p-group, metabelian, nilpotent (class 3), monomial
Aliases: C42.481C23, C4.772+ 1+4, C4⋊D8⋊40C2, C8⋊2D4⋊30C2, C8⋊6D4⋊19C2, C4⋊C4.377D4, Q8⋊7(C4○D4), Q8⋊6D4⋊9C2, D4⋊6D4⋊13C2, D4⋊D4⋊50C2, Q8⋊Q8⋊20C2, (C4×SD16)⋊60C2, (C2×D4).327D4, C4.4D8⋊35C2, C22⋊C4.60D4, C4⋊C8.114C22, C4⋊C4.424C23, C4.76(C8⋊C22), (C2×C4).524C24, (C2×C8).359C23, (C4×C8).295C22, (C2×D8).88C22, C23.341(C2×D4), C4⋊Q8.159C22, C4.Q8.64C22, C2.85(D4○SD16), (C4×D4).173C22, (C2×D4).246C23, C4⋊1D4.92C22, C4⋊D4.95C22, C22⋊C8.92C22, (C4×Q8).168C22, (C2×Q8).401C23, C2.160(D4⋊5D4), C23.46D4⋊19C2, C23.36D4⋊27C2, (C22×C4).337C23, Q8⋊C4.74C22, C22.784(C22×D4), D4⋊C4.123C22, (C2×SD16).163C22, (C2×M4(2)).126C22, C4.249(C2×C4○D4), (C2×C4).617(C2×D4), C2.81(C2×C8⋊C22), (C2×C4⋊C4).676C22, (C2×C4○D4).222C22, SmallGroup(128,2064)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C42.481C23
G = < a,b,c,d,e | a4=b4=e2=1, c2=a2b2, d2=b2, ab=ba, cac-1=a-1, dad-1=ab2, eae=a-1b2, cbc-1=dbd-1=b-1, be=eb, dcd-1=bc, ece=a2b2c, ede=b2d >
Subgroups: 472 in 215 conjugacy classes, 88 normal (38 characteristic)
C1, C2, C2, C4, C4, C4, C22, C22, C8, C2×C4, C2×C4, C2×C4, D4, Q8, Q8, C23, C23, C42, C42, C22⋊C4, C22⋊C4, C4⋊C4, C4⋊C4, C4⋊C4, C2×C8, C2×C8, M4(2), D8, SD16, C22×C4, C22×C4, C2×D4, C2×D4, C2×D4, C2×Q8, C2×Q8, C4○D4, C4×C8, C22⋊C8, D4⋊C4, D4⋊C4, Q8⋊C4, Q8⋊C4, C4⋊C8, C4.Q8, C4.Q8, C2×C4⋊C4, C4×D4, C4×D4, C4×Q8, C4⋊D4, C4⋊D4, C22⋊Q8, C22.D4, C4⋊1D4, C4⋊1D4, C4⋊Q8, C2×M4(2), C2×D8, C2×SD16, C2×C4○D4, C2×C4○D4, C23.36D4, C8⋊6D4, C4×SD16, D4⋊D4, C4⋊D8, C8⋊2D4, Q8⋊Q8, C23.46D4, C4.4D8, D4⋊6D4, Q8⋊6D4, C42.481C23
Quotients: C1, C2, C22, D4, C23, C2×D4, C4○D4, C24, C8⋊C22, C22×D4, C2×C4○D4, 2+ 1+4, D4⋊5D4, C2×C8⋊C22, D4○SD16, C42.481C23
Character table of C42.481C23
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 2H | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 4K | 4L | 4M | 4N | 8A | 8B | 8C | 8D | 8E | 8F | |
| size | 1 | 1 | 1 | 1 | 4 | 4 | 8 | 8 | 8 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 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 | 0 | 2 | -2 | -2 | 2 | 0 | -2 | -2 | 0 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ18 | 2 | 2 | 2 | 2 | -2 | 2 | 0 | 0 | 0 | -2 | -2 | -2 | -2 | 0 | 2 | 2 | 0 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ19 | 2 | 2 | 2 | 2 | -2 | -2 | 0 | 0 | 0 | 2 | -2 | -2 | 2 | 0 | -2 | 2 | 0 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ20 | 2 | 2 | 2 | 2 | 2 | -2 | 0 | 0 | 0 | -2 | -2 | -2 | -2 | 0 | 2 | -2 | 0 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ21 | 2 | -2 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | -2 | 2 | 0 | 2 | 0 | 0 | -2 | 0 | 2i | -2i | 0 | 0 | 0 | -2i | 0 | 2i | 0 | 0 | 0 | complex lifted from C4○D4 |
| ρ22 | 2 | -2 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | -2 | 2 | 0 | -2 | 0 | 0 | 2 | 0 | -2i | 2i | 0 | 0 | 0 | -2i | 0 | 2i | 0 | 0 | 0 | complex lifted from C4○D4 |
| ρ23 | 2 | -2 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | -2 | 2 | 0 | -2 | 0 | 0 | 2 | 0 | 2i | -2i | 0 | 0 | 0 | 2i | 0 | -2i | 0 | 0 | 0 | complex lifted from C4○D4 |
| ρ24 | 2 | -2 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | -2 | 2 | 0 | 2 | 0 | 0 | -2 | 0 | -2i | 2i | 0 | 0 | 0 | 2i | 0 | -2i | 0 | 0 | 0 | complex lifted from C4○D4 |
| ρ25 | 4 | -4 | -4 | 4 | 0 | 0 | 0 | 0 | 0 | -4 | 0 | 0 | 4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from C8⋊C22 |
| ρ26 | 4 | -4 | -4 | 4 | 0 | 0 | 0 | 0 | 0 | 4 | 0 | 0 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from C8⋊C22 |
| ρ27 | 4 | -4 | 4 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | 4 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from 2+ 1+4 |
| 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 | 0 | 2√-2 | 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 | 0 | -2√-2 | 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 48 18 43)(2 45 19 44)(3 46 20 41)(4 47 17 42)(5 36 61 37)(6 33 62 38)(7 34 63 39)(8 35 64 40)(9 21 13 50)(10 22 14 51)(11 23 15 52)(12 24 16 49)(25 56 32 59)(26 53 29 60)(27 54 30 57)(28 55 31 58)
(1 60 20 55)(2 59 17 54)(3 58 18 53)(4 57 19 56)(5 52 63 21)(6 51 64 24)(7 50 61 23)(8 49 62 22)(9 36 15 39)(10 35 16 38)(11 34 13 37)(12 33 14 40)(25 47 30 44)(26 46 31 43)(27 45 32 42)(28 48 29 41)
(1 13 18 9)(2 10 19 14)(3 15 20 11)(4 12 17 16)(5 60 61 53)(6 54 62 57)(7 58 63 55)(8 56 64 59)(21 43 50 48)(22 45 51 44)(23 41 52 46)(24 47 49 42)(25 40 32 35)(26 36 29 37)(27 38 30 33)(28 34 31 39)
(1 12)(2 15)(3 10)(4 13)(5 32)(6 28)(7 30)(8 26)(9 17)(11 19)(14 20)(16 18)(21 42)(22 46)(23 44)(24 48)(25 61)(27 63)(29 64)(31 62)(33 55)(34 57)(35 53)(36 59)(37 56)(38 58)(39 54)(40 60)(41 51)(43 49)(45 52)(47 50)
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), (1,48,18,43)(2,45,19,44)(3,46,20,41)(4,47,17,42)(5,36,61,37)(6,33,62,38)(7,34,63,39)(8,35,64,40)(9,21,13,50)(10,22,14,51)(11,23,15,52)(12,24,16,49)(25,56,32,59)(26,53,29,60)(27,54,30,57)(28,55,31,58), (1,60,20,55)(2,59,17,54)(3,58,18,53)(4,57,19,56)(5,52,63,21)(6,51,64,24)(7,50,61,23)(8,49,62,22)(9,36,15,39)(10,35,16,38)(11,34,13,37)(12,33,14,40)(25,47,30,44)(26,46,31,43)(27,45,32,42)(28,48,29,41), (1,13,18,9)(2,10,19,14)(3,15,20,11)(4,12,17,16)(5,60,61,53)(6,54,62,57)(7,58,63,55)(8,56,64,59)(21,43,50,48)(22,45,51,44)(23,41,52,46)(24,47,49,42)(25,40,32,35)(26,36,29,37)(27,38,30,33)(28,34,31,39), (1,12)(2,15)(3,10)(4,13)(5,32)(6,28)(7,30)(8,26)(9,17)(11,19)(14,20)(16,18)(21,42)(22,46)(23,44)(24,48)(25,61)(27,63)(29,64)(31,62)(33,55)(34,57)(35,53)(36,59)(37,56)(38,58)(39,54)(40,60)(41,51)(43,49)(45,52)(47,50)>;
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), (1,48,18,43)(2,45,19,44)(3,46,20,41)(4,47,17,42)(5,36,61,37)(6,33,62,38)(7,34,63,39)(8,35,64,40)(9,21,13,50)(10,22,14,51)(11,23,15,52)(12,24,16,49)(25,56,32,59)(26,53,29,60)(27,54,30,57)(28,55,31,58), (1,60,20,55)(2,59,17,54)(3,58,18,53)(4,57,19,56)(5,52,63,21)(6,51,64,24)(7,50,61,23)(8,49,62,22)(9,36,15,39)(10,35,16,38)(11,34,13,37)(12,33,14,40)(25,47,30,44)(26,46,31,43)(27,45,32,42)(28,48,29,41), (1,13,18,9)(2,10,19,14)(3,15,20,11)(4,12,17,16)(5,60,61,53)(6,54,62,57)(7,58,63,55)(8,56,64,59)(21,43,50,48)(22,45,51,44)(23,41,52,46)(24,47,49,42)(25,40,32,35)(26,36,29,37)(27,38,30,33)(28,34,31,39), (1,12)(2,15)(3,10)(4,13)(5,32)(6,28)(7,30)(8,26)(9,17)(11,19)(14,20)(16,18)(21,42)(22,46)(23,44)(24,48)(25,61)(27,63)(29,64)(31,62)(33,55)(34,57)(35,53)(36,59)(37,56)(38,58)(39,54)(40,60)(41,51)(43,49)(45,52)(47,50) );
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)], [(1,48,18,43),(2,45,19,44),(3,46,20,41),(4,47,17,42),(5,36,61,37),(6,33,62,38),(7,34,63,39),(8,35,64,40),(9,21,13,50),(10,22,14,51),(11,23,15,52),(12,24,16,49),(25,56,32,59),(26,53,29,60),(27,54,30,57),(28,55,31,58)], [(1,60,20,55),(2,59,17,54),(3,58,18,53),(4,57,19,56),(5,52,63,21),(6,51,64,24),(7,50,61,23),(8,49,62,22),(9,36,15,39),(10,35,16,38),(11,34,13,37),(12,33,14,40),(25,47,30,44),(26,46,31,43),(27,45,32,42),(28,48,29,41)], [(1,13,18,9),(2,10,19,14),(3,15,20,11),(4,12,17,16),(5,60,61,53),(6,54,62,57),(7,58,63,55),(8,56,64,59),(21,43,50,48),(22,45,51,44),(23,41,52,46),(24,47,49,42),(25,40,32,35),(26,36,29,37),(27,38,30,33),(28,34,31,39)], [(1,12),(2,15),(3,10),(4,13),(5,32),(6,28),(7,30),(8,26),(9,17),(11,19),(14,20),(16,18),(21,42),(22,46),(23,44),(24,48),(25,61),(27,63),(29,64),(31,62),(33,55),(34,57),(35,53),(36,59),(37,56),(38,58),(39,54),(40,60),(41,51),(43,49),(45,52),(47,50)]])
Matrix representation of C42.481C23 ►in GL6(𝔽17)
| 16 | 15 | 0 | 0 | 0 | 0 |
| 1 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 16 | 2 | 0 | 0 |
| 0 | 0 | 16 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 16 | 2 |
| 0 | 0 | 0 | 0 | 16 | 1 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 15 | 0 | 0 |
| 0 | 0 | 1 | 16 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 15 |
| 0 | 0 | 0 | 0 | 1 | 16 |
| 4 | 0 | 0 | 0 | 0 | 0 |
| 13 | 13 | 0 | 0 | 0 | 0 |
| 0 | 0 | 7 | 10 | 2 | 15 |
| 0 | 0 | 12 | 10 | 1 | 15 |
| 0 | 0 | 2 | 15 | 10 | 7 |
| 0 | 0 | 1 | 15 | 5 | 7 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 16 | 0 |
| 0 | 0 | 0 | 0 | 16 | 1 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 1 | 16 | 0 | 0 |
| 16 | 15 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 15 |
| 0 | 0 | 0 | 0 | 1 | 16 |
| 0 | 0 | 16 | 2 | 0 | 0 |
| 0 | 0 | 16 | 1 | 0 | 0 |
G:=sub<GL(6,GF(17))| [16,1,0,0,0,0,15,1,0,0,0,0,0,0,16,16,0,0,0,0,2,1,0,0,0,0,0,0,16,16,0,0,0,0,2,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,1,0,0,0,0,15,16,0,0,0,0,0,0,1,1,0,0,0,0,15,16],[4,13,0,0,0,0,0,13,0,0,0,0,0,0,7,12,2,1,0,0,10,10,15,15,0,0,2,1,10,5,0,0,15,15,7,7],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,1,0,0,0,0,0,16,0,0,16,16,0,0,0,0,0,1,0,0],[16,0,0,0,0,0,15,1,0,0,0,0,0,0,0,0,16,16,0,0,0,0,2,1,0,0,1,1,0,0,0,0,15,16,0,0] >;
C42.481C23 in GAP, Magma, Sage, TeX
C_4^2._{481}C_2^3 % in TeX
G:=Group("C4^2.481C2^3"); // GroupNames label
G:=SmallGroup(128,2064);
// by ID
G=gap.SmallGroup(128,2064);
# by ID
G:=PCGroup([7,-2,2,2,2,-2,2,-2,112,253,758,723,352,2019,346,248,4037,1027,124]);
// Polycyclic
G:=Group<a,b,c,d,e|a^4=b^4=e^2=1,c^2=a^2*b^2,d^2=b^2,a*b=b*a,c*a*c^-1=a^-1,d*a*d^-1=a*b^2,e*a*e=a^-1*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=a^2*b^2*c,e*d*e=b^2*d>;
// generators/relations
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