p-group, metabelian, nilpotent (class 2), monomial, rational
Aliases: C22.58C24, C42.57C22, C2.172- 1+4, C4⋊C4.41C22, (C2×C4).40C23, C42.C2.7C2, 2-Sylow(GU(3,4)), SmallGroup(64,245)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C22.58C24
G = < a,b,c,d,e,f | a2=b2=1, c2=f2=a, d2=e2=b, ab=ba, dcd-1=ac=ca, fdf-1=ad=da, ae=ea, af=fa, ece-1=bc=cb, bd=db, be=eb, bf=fb, fcf-1=abc, ede-1=abd, ef=fe >
Subgroups: 101 in 86 conjugacy classes, 71 normal (3 characteristic)
C1, C2, C4, C22, C2×C4, C42, C4⋊C4, C42.C2, C22.58C24
Quotients: C1, C2, C22, C23, C24, 2- 1+4, C22.58C24
Character table of C22.58C24
| class | 1 | 2A | 2B | 2C | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 4K | 4L | 4M | 4N | 4O | |
| size | 1 | 1 | 1 | 1 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | |
| ρ1 | 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 | linear of order 2 |
| ρ3 | 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 | linear of order 2 |
| ρ5 | 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 | linear of order 2 |
| ρ7 | 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 | linear of order 2 |
| ρ9 | 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 | linear of order 2 |
| ρ11 | 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 | linear of order 2 |
| ρ13 | 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 | linear of order 2 |
| ρ15 | 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 | linear of order 2 |
| ρ17 | 4 | -4 | -4 | 4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | symplectic lifted from 2- 1+4, Schur index 2 |
| ρ18 | 4 | -4 | 4 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | symplectic lifted from 2- 1+4, Schur index 2 |
| ρ19 | 4 | 4 | -4 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | symplectic lifted from 2- 1+4, Schur index 2 |
►Regular action on 64 points
Generators in S64
(1 3)(2 4)(5 7)(6 8)(9 11)(10 12)(13 15)(14 16)(17 19)(18 20)(21 23)(22 24)(25 27)(26 28)(29 31)(30 32)(33 35)(34 36)(37 39)(38 40)(41 43)(42 44)(45 47)(46 48)(49 51)(50 52)(53 55)(54 56)(57 59)(58 60)(61 63)(62 64)
(1 51)(2 52)(3 49)(4 50)(5 36)(6 33)(7 34)(8 35)(9 53)(10 54)(11 55)(12 56)(13 57)(14 58)(15 59)(16 60)(17 61)(18 62)(19 63)(20 64)(21 37)(22 38)(23 39)(24 40)(25 41)(26 42)(27 43)(28 44)(29 45)(30 46)(31 47)(32 48)
(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 59 51 15)(2 58 52 14)(3 57 49 13)(4 60 50 16)(5 41 36 25)(6 44 33 28)(7 43 34 27)(8 42 35 26)(9 61 53 17)(10 64 54 20)(11 63 55 19)(12 62 56 18)(21 45 37 29)(22 48 38 32)(23 47 39 31)(24 46 40 30)
(1 55 51 11)(2 12 52 56)(3 53 49 9)(4 10 50 54)(5 31 36 47)(6 48 33 32)(7 29 34 45)(8 46 35 30)(13 19 57 63)(14 64 58 20)(15 17 59 61)(16 62 60 18)(21 41 37 25)(22 26 38 42)(23 43 39 27)(24 28 40 44)
(1 37 3 39)(2 24 4 22)(5 19 7 17)(6 62 8 64)(9 43 11 41)(10 26 12 28)(13 45 15 47)(14 32 16 30)(18 35 20 33)(21 49 23 51)(25 53 27 55)(29 59 31 57)(34 61 36 63)(38 52 40 50)(42 56 44 54)(46 58 48 60)
G:=sub<Sym(64)| (1,3)(2,4)(5,7)(6,8)(9,11)(10,12)(13,15)(14,16)(17,19)(18,20)(21,23)(22,24)(25,27)(26,28)(29,31)(30,32)(33,35)(34,36)(37,39)(38,40)(41,43)(42,44)(45,47)(46,48)(49,51)(50,52)(53,55)(54,56)(57,59)(58,60)(61,63)(62,64), (1,51)(2,52)(3,49)(4,50)(5,36)(6,33)(7,34)(8,35)(9,53)(10,54)(11,55)(12,56)(13,57)(14,58)(15,59)(16,60)(17,61)(18,62)(19,63)(20,64)(21,37)(22,38)(23,39)(24,40)(25,41)(26,42)(27,43)(28,44)(29,45)(30,46)(31,47)(32,48), (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), (1,59,51,15)(2,58,52,14)(3,57,49,13)(4,60,50,16)(5,41,36,25)(6,44,33,28)(7,43,34,27)(8,42,35,26)(9,61,53,17)(10,64,54,20)(11,63,55,19)(12,62,56,18)(21,45,37,29)(22,48,38,32)(23,47,39,31)(24,46,40,30), (1,55,51,11)(2,12,52,56)(3,53,49,9)(4,10,50,54)(5,31,36,47)(6,48,33,32)(7,29,34,45)(8,46,35,30)(13,19,57,63)(14,64,58,20)(15,17,59,61)(16,62,60,18)(21,41,37,25)(22,26,38,42)(23,43,39,27)(24,28,40,44), (1,37,3,39)(2,24,4,22)(5,19,7,17)(6,62,8,64)(9,43,11,41)(10,26,12,28)(13,45,15,47)(14,32,16,30)(18,35,20,33)(21,49,23,51)(25,53,27,55)(29,59,31,57)(34,61,36,63)(38,52,40,50)(42,56,44,54)(46,58,48,60)>;
G:=Group( (1,3)(2,4)(5,7)(6,8)(9,11)(10,12)(13,15)(14,16)(17,19)(18,20)(21,23)(22,24)(25,27)(26,28)(29,31)(30,32)(33,35)(34,36)(37,39)(38,40)(41,43)(42,44)(45,47)(46,48)(49,51)(50,52)(53,55)(54,56)(57,59)(58,60)(61,63)(62,64), (1,51)(2,52)(3,49)(4,50)(5,36)(6,33)(7,34)(8,35)(9,53)(10,54)(11,55)(12,56)(13,57)(14,58)(15,59)(16,60)(17,61)(18,62)(19,63)(20,64)(21,37)(22,38)(23,39)(24,40)(25,41)(26,42)(27,43)(28,44)(29,45)(30,46)(31,47)(32,48), (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), (1,59,51,15)(2,58,52,14)(3,57,49,13)(4,60,50,16)(5,41,36,25)(6,44,33,28)(7,43,34,27)(8,42,35,26)(9,61,53,17)(10,64,54,20)(11,63,55,19)(12,62,56,18)(21,45,37,29)(22,48,38,32)(23,47,39,31)(24,46,40,30), (1,55,51,11)(2,12,52,56)(3,53,49,9)(4,10,50,54)(5,31,36,47)(6,48,33,32)(7,29,34,45)(8,46,35,30)(13,19,57,63)(14,64,58,20)(15,17,59,61)(16,62,60,18)(21,41,37,25)(22,26,38,42)(23,43,39,27)(24,28,40,44), (1,37,3,39)(2,24,4,22)(5,19,7,17)(6,62,8,64)(9,43,11,41)(10,26,12,28)(13,45,15,47)(14,32,16,30)(18,35,20,33)(21,49,23,51)(25,53,27,55)(29,59,31,57)(34,61,36,63)(38,52,40,50)(42,56,44,54)(46,58,48,60) );
G=PermutationGroup([[(1,3),(2,4),(5,7),(6,8),(9,11),(10,12),(13,15),(14,16),(17,19),(18,20),(21,23),(22,24),(25,27),(26,28),(29,31),(30,32),(33,35),(34,36),(37,39),(38,40),(41,43),(42,44),(45,47),(46,48),(49,51),(50,52),(53,55),(54,56),(57,59),(58,60),(61,63),(62,64)], [(1,51),(2,52),(3,49),(4,50),(5,36),(6,33),(7,34),(8,35),(9,53),(10,54),(11,55),(12,56),(13,57),(14,58),(15,59),(16,60),(17,61),(18,62),(19,63),(20,64),(21,37),(22,38),(23,39),(24,40),(25,41),(26,42),(27,43),(28,44),(29,45),(30,46),(31,47),(32,48)], [(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)], [(1,59,51,15),(2,58,52,14),(3,57,49,13),(4,60,50,16),(5,41,36,25),(6,44,33,28),(7,43,34,27),(8,42,35,26),(9,61,53,17),(10,64,54,20),(11,63,55,19),(12,62,56,18),(21,45,37,29),(22,48,38,32),(23,47,39,31),(24,46,40,30)], [(1,55,51,11),(2,12,52,56),(3,53,49,9),(4,10,50,54),(5,31,36,47),(6,48,33,32),(7,29,34,45),(8,46,35,30),(13,19,57,63),(14,64,58,20),(15,17,59,61),(16,62,60,18),(21,41,37,25),(22,26,38,42),(23,43,39,27),(24,28,40,44)], [(1,37,3,39),(2,24,4,22),(5,19,7,17),(6,62,8,64),(9,43,11,41),(10,26,12,28),(13,45,15,47),(14,32,16,30),(18,35,20,33),(21,49,23,51),(25,53,27,55),(29,59,31,57),(34,61,36,63),(38,52,40,50),(42,56,44,54),(46,58,48,60)]])
C22.58C24 is a maximal subgroup of
C42.4C23 C42.9C23 C22.142C25 C22.156C25 C42.A4 C22.58C24⋊C5
C2p.2- 1+4: C22.120C25 C22.145C25 C22.152C25 C42.147D6 C42.147D10 C42.147D14 ...
C22.58C24 is a maximal quotient of
C23.264C24 C23.619C24 C23.626C24 C23.667C24 C23.710C24 C23.739C24 C42.40Q8
C42.D2p: C42.201D4 C42.147D6 C42.147D10 C42.147D14 ...
Matrix representation of C22.58C24 ►in GL8(𝔽5)
| 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 4 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 4 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 4 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 4 |
| 4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 4 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 4 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 4 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 4 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 4 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 4 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 4 |
| 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 | 3 | 2 | 2 | 1 |
| 0 | 0 | 0 | 0 | 0 | 4 | 1 | 3 |
| 0 | 0 | 0 | 0 | 1 | 1 | 1 | 2 |
| 0 | 0 | 0 | 0 | 3 | 4 | 0 | 2 |
| 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 4 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 0 | 3 | 3 | 2 |
| 0 | 0 | 0 | 0 | 4 | 0 | 2 | 3 |
| 0 | 0 | 0 | 0 | 4 | 0 | 0 | 0 |
| 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 3 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 3 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 2 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 1 | 2 | 2 | 0 |
| 0 | 0 | 0 | 0 | 4 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 4 | 0 | 2 | 3 |
| 4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 4 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 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 | 4 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 2 | 2 | 0 |
| 0 | 0 | 0 | 0 | 0 | 2 | 2 | 3 |
G:=sub<GL(8,GF(5))| [1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4],[4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4],[0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,3,0,1,3,0,0,0,0,2,4,1,4,0,0,0,0,2,1,1,0,0,0,0,0,1,3,2,2],[0,4,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,4,4,0,0,0,0,0,3,0,0,0,0,0,0,0,3,2,0,0,0,0,0,1,2,3,0],[2,0,0,0,0,0,0,0,0,3,0,0,0,0,0,0,0,0,3,0,0,0,0,0,0,0,0,2,0,0,0,0,0,0,0,0,0,1,4,4,0,0,0,0,0,2,0,0,0,0,0,0,1,2,0,2,0,0,0,0,0,0,0,3],[4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,4,1,0,0,0,0,0,1,0,2,2,0,0,0,0,0,0,2,2,0,0,0,0,0,0,0,3] >;
C22.58C24 in GAP, Magma, Sage, TeX
C_2^2._{58}C_2^4 % in TeX
G:=Group("C2^2.58C2^4"); // GroupNames label
G:=SmallGroup(64,245);
// by ID
G=gap.SmallGroup(64,245);
# by ID
G:=PCGroup([6,-2,2,2,2,-2,2,96,217,199,650,476,158,1347,297,69]);
// Polycyclic
G:=Group<a,b,c,d,e,f|a^2=b^2=1,c^2=f^2=a,d^2=e^2=b,a*b=b*a,d*c*d^-1=a*c=c*a,f*d*f^-1=a*d=d*a,a*e=e*a,a*f=f*a,e*c*e^-1=b*c=c*b,b*d=d*b,b*e=e*b,b*f=f*b,f*c*f^-1=a*b*c,e*d*e^-1=a*b*d,e*f=f*e>;
// generators/relations
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