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G = C22.58C24order 64 = 26

44th central stem extension by C22 of C24

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

C1C22 — C22.58C24
C1C2C22C2×C4C42C42.C2 — C22.58C24
C1C22 — C22.58C24
C1C22 — C22.58C24
C1C22 — C22.58C24

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 [×3], C4 [×15], C22, C2×C4 [×15], C42 [×5], C4⋊C4 [×30], C42.C2 [×15], C22.58C24
Quotients: C1, C2 [×15], C22 [×35], C23 [×15], C24, 2- 1+4 [×3], C22.58C24

Character table of C22.58C24

 class 12A2B2C4A4B4C4D4E4F4G4H4I4J4K4L4M4N4O
 size 1111444444444444444
ρ11111111111111111111    trivial
ρ21111-1-1-1-11111-1-1-1-1111    linear of order 2
ρ31111-111-11-1-11-111-1-1-11    linear of order 2
ρ411111-1-111-1-111-1-11-1-11    linear of order 2
ρ5111111-1-1-1-111-1-1111-1-1    linear of order 2
ρ61111-1-111-1-11111-1-11-1-1    linear of order 2
ρ71111-11-11-11-111-11-1-11-1    linear of order 2
ρ811111-11-1-11-11-11-11-11-1    linear of order 2
ρ911111111-1-1-1-1-1-1-1-1111    linear of order 2
ρ101111-1-1-1-1-1-1-1-11111111    linear of order 2
ρ111111-111-1-111-11-1-11-1-11    linear of order 2
ρ1211111-1-11-111-1-111-1-1-11    linear of order 2
ρ13111111-1-111-1-111-1-11-1-1    linear of order 2
ρ141111-1-11111-1-1-1-1111-1-1    linear of order 2
ρ151111-11-111-11-1-11-11-11-1    linear of order 2
ρ1611111-11-11-11-11-11-1-11-1    linear of order 2
ρ174-4-44000000000000000    symplectic lifted from 2- 1+4, Schur index 2
ρ184-44-4000000000000000    symplectic lifted from 2- 1+4, Schur index 2
ρ1944-4-4000000000000000    symplectic lifted from 2- 1+4, Schur index 2

Smallest permutation representation of C22.58C24
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)

10000000
01000000
00100000
00010000
00004000
00000400
00000040
00000004
,
40000000
04000000
00400000
00040000
00004000
00000400
00000040
00000004
,
00100000
00010000
10000000
01000000
00003221
00000413
00001112
00003402
,
01000000
40000000
00010000
00400000
00000001
00000332
00004023
00004000
,
20000000
03000000
00300000
00020000
00000010
00001220
00004000
00004023
,
40000000
04000000
00100000
00010000
00000100
00004000
00001220
00000223

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

Export

Character table of C22.58C24 in TeX

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