -2 | 2 | -2 | 0 | 0 | 0 | 0 | 2 | -2 | 0 | 0 | 0 | 0 | -2 | 2i | 0 | -2i | 2 | 0 | 0 | 0 | 0 | 2i | 0 | 0 | -2i | 0 | 0 | complex lifted from C4○D4 |
| ρ22 | 2 | -2 | 2 | -2 | 0 | 0 | 0 | 0 | 2 | -2 | 0 | 0 | 0 | 0 | 2 | 2i | 0 | -2i | -2 | 0 | 0 | 0 | 0 | -2i | 0 | 0 | 2i | 0 | 0 | complex lifted from C4○D4 |
| ρ23 | 2 | -2 | 2 | -2 | 0 | 0 | 0 | 0 | 2 | -2 | 0 | 0 | 0 | 0 | 2 | -2i | 0 | 2i | -2 | 0 | 0 | 0 | 0 | 2i | 0 | 0 | -2i | 0 | 0 | complex lifted from C4○D4 |
| ρ24 | 2 | -2 | 2 | -2 | 0 | 0 | 0 | 0 | 2 | -2 | 0 | 0 | 0 | 0 | -2 | -2i | 0 | 2i | 2 | 0 | 0 | 0 | 0 | -2i | 0 | 0 | 2i | 0 | 0 | complex lifted from C4○D4 |
| ρ25 | 4 | -4 | 4 | -4 | 0 | 0 | 0 | 0 | -4 | 4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from 2+ 1+4 |
| ρ26 | 4 | -4 | -4 | 4 | 0 | 0 | 0 | 0 | 0 | 0 | 4i | -4i | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from D8⋊C22 |
| ρ27 | 4 | -4 | -4 | 4 | 0 | 0 | 0 | 0 | 0 | 0 | -4i | 4i | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from D8⋊C22 |
| ρ28 | 4 | 4 | -4 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -2√-2 | 2√-2 | 0 | 0 | 0 | complex lifted from D4○SD16 |
| ρ29 | 4 | 4 | -4 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 2√-2 | -2√-2 | 0 | 0 | 0 | complex lifted from D4○SD16 |
►On 64 points
(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 42 46 17)(2 43 47 18)(3 44 48 19)(4 41 45 20)(5 62 33 39)(6 63 34 40)(7 64 35 37)(8 61 36 38)(9 15 23 49)(10 16 24 50)(11 13 21 51)(12 14 22 52)(25 56 60 31)(26 53 57 32)(27 54 58 29)(28 55 59 30)
(1 57 48 28)(2 60 45 27)(3 59 46 26)(4 58 47 25)(5 52 35 16)(6 51 36 15)(7 50 33 14)(8 49 34 13)(9 63 21 38)(10 62 22 37)(11 61 23 40)(12 64 24 39)(17 32 44 55)(18 31 41 54)(19 30 42 53)(20 29 43 56)
(1 11 46 21)(2 22 47 12)(3 9 48 23)(4 24 45 10)(5 58 33 27)(6 28 34 59)(7 60 35 25)(8 26 36 57)(13 42 51 17)(14 18 52 43)(15 44 49 19)(16 20 50 41)(29 62 54 39)(30 40 55 63)(31 64 56 37)(32 38 53 61)
(1 10 46 24)(2 9 47 23)(3 12 48 22)(4 11 45 21)(5 30 33 55)(6 29 34 54)(7 32 35 53)(8 31 36 56)(13 20 51 41)(14 19 52 44)(15 18 49 43)(16 17 50 42)(25 38 60 61)(26 37 57 64)(27 40 58 63)(28 39 59 62)
G:=sub<Sym(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), (1,42,46,17)(2,43,47,18)(3,44,48,19)(4,41,45,20)(5,62,33,39)(6,63,34,40)(7,64,35,37)(8,61,36,38)(9,15,23,49)(10,16,24,50)(11,13,21,51)(12,14,22,52)(25,56,60,31)(26,53,57,32)(27,54,58,29)(28,55,59,30), (1,57,48,28)(2,60,45,27)(3,59,46,26)(4,58,47,25)(5,52,35,16)(6,51,36,15)(7,50,33,14)(8,49,34,13)(9,63,21,38)(10,62,22,37)(11,61,23,40)(12,64,24,39)(17,32,44,55)(18,31,41,54)(19,30,42,53)(20,29,43,56), (1,11,46,21)(2,22,47,12)(3,9,48,23)(4,24,45,10)(5,58,33,27)(6,28,34,59)(7,60,35,25)(8,26,36,57)(13,42,51,17)(14,18,52,43)(15,44,49,19)(16,20,50,41)(29,62,54,39)(30,40,55,63)(31,64,56,37)(32,38,53,61), (1,10,46,24)(2,9,47,23)(3,12,48,22)(4,11,45,21)(5,30,33,55)(6,29,34,54)(7,32,35,53)(8,31,36,56)(13,20,51,41)(14,19,52,44)(15,18,49,43)(16,17,50,42)(25,38,60,61)(26,37,57,64)(27,40,58,63)(28,39,59,62)>;
G:=Group( (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), (1,42,46,17)(2,43,47,18)(3,44,48,19)(4,41,45,20)(5,62,33,39)(6,63,34,40)(7,64,35,37)(8,61,36,38)(9,15,23,49)(10,16,24,50)(11,13,21,51)(12,14,22,52)(25,56,60,31)(26,53,57,32)(27,54,58,29)(28,55,59,30), (1,57,48,28)(2,60,45,27)(3,59,46,26)(4,58,47,25)(5,52,35,16)(6,51,36,15)(7,50,33,14)(8,49,34,13)(9,63,21,38)(10,62,22,37)(11,61,23,40)(12,64,24,39)(17,32,44,55)(18,31,41,54)(19,30,42,53)(20,29,43,56), (1,11,46,21)(2,22,47,12)(3,9,48,23)(4,24,45,10)(5,58,33,27)(6,28,34,59)(7,60,35,25)(8,26,36,57)(13,42,51,17)(14,18,52,43)(15,44,49,19)(16,20,50,41)(29,62,54,39)(30,40,55,63)(31,64,56,37)(32,38,53,61), (1,10,46,24)(2,9,47,23)(3,12,48,22)(4,11,45,21)(5,30,33,55)(6,29,34,54)(7,32,35,53)(8,31,36,56)(13,20,51,41)(14,19,52,44)(15,18,49,43)(16,17,50,42)(25,38,60,61)(26,37,57,64)(27,40,58,63)(28,39,59,62) );
G=PermutationGroup([[(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)], [(1,42,46,17),(2,43,47,18),(3,44,48,19),(4,41,45,20),(5,62,33,39),(6,63,34,40),(7,64,35,37),(8,61,36,38),(9,15,23,49),(10,16,24,50),(11,13,21,51),(12,14,22,52),(25,56,60,31),(26,53,57,32),(27,54,58,29),(28,55,59,30)], [(1,57,48,28),(2,60,45,27),(3,59,46,26),(4,58,47,25),(5,52,35,16),(6,51,36,15),(7,50,33,14),(8,49,34,13),(9,63,21,38),(10,62,22,37),(11,61,23,40),(12,64,24,39),(17,32,44,55),(18,31,41,54),(19,30,42,53),(20,29,43,56)], [(1,11,46,21),(2,22,47,12),(3,9,48,23),(4,24,45,10),(5,58,33,27),(6,28,34,59),(7,60,35,25),(8,26,36,57),(13,42,51,17),(14,18,52,43),(15,44,49,19),(16,20,50,41),(29,62,54,39),(30,40,55,63),(31,64,56,37),(32,38,53,61)], [(1,10,46,24),(2,9,47,23),(3,12,48,22),(4,11,45,21),(5,30,33,55),(6,29,34,54),(7,32,35,53),(8,31,36,56),(13,20,51,41),(14,19,52,44),(15,18,49,43),(16,17,50,42),(25,38,60,61),(26,37,57,64),(27,40,58,63),(28,39,59,62)]])
Matrix representation of C42.56C23 ►in GL6(𝔽17)
| 1 | 1 | 0 | 0 | 0 | 0 |
| 15 | 16 | 0 | 0 | 0 | 0 |
| 0 | 0 | 10 | 0 | 1 | 0 |
| 0 | 0 | 0 | 10 | 0 | 1 |
| 0 | 0 | 1 | 0 | 7 | 0 |
| 0 | 0 | 0 | 1 | 0 | 7 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 16 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 16 | 0 |
| 4 | 0 | 0 | 0 | 0 | 0 |
| 9 | 13 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 12 | 5 |
| 0 | 0 | 0 | 0 | 5 | 5 |
| 0 | 0 | 5 | 12 | 0 | 0 |
| 0 | 0 | 12 | 12 | 0 | 0 |
| 16 | 0 | 0 | 0 | 0 | 0 |
| 0 | 16 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 16 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 16 | 0 | 0 |
| 16 | 16 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 16 | 0 | 10 | 0 |
| 0 | 0 | 0 | 16 | 0 | 10 |
| 0 | 0 | 10 | 0 | 1 | 0 |
| 0 | 0 | 0 | 10 | 0 | 1 |
G:=sub<GL(6,GF(17))| [1,15,0,0,0,0,1,16,0,0,0,0,0,0,10,0,1,0,0,0,0,10,0,1,0,0,1,0,7,0,0,0,0,1,0,7],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,16,0,0,0,0,1,0,0,0,0,0,0,0,0,16,0,0,0,0,1,0],[4,9,0,0,0,0,0,13,0,0,0,0,0,0,0,0,5,12,0,0,0,0,12,12,0,0,12,5,0,0,0,0,5,5,0,0],[16,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,16,0,0,16,0,0,0,0,0,0,1,0,0],[16,0,0,0,0,0,16,1,0,0,0,0,0,0,16,0,10,0,0,0,0,16,0,10,0,0,10,0,1,0,0,0,0,10,0,1] >;
C42.56C23 in GAP, Magma, Sage, TeX
C_4^2._{56}C_2^3 % in TeX
G:=Group("C4^2.56C2^3"); // GroupNames label
G:=SmallGroup(128,2053);
// by ID
G=gap.SmallGroup(128,2053);
# by ID
G:=PCGroup([7,-2,2,2,2,-2,2,-2,560,253,456,758,723,352,346,248,4037,1027,124]);
// Polycyclic
G:=Group<a,b,c,d,e|a^4=b^4=1,c^2=a^2*b^2,d^2=e^2=b^2,a*b=b*a,c*a*c^-1=e*a*e^-1=a^-1,d*a*d^-1=a*b^2,c*b*c^-1=d*b*d^-1=b^-1,b*e=e*b,d*c*d^-1=b*c,e*c*e^-1=a^2*c,e*d*e^-1=b^2*d>;
// generators/relations
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