p-group, metabelian, nilpotent (class 3), monomial
Aliases: C42.60C23, C4.812- 1+4, C8⋊Q8⋊26C2, C8⋊D4⋊51C2, C8⋊9D4⋊26C2, C8⋊8D4⋊22C2, C4⋊C4.168D4, D4⋊3Q8⋊8C2, Q8.Q8⋊41C2, D4⋊Q8⋊38C2, (C2×D4).332D4, C8.33(C4○D4), C2.57(D4○D8), C4⋊C4.251C23, C4⋊C8.119C22, (C2×C8).201C23, (C2×C4).538C24, C22⋊C4.177D4, C23.482(C2×D4), C4⋊Q8.170C22, SD16⋊C4⋊40C2, C2.91(D4⋊6D4), C4.Q8.67C22, C8⋊C4.52C22, (C4×D4).178C22, (C2×D4).256C23, C22.D8⋊33C2, C22⋊C8.97C22, (C2×Q8).241C23, (C4×Q8).178C22, M4(2)⋊C4⋊33C2, C2.D8.131C22, D4⋊C4.80C22, C4⋊D4.105C22, C23.20D4⋊44C2, C23.48D4⋊33C2, C23.46D4⋊21C2, (C22×C8).289C22, (C22×C4).342C23, Q8⋊C4.78C22, (C2×SD16).64C22, C22.798(C22×D4), C22.9(C8.C22), C22⋊Q8.104C22, C42.C2.51C22, C42⋊C2.209C22, (C2×M4(2)).131C22, C22.47C24.3C2, (C2×C2.D8)⋊43C2, C4.120(C2×C4○D4), (C2×C4).622(C2×D4), C2.83(C2×C8.C22), (C2×C4⋊C4).687C22, SmallGroup(128,2078)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C42.60C23
G = < a,b,c,d,e | a4=b4=e2=1, c2=d2=a2b2, ab=ba, cac-1=eae=a-1, dad-1=ab2, cbc-1=dbd-1=b-1, be=eb, dcd-1=bc, ece=a2c, ede=b2d >
Subgroups: 336 in 178 conjugacy classes, 88 normal (84 characteristic)
C1, C2, C2, C4, C4, C22, C22, C22, C8, C8, C2×C4, C2×C4, D4, Q8, C23, C23, C42, C42, C22⋊C4, C22⋊C4, C4⋊C4, C4⋊C4, C2×C8, C2×C8, M4(2), SD16, C22×C4, C22×C4, C2×D4, C2×D4, C2×Q8, C2×Q8, C8⋊C4, C22⋊C8, D4⋊C4, Q8⋊C4, C4⋊C8, C4.Q8, C2.D8, C2×C4⋊C4, C42⋊C2, C4×D4, C4×D4, C4×Q8, C4⋊D4, C4⋊D4, C22⋊Q8, C22⋊Q8, C22.D4, C42.C2, C42.C2, C42⋊2C2, C4⋊Q8, C22×C8, C2×M4(2), C2×SD16, C2×C2.D8, M4(2)⋊C4, C8⋊9D4, SD16⋊C4, C8⋊8D4, C8⋊D4, D4⋊Q8, Q8.Q8, C22.D8, C23.46D4, C23.48D4, C23.20D4, C8⋊Q8, C22.47C24, D4⋊3Q8, C42.60C23
Quotients: C1, C2, C22, D4, C23, C2×D4, C4○D4, C24, C8.C22, C22×D4, C2×C4○D4, 2- 1+4, D4⋊6D4, C2×C8.C22, D4○D8, C42.60C23
Character table of C42.60C23
| 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 | 2 | 2 | 4 | 8 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 8 | 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 | -2 | 0 | -2 | -2 | 0 | 2 | 2 | 0 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ18 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 0 | -2 | -2 | 0 | 2 | -2 | 0 | -2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ19 | 2 | 2 | 2 | 2 | 2 | 2 | -2 | 0 | -2 | -2 | 0 | -2 | -2 | 0 | 2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ20 | 2 | 2 | 2 | 2 | -2 | -2 | 2 | 0 | -2 | -2 | 0 | -2 | 2 | 0 | -2 | 2 | 0 | 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 | -2i | 0 | 0 | 2i | 0 | 0 | 2i | -2i | 0 | 0 | 0 | 0 | 0 | 0 | 2 | -2 | 0 | 0 | 0 | complex lifted from C4○D4 |
| ρ22 | 2 | -2 | 2 | -2 | 0 | 0 | 0 | 0 | 2 | -2 | 2i | 0 | 0 | -2i | 0 | 0 | -2i | 2i | 0 | 0 | 0 | 0 | 0 | 0 | 2 | -2 | 0 | 0 | 0 | complex lifted from C4○D4 |
| ρ23 | 2 | -2 | 2 | -2 | 0 | 0 | 0 | 0 | 2 | -2 | 2i | 0 | 0 | 2i | 0 | 0 | -2i | -2i | 0 | 0 | 0 | 0 | 0 | 0 | -2 | 2 | 0 | 0 | 0 | complex lifted from C4○D4 |
| ρ24 | 2 | -2 | 2 | -2 | 0 | 0 | 0 | 0 | 2 | -2 | -2i | 0 | 0 | -2i | 0 | 0 | 2i | 2i | 0 | 0 | 0 | 0 | 0 | 0 | -2 | 2 | 0 | 0 | 0 | complex lifted from C4○D4 |
| ρ25 | 4 | 4 | -4 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -2√2 | 0 | 0 | 2√2 | 0 | 0 | orthogonal lifted from D4○D8 |
| ρ26 | 4 | 4 | -4 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 2√2 | 0 | 0 | -2√2 | 0 | 0 | orthogonal lifted from D4○D8 |
| ρ27 | 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 | symplectic lifted from 2- 1+4, Schur index 2 |
| ρ28 | 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 | 0 | 0 | 0 | symplectic lifted from C8.C22, Schur index 2 |
| ρ29 | 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 | 0 | 0 | 0 | symplectic lifted from C8.C22, Schur index 2 |
►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 25 31 36)(2 26 32 33)(3 27 29 34)(4 28 30 35)(5 42 56 19)(6 43 53 20)(7 44 54 17)(8 41 55 18)(9 15 51 40)(10 16 52 37)(11 13 49 38)(12 14 50 39)(21 62 46 60)(22 63 47 57)(23 64 48 58)(24 61 45 59)
(1 63 29 59)(2 62 30 58)(3 61 31 57)(4 64 32 60)(5 52 54 12)(6 51 55 11)(7 50 56 10)(8 49 53 9)(13 20 40 41)(14 19 37 44)(15 18 38 43)(16 17 39 42)(21 35 48 26)(22 34 45 25)(23 33 46 28)(24 36 47 27)
(1 30 29 2)(3 32 31 4)(5 18 54 43)(6 42 55 17)(7 20 56 41)(8 44 53 19)(9 52 49 12)(10 11 50 51)(13 14 40 37)(15 16 38 39)(21 63 48 59)(22 58 45 62)(23 61 46 57)(24 60 47 64)(25 28 34 33)(26 36 35 27)
(1 10)(2 9)(3 12)(4 11)(5 63)(6 62)(7 61)(8 64)(13 28)(14 27)(15 26)(16 25)(17 24)(18 23)(19 22)(20 21)(29 50)(30 49)(31 52)(32 51)(33 40)(34 39)(35 38)(36 37)(41 48)(42 47)(43 46)(44 45)(53 60)(54 59)(55 58)(56 57)
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), (1,25,31,36)(2,26,32,33)(3,27,29,34)(4,28,30,35)(5,42,56,19)(6,43,53,20)(7,44,54,17)(8,41,55,18)(9,15,51,40)(10,16,52,37)(11,13,49,38)(12,14,50,39)(21,62,46,60)(22,63,47,57)(23,64,48,58)(24,61,45,59), (1,63,29,59)(2,62,30,58)(3,61,31,57)(4,64,32,60)(5,52,54,12)(6,51,55,11)(7,50,56,10)(8,49,53,9)(13,20,40,41)(14,19,37,44)(15,18,38,43)(16,17,39,42)(21,35,48,26)(22,34,45,25)(23,33,46,28)(24,36,47,27), (1,30,29,2)(3,32,31,4)(5,18,54,43)(6,42,55,17)(7,20,56,41)(8,44,53,19)(9,52,49,12)(10,11,50,51)(13,14,40,37)(15,16,38,39)(21,63,48,59)(22,58,45,62)(23,61,46,57)(24,60,47,64)(25,28,34,33)(26,36,35,27), (1,10)(2,9)(3,12)(4,11)(5,63)(6,62)(7,61)(8,64)(13,28)(14,27)(15,26)(16,25)(17,24)(18,23)(19,22)(20,21)(29,50)(30,49)(31,52)(32,51)(33,40)(34,39)(35,38)(36,37)(41,48)(42,47)(43,46)(44,45)(53,60)(54,59)(55,58)(56,57)>;
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), (1,25,31,36)(2,26,32,33)(3,27,29,34)(4,28,30,35)(5,42,56,19)(6,43,53,20)(7,44,54,17)(8,41,55,18)(9,15,51,40)(10,16,52,37)(11,13,49,38)(12,14,50,39)(21,62,46,60)(22,63,47,57)(23,64,48,58)(24,61,45,59), (1,63,29,59)(2,62,30,58)(3,61,31,57)(4,64,32,60)(5,52,54,12)(6,51,55,11)(7,50,56,10)(8,49,53,9)(13,20,40,41)(14,19,37,44)(15,18,38,43)(16,17,39,42)(21,35,48,26)(22,34,45,25)(23,33,46,28)(24,36,47,27), (1,30,29,2)(3,32,31,4)(5,18,54,43)(6,42,55,17)(7,20,56,41)(8,44,53,19)(9,52,49,12)(10,11,50,51)(13,14,40,37)(15,16,38,39)(21,63,48,59)(22,58,45,62)(23,61,46,57)(24,60,47,64)(25,28,34,33)(26,36,35,27), (1,10)(2,9)(3,12)(4,11)(5,63)(6,62)(7,61)(8,64)(13,28)(14,27)(15,26)(16,25)(17,24)(18,23)(19,22)(20,21)(29,50)(30,49)(31,52)(32,51)(33,40)(34,39)(35,38)(36,37)(41,48)(42,47)(43,46)(44,45)(53,60)(54,59)(55,58)(56,57) );
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)], [(1,25,31,36),(2,26,32,33),(3,27,29,34),(4,28,30,35),(5,42,56,19),(6,43,53,20),(7,44,54,17),(8,41,55,18),(9,15,51,40),(10,16,52,37),(11,13,49,38),(12,14,50,39),(21,62,46,60),(22,63,47,57),(23,64,48,58),(24,61,45,59)], [(1,63,29,59),(2,62,30,58),(3,61,31,57),(4,64,32,60),(5,52,54,12),(6,51,55,11),(7,50,56,10),(8,49,53,9),(13,20,40,41),(14,19,37,44),(15,18,38,43),(16,17,39,42),(21,35,48,26),(22,34,45,25),(23,33,46,28),(24,36,47,27)], [(1,30,29,2),(3,32,31,4),(5,18,54,43),(6,42,55,17),(7,20,56,41),(8,44,53,19),(9,52,49,12),(10,11,50,51),(13,14,40,37),(15,16,38,39),(21,63,48,59),(22,58,45,62),(23,61,46,57),(24,60,47,64),(25,28,34,33),(26,36,35,27)], [(1,10),(2,9),(3,12),(4,11),(5,63),(6,62),(7,61),(8,64),(13,28),(14,27),(15,26),(16,25),(17,24),(18,23),(19,22),(20,21),(29,50),(30,49),(31,52),(32,51),(33,40),(34,39),(35,38),(36,37),(41,48),(42,47),(43,46),(44,45),(53,60),(54,59),(55,58),(56,57)]])
Matrix representation of C42.60C23 ►in GL6(𝔽17)
| 13 | 0 | 0 | 0 | 0 | 0 |
| 0 | 4 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 16 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 16 | 0 | 0 |
| 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 | 16 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 13 | 0 | 0 | 0 | 0 |
| 13 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 2 | 15 | 15 | 2 |
| 0 | 0 | 15 | 15 | 15 | 15 |
| 0 | 0 | 15 | 15 | 2 | 2 |
| 0 | 0 | 2 | 15 | 2 | 15 |
| 13 | 0 | 0 | 0 | 0 | 0 |
| 0 | 13 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 16 | 0 | 0 | 0 |
| 0 | 0 | 0 | 16 | 0 | 0 |
| 0 | 4 | 0 | 0 | 0 | 0 |
| 13 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 16 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 16 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
G:=sub<GL(6,GF(17))| [13,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,16,0,0,1,0,0,0,0,0,0,16,0,0],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,16,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,16,0],[0,13,0,0,0,0,13,0,0,0,0,0,0,0,2,15,15,2,0,0,15,15,15,15,0,0,15,15,2,2,0,0,2,15,2,15],[13,0,0,0,0,0,0,13,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,1,0,0,0,0,0,0,1,0,0],[0,13,0,0,0,0,4,0,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,1,0,0,16,0,0,0,0,0,0,1,0,0] >;
C42.60C23 in GAP, Magma, Sage, TeX
C_4^2._{60}C_2^3 % in TeX
G:=Group("C4^2.60C2^3"); // GroupNames label
G:=SmallGroup(128,2078);
// by ID
G=gap.SmallGroup(128,2078);
# by ID
G:=PCGroup([7,-2,2,2,2,-2,2,-2,560,253,758,723,436,346,4037,1027,124]);
// Polycyclic
G:=Group<a,b,c,d,e|a^4=b^4=e^2=1,c^2=d^2=a^2*b^2,a*b=b*a,c*a*c^-1=e*a*e=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=a^2*c,e*d*e=b^2*d>;
// generators/relations
Export