p-group, metabelian, nilpotent (class 3), monomial
Aliases: C4.162+ 1+4, C8⋊8D4⋊4C2, C8⋊D4⋊18C2, Q8⋊D4⋊10C2, (C2×D4).156D4, C8.D4⋊12C2, (C2×Q8).132D4, C2.27(Q8○D8), D4.7D4⋊24C2, C8.18D4⋊18C2, C4⋊C4.140C23, (C2×C8).322C23, (C2×C4).399C24, C22⋊Q16⋊19C2, C23.281(C2×D4), C4.Q8.81C22, C2.43(D4○SD16), (C2×D4).150C23, C4⋊D4.40C22, C22⋊C8.42C22, (C2×Q8).137C23, (C2×Q16).22C22, Q8⋊C4.1C22, C2.D8.101C22, C22⋊Q8.39C22, D4⋊C4.40C22, C23.47D4⋊10C2, C23.48D4⋊21C2, C23.19D4⋊27C2, C23.20D4⋊26C2, C2.80(C23⋊3D4), (C22×C8).187C22, (C22×C4).302C23, (C2×SD16).30C22, C22.659(C22×D4), (C2×M4(2)).83C22, (C22×Q8).317C22, C42⋊C2.155C22, C23.38C23⋊15C2, C22.31C24.10C2, (C2×C4).151(C2×D4), (C22×C8)⋊C2⋊15C2, (C2×C4⋊C4).642C22, (C2×C4○D4).167C22, SmallGroup(128,1933)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C4.162+ 1+4
G = < a,b,c,d,e | a4=c2=e2=1, b4=a2, d2=ab2, dbd-1=ab=ba, ac=ca, dad-1=a-1, ae=ea, cbc=ab3, be=eb, dcd-1=ece=a2c, ede=a-1b2d >
Subgroups: 404 in 195 conjugacy classes, 84 normal (all characteristic)
C1, C2, C2, C4, C4, C22, C22, C8, C2×C4, C2×C4, D4, Q8, C23, C23, C42, C22⋊C4, C4⋊C4, C4⋊C4, C2×C8, C2×C8, M4(2), SD16, Q16, C22×C4, C22×C4, C2×D4, C2×D4, C2×Q8, C2×Q8, C4○D4, C22⋊C8, D4⋊C4, Q8⋊C4, C4.Q8, C2.D8, C2×C4⋊C4, C42⋊C2, C4⋊D4, C4⋊D4, C22⋊Q8, C22⋊Q8, C22.D4, C4.4D4, C4⋊Q8, C22×C8, C2×M4(2), C2×SD16, C2×Q16, C22×Q8, C2×C4○D4, (C22×C8)⋊C2, Q8⋊D4, C22⋊Q16, D4.7D4, C8⋊8D4, C8.18D4, C8⋊D4, C8.D4, C23.19D4, C23.47D4, C23.48D4, C23.20D4, C23.38C23, C22.31C24, C4.162+ 1+4
Quotients: C1, C2, C22, D4, C23, C2×D4, C24, C22×D4, 2+ 1+4, C23⋊3D4, D4○SD16, Q8○D8, C4.162+ 1+4
Character table of C4.162+ 1+4
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 4K | 4L | 8A | 8B | 8C | 8D | 8E | 8F | |
| size | 1 | 1 | 1 | 1 | 4 | 4 | 4 | 8 | 2 | 2 | 4 | 4 | 4 | 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 | 0 | -2 | -2 | 2 | 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 | -2 | 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 | 2 | -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 | -2 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ21 | 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 | orthogonal lifted from 2+ 1+4 |
| ρ22 | 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 | orthogonal lifted from 2+ 1+4 |
| 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 63 5 59)(2 64 6 60)(3 57 7 61)(4 58 8 62)(9 28 13 32)(10 29 14 25)(11 30 15 26)(12 31 16 27)(17 52 21 56)(18 53 22 49)(19 54 23 50)(20 55 24 51)(33 48 37 44)(34 41 38 45)(35 42 39 46)(36 43 40 47)
(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 55)(2 19)(3 49)(4 21)(5 51)(6 23)(7 53)(8 17)(9 34)(10 44)(11 36)(12 46)(13 38)(14 48)(15 40)(16 42)(18 57)(20 59)(22 61)(24 63)(25 37)(26 47)(27 39)(28 41)(29 33)(30 43)(31 35)(32 45)(50 60)(52 62)(54 64)(56 58)
(1 62 57 6)(2 5 58 61)(3 60 59 4)(7 64 63 8)(9 12 30 25)(10 32 31 11)(13 16 26 29)(14 28 27 15)(17 49 54 20)(18 19 55 56)(21 53 50 24)(22 23 51 52)(33 34 42 43)(35 40 44 41)(36 48 45 39)(37 38 46 47)
(1 33)(2 34)(3 35)(4 36)(5 37)(6 38)(7 39)(8 40)(9 23)(10 24)(11 17)(12 18)(13 19)(14 20)(15 21)(16 22)(25 55)(26 56)(27 49)(28 50)(29 51)(30 52)(31 53)(32 54)(41 64)(42 57)(43 58)(44 59)(45 60)(46 61)(47 62)(48 63)
G:=sub<Sym(64)| (1,63,5,59)(2,64,6,60)(3,57,7,61)(4,58,8,62)(9,28,13,32)(10,29,14,25)(11,30,15,26)(12,31,16,27)(17,52,21,56)(18,53,22,49)(19,54,23,50)(20,55,24,51)(33,48,37,44)(34,41,38,45)(35,42,39,46)(36,43,40,47), (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), (1,55)(2,19)(3,49)(4,21)(5,51)(6,23)(7,53)(8,17)(9,34)(10,44)(11,36)(12,46)(13,38)(14,48)(15,40)(16,42)(18,57)(20,59)(22,61)(24,63)(25,37)(26,47)(27,39)(28,41)(29,33)(30,43)(31,35)(32,45)(50,60)(52,62)(54,64)(56,58), (1,62,57,6)(2,5,58,61)(3,60,59,4)(7,64,63,8)(9,12,30,25)(10,32,31,11)(13,16,26,29)(14,28,27,15)(17,49,54,20)(18,19,55,56)(21,53,50,24)(22,23,51,52)(33,34,42,43)(35,40,44,41)(36,48,45,39)(37,38,46,47), (1,33)(2,34)(3,35)(4,36)(5,37)(6,38)(7,39)(8,40)(9,23)(10,24)(11,17)(12,18)(13,19)(14,20)(15,21)(16,22)(25,55)(26,56)(27,49)(28,50)(29,51)(30,52)(31,53)(32,54)(41,64)(42,57)(43,58)(44,59)(45,60)(46,61)(47,62)(48,63)>;
G:=Group( (1,63,5,59)(2,64,6,60)(3,57,7,61)(4,58,8,62)(9,28,13,32)(10,29,14,25)(11,30,15,26)(12,31,16,27)(17,52,21,56)(18,53,22,49)(19,54,23,50)(20,55,24,51)(33,48,37,44)(34,41,38,45)(35,42,39,46)(36,43,40,47), (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), (1,55)(2,19)(3,49)(4,21)(5,51)(6,23)(7,53)(8,17)(9,34)(10,44)(11,36)(12,46)(13,38)(14,48)(15,40)(16,42)(18,57)(20,59)(22,61)(24,63)(25,37)(26,47)(27,39)(28,41)(29,33)(30,43)(31,35)(32,45)(50,60)(52,62)(54,64)(56,58), (1,62,57,6)(2,5,58,61)(3,60,59,4)(7,64,63,8)(9,12,30,25)(10,32,31,11)(13,16,26,29)(14,28,27,15)(17,49,54,20)(18,19,55,56)(21,53,50,24)(22,23,51,52)(33,34,42,43)(35,40,44,41)(36,48,45,39)(37,38,46,47), (1,33)(2,34)(3,35)(4,36)(5,37)(6,38)(7,39)(8,40)(9,23)(10,24)(11,17)(12,18)(13,19)(14,20)(15,21)(16,22)(25,55)(26,56)(27,49)(28,50)(29,51)(30,52)(31,53)(32,54)(41,64)(42,57)(43,58)(44,59)(45,60)(46,61)(47,62)(48,63) );
G=PermutationGroup([[(1,63,5,59),(2,64,6,60),(3,57,7,61),(4,58,8,62),(9,28,13,32),(10,29,14,25),(11,30,15,26),(12,31,16,27),(17,52,21,56),(18,53,22,49),(19,54,23,50),(20,55,24,51),(33,48,37,44),(34,41,38,45),(35,42,39,46),(36,43,40,47)], [(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)], [(1,55),(2,19),(3,49),(4,21),(5,51),(6,23),(7,53),(8,17),(9,34),(10,44),(11,36),(12,46),(13,38),(14,48),(15,40),(16,42),(18,57),(20,59),(22,61),(24,63),(25,37),(26,47),(27,39),(28,41),(29,33),(30,43),(31,35),(32,45),(50,60),(52,62),(54,64),(56,58)], [(1,62,57,6),(2,5,58,61),(3,60,59,4),(7,64,63,8),(9,12,30,25),(10,32,31,11),(13,16,26,29),(14,28,27,15),(17,49,54,20),(18,19,55,56),(21,53,50,24),(22,23,51,52),(33,34,42,43),(35,40,44,41),(36,48,45,39),(37,38,46,47)], [(1,33),(2,34),(3,35),(4,36),(5,37),(6,38),(7,39),(8,40),(9,23),(10,24),(11,17),(12,18),(13,19),(14,20),(15,21),(16,22),(25,55),(26,56),(27,49),(28,50),(29,51),(30,52),(31,53),(32,54),(41,64),(42,57),(43,58),(44,59),(45,60),(46,61),(47,62),(48,63)]])
Matrix representation of C4.162+ 1+4 ►in GL8(𝔽17)
| 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 | 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 | 0 | 16 |
| 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 7 | 0 | 4 | 0 | 0 | 0 | 0 |
| 10 | 0 | 13 | 0 | 0 | 0 | 0 | 0 |
| 0 | 13 | 0 | 10 | 0 | 0 | 0 | 0 |
| 4 | 0 | 7 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 14 | 14 | 0 | 0 |
| 0 | 0 | 0 | 0 | 3 | 14 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 14 | 14 |
| 0 | 0 | 0 | 0 | 0 | 0 | 3 | 14 |
| 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 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 | 7 | 0 | 4 | 0 | 0 | 0 | 0 |
| 7 | 0 | 4 | 0 | 0 | 0 | 0 | 0 |
| 0 | 13 | 0 | 10 | 0 | 0 | 0 | 0 |
| 13 | 0 | 10 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 3 | 14 | 0 | 0 |
| 0 | 0 | 0 | 0 | 14 | 14 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 14 | 3 |
| 0 | 0 | 0 | 0 | 0 | 0 | 3 | 3 |
| 0 | 13 | 0 | 10 | 0 | 0 | 0 | 0 |
| 13 | 0 | 10 | 0 | 0 | 0 | 0 | 0 |
| 0 | 7 | 0 | 4 | 0 | 0 | 0 | 0 |
| 7 | 0 | 4 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 | 10 | 0 |
| 0 | 0 | 0 | 0 | 16 | 0 | 0 | 10 |
| 0 | 0 | 0 | 0 | 7 | 0 | 0 | 16 |
| 0 | 0 | 0 | 0 | 0 | 7 | 1 | 0 |
G:=sub<GL(8,GF(17))| [16,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,16,0],[0,10,0,4,0,0,0,0,7,0,13,0,0,0,0,0,0,13,0,7,0,0,0,0,4,0,10,0,0,0,0,0,0,0,0,0,14,3,0,0,0,0,0,0,14,14,0,0,0,0,0,0,0,0,14,3,0,0,0,0,0,0,14,14],[0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0],[0,7,0,13,0,0,0,0,7,0,13,0,0,0,0,0,0,4,0,10,0,0,0,0,4,0,10,0,0,0,0,0,0,0,0,0,3,14,0,0,0,0,0,0,14,14,0,0,0,0,0,0,0,0,14,3,0,0,0,0,0,0,3,3],[0,13,0,7,0,0,0,0,13,0,7,0,0,0,0,0,0,10,0,4,0,0,0,0,10,0,4,0,0,0,0,0,0,0,0,0,0,16,7,0,0,0,0,0,1,0,0,7,0,0,0,0,10,0,0,1,0,0,0,0,0,10,16,0] >;
C4.162+ 1+4 in GAP, Magma, Sage, TeX
C_4._{16}2_+^{1+4} % in TeX
G:=Group("C4.16ES+(2,2)"); // GroupNames label
G:=SmallGroup(128,1933);
// by ID
G=gap.SmallGroup(128,1933);
# by ID
G:=PCGroup([7,-2,2,2,2,-2,2,-2,253,758,219,352,675,1018,4037,1027,124]);
// Polycyclic
G:=Group<a,b,c,d,e|a^4=c^2=e^2=1,b^4=a^2,d^2=a*b^2,d*b*d^-1=a*b=b*a,a*c=c*a,d*a*d^-1=a^-1,a*e=e*a,c*b*c=a*b^3,b*e=e*b,d*c*d^-1=e*c*e=a^2*c,e*d*e=a^-1*b^2*d>;
// generators/relations
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