p-group, metabelian, nilpotent (class 3), monomial
Aliases: C42.424C23, C4.662- 1+4, C8⋊Q8⋊20C2, C4⋊C4.136D4, Q8.Q8⋊25C2, C8.5Q8⋊6C2, Q8⋊Q8⋊11C2, C4.Q16⋊28C2, C4⋊C8.76C22, C2.32(Q8○D8), C22⋊C4.28D4, C4⋊C4.181C23, (C4×C8).118C22, (C2×C4).440C24, (C2×C8).336C23, C4.SD16⋊18C2, C23.303(C2×D4), C4⋊Q8.124C22, C4.Q8.43C22, C8⋊C4.33C22, C2.D8.41C22, C2.50(D4○SD16), C22⋊C8.67C22, (C2×Q8).171C23, (C4×Q8).118C22, C22⋊Q8.47C22, (C22×C4).313C23, Q8⋊C4.53C22, C23.48D4.4C2, C23.20D4.4C2, C23.47D4.3C2, C22.700(C22×D4), C42.C2.27C22, C42.30C22⋊7C2, C42.7C22.2C2, C42⋊C2.170C22, C22.35C24.3C2, C23.41C23.10C2, C2.88(C23.38C23), (C2×C4).564(C2×D4), (C2×C4⋊C4).655C22, SmallGroup(128,1974)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C42.424C23
G = < a,b,c,d,e | a4=b4=e2=1, c2=b2, d2=a2, ab=ba, cac-1=dad-1=a-1, eae=ab2, cbc-1=dbd-1=b-1, be=eb, dcd-1=bc, ece=a2c, de=ed >
Subgroups: 260 in 154 conjugacy classes, 84 normal (all characteristic)
C1, C2, C2, C4, C4, C22, C22, C8, C2×C4, C2×C4, Q8, C23, C42, C42, C22⋊C4, C22⋊C4, C4⋊C4, C4⋊C4, C2×C8, C22×C4, C22×C4, C2×Q8, C2×Q8, C4×C8, C8⋊C4, C22⋊C8, Q8⋊C4, C4⋊C8, C4.Q8, C2.D8, C2×C4⋊C4, C42⋊C2, C4×Q8, C22⋊Q8, C22⋊Q8, C42.C2, C42.C2, C42⋊2C2, C4⋊Q8, C4⋊Q8, C42.7C22, Q8⋊Q8, C4.Q16, Q8.Q8, C23.47D4, C23.48D4, C23.20D4, C4.SD16, C42.30C22, C8.5Q8, C8⋊Q8, C22.35C24, C23.41C23, C42.424C23
Quotients: C1, C2, C22, D4, C23, C2×D4, C24, C22×D4, 2- 1+4, C23.38C23, D4○SD16, Q8○D8, C42.424C23
Character table of C42.424C23
| class | 1 | 2A | 2B | 2C | 2D | 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 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 8 | 8 | 8 | 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 | 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 | 0 | 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 | 2 | 2 | -2 | -2 | -2 | 0 | 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 | -2 | -2 | -2 | 2 | 2 | 0 | 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 | -2 | 2 | 2 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ21 | 4 | -4 | 4 | -4 | 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 |
| ρ22 | 4 | -4 | 4 | -4 | 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 |
| ρ23 | 4 | 4 | -4 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 2√2 | 0 | -2√2 | 0 | 0 | symplectic lifted from Q8○D8, Schur index 2 |
| ρ24 | 4 | 4 | -4 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -2√2 | 0 | 2√2 | 0 | 0 | symplectic lifted from Q8○D8, Schur index 2 |
| ρ25 | 4 | -4 | -4 | 4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -2√-2 | 0 | 2√-2 | 0 | 0 | 0 | complex lifted from D4○SD16 |
| ρ26 | 4 | -4 | -4 | 4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 2√-2 | 0 | -2√-2 | 0 | 0 | 0 | complex lifted from D4○SD16 |
►On 64 points
Generators in S64
(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 39 29 9)(2 40 30 10)(3 37 31 11)(4 38 32 12)(5 60 44 23)(6 57 41 24)(7 58 42 21)(8 59 43 22)(13 49 35 25)(14 50 36 26)(15 51 33 27)(16 52 34 28)(17 46 54 62)(18 47 55 63)(19 48 56 64)(20 45 53 61)
(1 63 29 47)(2 62 30 46)(3 61 31 45)(4 64 32 48)(5 52 44 28)(6 51 41 27)(7 50 42 26)(8 49 43 25)(9 18 39 55)(10 17 40 54)(11 20 37 53)(12 19 38 56)(13 22 35 59)(14 21 36 58)(15 24 33 57)(16 23 34 60)
(1 27 3 25)(2 26 4 28)(5 54 7 56)(6 53 8 55)(9 15 11 13)(10 14 12 16)(17 42 19 44)(18 41 20 43)(21 64 23 62)(22 63 24 61)(29 51 31 49)(30 50 32 52)(33 37 35 39)(34 40 36 38)(45 59 47 57)(46 58 48 60)
(2 30)(4 32)(5 42)(6 8)(7 44)(10 40)(12 38)(14 36)(16 34)(17 56)(18 20)(19 54)(21 60)(22 24)(23 58)(26 50)(28 52)(41 43)(45 47)(46 64)(48 62)(53 55)(57 59)(61 63)
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), (1,39,29,9)(2,40,30,10)(3,37,31,11)(4,38,32,12)(5,60,44,23)(6,57,41,24)(7,58,42,21)(8,59,43,22)(13,49,35,25)(14,50,36,26)(15,51,33,27)(16,52,34,28)(17,46,54,62)(18,47,55,63)(19,48,56,64)(20,45,53,61), (1,63,29,47)(2,62,30,46)(3,61,31,45)(4,64,32,48)(5,52,44,28)(6,51,41,27)(7,50,42,26)(8,49,43,25)(9,18,39,55)(10,17,40,54)(11,20,37,53)(12,19,38,56)(13,22,35,59)(14,21,36,58)(15,24,33,57)(16,23,34,60), (1,27,3,25)(2,26,4,28)(5,54,7,56)(6,53,8,55)(9,15,11,13)(10,14,12,16)(17,42,19,44)(18,41,20,43)(21,64,23,62)(22,63,24,61)(29,51,31,49)(30,50,32,52)(33,37,35,39)(34,40,36,38)(45,59,47,57)(46,58,48,60), (2,30)(4,32)(5,42)(6,8)(7,44)(10,40)(12,38)(14,36)(16,34)(17,56)(18,20)(19,54)(21,60)(22,24)(23,58)(26,50)(28,52)(41,43)(45,47)(46,64)(48,62)(53,55)(57,59)(61,63)>;
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), (1,39,29,9)(2,40,30,10)(3,37,31,11)(4,38,32,12)(5,60,44,23)(6,57,41,24)(7,58,42,21)(8,59,43,22)(13,49,35,25)(14,50,36,26)(15,51,33,27)(16,52,34,28)(17,46,54,62)(18,47,55,63)(19,48,56,64)(20,45,53,61), (1,63,29,47)(2,62,30,46)(3,61,31,45)(4,64,32,48)(5,52,44,28)(6,51,41,27)(7,50,42,26)(8,49,43,25)(9,18,39,55)(10,17,40,54)(11,20,37,53)(12,19,38,56)(13,22,35,59)(14,21,36,58)(15,24,33,57)(16,23,34,60), (1,27,3,25)(2,26,4,28)(5,54,7,56)(6,53,8,55)(9,15,11,13)(10,14,12,16)(17,42,19,44)(18,41,20,43)(21,64,23,62)(22,63,24,61)(29,51,31,49)(30,50,32,52)(33,37,35,39)(34,40,36,38)(45,59,47,57)(46,58,48,60), (2,30)(4,32)(5,42)(6,8)(7,44)(10,40)(12,38)(14,36)(16,34)(17,56)(18,20)(19,54)(21,60)(22,24)(23,58)(26,50)(28,52)(41,43)(45,47)(46,64)(48,62)(53,55)(57,59)(61,63) );
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)], [(1,39,29,9),(2,40,30,10),(3,37,31,11),(4,38,32,12),(5,60,44,23),(6,57,41,24),(7,58,42,21),(8,59,43,22),(13,49,35,25),(14,50,36,26),(15,51,33,27),(16,52,34,28),(17,46,54,62),(18,47,55,63),(19,48,56,64),(20,45,53,61)], [(1,63,29,47),(2,62,30,46),(3,61,31,45),(4,64,32,48),(5,52,44,28),(6,51,41,27),(7,50,42,26),(8,49,43,25),(9,18,39,55),(10,17,40,54),(11,20,37,53),(12,19,38,56),(13,22,35,59),(14,21,36,58),(15,24,33,57),(16,23,34,60)], [(1,27,3,25),(2,26,4,28),(5,54,7,56),(6,53,8,55),(9,15,11,13),(10,14,12,16),(17,42,19,44),(18,41,20,43),(21,64,23,62),(22,63,24,61),(29,51,31,49),(30,50,32,52),(33,37,35,39),(34,40,36,38),(45,59,47,57),(46,58,48,60)], [(2,30),(4,32),(5,42),(6,8),(7,44),(10,40),(12,38),(14,36),(16,34),(17,56),(18,20),(19,54),(21,60),(22,24),(23,58),(26,50),(28,52),(41,43),(45,47),(46,64),(48,62),(53,55),(57,59),(61,63)]])
Matrix representation of C42.424C23 ►in GL8(𝔽17)
| 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 7 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 10 | 0 | 0 |
| 0 | 0 | 0 | 0 | 16 | 8 | 11 | 2 |
| 0 | 0 | 0 | 0 | 5 | 10 | 7 | 6 |
| 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 16 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 16 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 16 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 16 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 16 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 16 |
| 5 | 12 | 0 | 0 | 0 | 0 | 0 | 0 |
| 12 | 12 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 5 | 12 | 0 | 0 | 0 | 0 |
| 0 | 0 | 12 | 12 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 11 | 13 | 1 | 0 |
| 0 | 0 | 0 | 0 | 4 | 15 | 16 | 15 |
| 0 | 0 | 0 | 0 | 15 | 2 | 2 | 9 |
| 0 | 0 | 0 | 0 | 2 | 1 | 2 | 6 |
| 16 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 16 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 16 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 9 | 8 | 1 | 2 |
| 0 | 0 | 0 | 0 | 8 | 0 | 16 | 16 |
| 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 16 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 16 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 12 | 8 | 16 | 0 |
| 0 | 0 | 0 | 0 | 15 | 11 | 0 | 16 |
G:=sub<GL(8,GF(17))| [0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,7,1,16,5,0,0,0,0,1,10,8,10,0,0,0,0,0,0,11,7,0,0,0,0,0,0,2,6],[0,16,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,16],[5,12,0,0,0,0,0,0,12,12,0,0,0,0,0,0,0,0,5,12,0,0,0,0,0,0,12,12,0,0,0,0,0,0,0,0,11,4,15,2,0,0,0,0,13,15,2,1,0,0,0,0,1,16,2,2,0,0,0,0,0,15,9,6],[16,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,16,9,8,0,0,0,0,1,0,8,0,0,0,0,0,0,0,1,16,0,0,0,0,0,0,2,16],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,1,0,12,15,0,0,0,0,0,1,8,11,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,16] >;
C42.424C23 in GAP, Magma, Sage, TeX
C_4^2._{424}C_2^3 % in TeX
G:=Group("C4^2.424C2^3"); // GroupNames label
G:=SmallGroup(128,1974);
// by ID
G=gap.SmallGroup(128,1974);
# by ID
G:=PCGroup([7,-2,2,2,2,-2,2,-2,448,253,120,758,219,100,675,1018,4037,1027,124]);
// Polycyclic
G:=Group<a,b,c,d,e|a^4=b^4=e^2=1,c^2=b^2,d^2=a^2,a*b=b*a,c*a*c^-1=d*a*d^-1=a^-1,e*a*e=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,d*e=e*d>;
// generators/relations
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