Copied to
clipboard

G = C23.84C23order 64 = 26

10th central stem extension by C23 of C23

p-group, metabelian, nilpotent (class 2), monomial

Aliases: C23.84C23, C2.7(C422C2), C22.44(C4○D4), C2.C42.3C2, (C22×C4).12C22, 2-Sylow(Sz8), SmallGroup(64,82)

Series: Derived Chief Lower central Upper central Jennings

C1C23 — C23.84C23
C1C2C22C23C22×C4C2.C42 — C23.84C23
C1C23 — C23.84C23
C1C23 — C23.84C23
C1C23 — C23.84C23

Generators and relations for C23.84C23
 G = < a,b,c,d,e,f | a2=b2=c2=1, d2=abc, e2=ba=ab, f2=a, ac=ca, ede-1=ad=da, ae=ea, af=fa, bc=cb, fdf-1=bd=db, be=eb, bf=fb, cd=dc, fef-1=ce=ec, cf=fc >

4C4
4C4
4C4
4C4
4C4
4C4
4C4
2C2×C4
2C2×C4
2C2×C4
2C2×C4
2C2×C4
2C2×C4
2C2×C4
2C2×C4
2C2×C4
2C2×C4
2C2×C4
2C2×C4
2C2×C4
2C2×C4
2C2×C4
2C2×C4
2C2×C4
2C2×C4
2C2×C4
2C2×C4
2C2×C4

Character table of C23.84C23

 class 12A2B2C2D2E2F2G4A4B4C4D4E4F4G4H4I4J4K4L4M4N
 size 1111111144444444444444
ρ11111111111111111111111    trivial
ρ211111111-1-11-111-1-1-11-111-1    linear of order 2
ρ311111111111-1-1-1-1111-1-1-1-1    linear of order 2
ρ411111111-1-111-1-11-1-111-1-11    linear of order 2
ρ511111111-11-1-11-11-11-1-11-11    linear of order 2
ρ6111111111-1-111-1-11-1-111-1-1    linear of order 2
ρ711111111-11-11-11-1-11-11-11-1    linear of order 2
ρ8111111111-1-1-1-1111-1-1-1-111    linear of order 2
ρ92-2-22-222-20000002i000000-2i    complex lifted from C4○D4
ρ102-2222-2-2-2-2i0000002i000000    complex lifted from C4○D4
ρ1122-2-222-2-202i000000-2i00000    complex lifted from C4○D4
ρ122-2-2-22-222002i000000-2i0000    complex lifted from C4○D4
ρ132-22-2-22-220000-2i0000002i00    complex lifted from C4○D4
ρ142-2222-2-2-22i000000-2i000000    complex lifted from C4○D4
ρ1522-22-2-2-22000002i000000-2i0    complex lifted from C4○D4
ρ162-2-22-222-2000000-2i0000002i    complex lifted from C4○D4
ρ1722-22-2-2-2200000-2i0000002i0    complex lifted from C4○D4
ρ18222-2-2-22-2000-2i0000002i000    complex lifted from C4○D4
ρ1922-2-222-2-20-2i0000002i00000    complex lifted from C4○D4
ρ202-22-2-22-2200002i000000-2i00    complex lifted from C4○D4
ρ21222-2-2-22-20002i000000-2i000    complex lifted from C4○D4
ρ222-2-2-22-22200-2i0000002i0000    complex lifted from C4○D4

Smallest permutation representation of C23.84C23
Regular action on 64 points
Generators in S64
(1 55)(2 56)(3 53)(4 54)(5 37)(6 38)(7 39)(8 40)(9 43)(10 44)(11 41)(12 42)(13 45)(14 46)(15 47)(16 48)(17 49)(18 50)(19 51)(20 52)(21 26)(22 27)(23 28)(24 25)(29 58)(30 59)(31 60)(32 57)(33 64)(34 61)(35 62)(36 63)
(1 12)(2 9)(3 10)(4 11)(5 28)(6 25)(7 26)(8 27)(13 31)(14 32)(15 29)(16 30)(17 35)(18 36)(19 33)(20 34)(21 39)(22 40)(23 37)(24 38)(41 54)(42 55)(43 56)(44 53)(45 60)(46 57)(47 58)(48 59)(49 62)(50 63)(51 64)(52 61)
(1 44)(2 41)(3 42)(4 43)(5 21)(6 22)(7 23)(8 24)(9 54)(10 55)(11 56)(12 53)(13 58)(14 59)(15 60)(16 57)(17 64)(18 61)(19 62)(20 63)(25 40)(26 37)(27 38)(28 39)(29 45)(30 46)(31 47)(32 48)(33 49)(34 50)(35 51)(36 52)
(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 50 42 36)(2 19 43 64)(3 52 44 34)(4 17 41 62)(5 58 23 15)(6 30 24 48)(7 60 21 13)(8 32 22 46)(9 33 56 51)(10 61 53 20)(11 35 54 49)(12 63 55 18)(14 40 57 27)(16 38 59 25)(26 45 39 31)(28 47 37 29)
(1 60 55 31)(2 46 56 14)(3 58 53 29)(4 48 54 16)(5 18 37 50)(6 33 38 64)(7 20 39 52)(8 35 40 62)(9 57 43 32)(10 47 44 15)(11 59 41 30)(12 45 42 13)(17 22 49 27)(19 24 51 25)(21 61 26 34)(23 63 28 36)

G:=sub<Sym(64)| (1,55)(2,56)(3,53)(4,54)(5,37)(6,38)(7,39)(8,40)(9,43)(10,44)(11,41)(12,42)(13,45)(14,46)(15,47)(16,48)(17,49)(18,50)(19,51)(20,52)(21,26)(22,27)(23,28)(24,25)(29,58)(30,59)(31,60)(32,57)(33,64)(34,61)(35,62)(36,63), (1,12)(2,9)(3,10)(4,11)(5,28)(6,25)(7,26)(8,27)(13,31)(14,32)(15,29)(16,30)(17,35)(18,36)(19,33)(20,34)(21,39)(22,40)(23,37)(24,38)(41,54)(42,55)(43,56)(44,53)(45,60)(46,57)(47,58)(48,59)(49,62)(50,63)(51,64)(52,61), (1,44)(2,41)(3,42)(4,43)(5,21)(6,22)(7,23)(8,24)(9,54)(10,55)(11,56)(12,53)(13,58)(14,59)(15,60)(16,57)(17,64)(18,61)(19,62)(20,63)(25,40)(26,37)(27,38)(28,39)(29,45)(30,46)(31,47)(32,48)(33,49)(34,50)(35,51)(36,52), (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), (1,50,42,36)(2,19,43,64)(3,52,44,34)(4,17,41,62)(5,58,23,15)(6,30,24,48)(7,60,21,13)(8,32,22,46)(9,33,56,51)(10,61,53,20)(11,35,54,49)(12,63,55,18)(14,40,57,27)(16,38,59,25)(26,45,39,31)(28,47,37,29), (1,60,55,31)(2,46,56,14)(3,58,53,29)(4,48,54,16)(5,18,37,50)(6,33,38,64)(7,20,39,52)(8,35,40,62)(9,57,43,32)(10,47,44,15)(11,59,41,30)(12,45,42,13)(17,22,49,27)(19,24,51,25)(21,61,26,34)(23,63,28,36)>;

G:=Group( (1,55)(2,56)(3,53)(4,54)(5,37)(6,38)(7,39)(8,40)(9,43)(10,44)(11,41)(12,42)(13,45)(14,46)(15,47)(16,48)(17,49)(18,50)(19,51)(20,52)(21,26)(22,27)(23,28)(24,25)(29,58)(30,59)(31,60)(32,57)(33,64)(34,61)(35,62)(36,63), (1,12)(2,9)(3,10)(4,11)(5,28)(6,25)(7,26)(8,27)(13,31)(14,32)(15,29)(16,30)(17,35)(18,36)(19,33)(20,34)(21,39)(22,40)(23,37)(24,38)(41,54)(42,55)(43,56)(44,53)(45,60)(46,57)(47,58)(48,59)(49,62)(50,63)(51,64)(52,61), (1,44)(2,41)(3,42)(4,43)(5,21)(6,22)(7,23)(8,24)(9,54)(10,55)(11,56)(12,53)(13,58)(14,59)(15,60)(16,57)(17,64)(18,61)(19,62)(20,63)(25,40)(26,37)(27,38)(28,39)(29,45)(30,46)(31,47)(32,48)(33,49)(34,50)(35,51)(36,52), (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), (1,50,42,36)(2,19,43,64)(3,52,44,34)(4,17,41,62)(5,58,23,15)(6,30,24,48)(7,60,21,13)(8,32,22,46)(9,33,56,51)(10,61,53,20)(11,35,54,49)(12,63,55,18)(14,40,57,27)(16,38,59,25)(26,45,39,31)(28,47,37,29), (1,60,55,31)(2,46,56,14)(3,58,53,29)(4,48,54,16)(5,18,37,50)(6,33,38,64)(7,20,39,52)(8,35,40,62)(9,57,43,32)(10,47,44,15)(11,59,41,30)(12,45,42,13)(17,22,49,27)(19,24,51,25)(21,61,26,34)(23,63,28,36) );

G=PermutationGroup([(1,55),(2,56),(3,53),(4,54),(5,37),(6,38),(7,39),(8,40),(9,43),(10,44),(11,41),(12,42),(13,45),(14,46),(15,47),(16,48),(17,49),(18,50),(19,51),(20,52),(21,26),(22,27),(23,28),(24,25),(29,58),(30,59),(31,60),(32,57),(33,64),(34,61),(35,62),(36,63)], [(1,12),(2,9),(3,10),(4,11),(5,28),(6,25),(7,26),(8,27),(13,31),(14,32),(15,29),(16,30),(17,35),(18,36),(19,33),(20,34),(21,39),(22,40),(23,37),(24,38),(41,54),(42,55),(43,56),(44,53),(45,60),(46,57),(47,58),(48,59),(49,62),(50,63),(51,64),(52,61)], [(1,44),(2,41),(3,42),(4,43),(5,21),(6,22),(7,23),(8,24),(9,54),(10,55),(11,56),(12,53),(13,58),(14,59),(15,60),(16,57),(17,64),(18,61),(19,62),(20,63),(25,40),(26,37),(27,38),(28,39),(29,45),(30,46),(31,47),(32,48),(33,49),(34,50),(35,51),(36,52)], [(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)], [(1,50,42,36),(2,19,43,64),(3,52,44,34),(4,17,41,62),(5,58,23,15),(6,30,24,48),(7,60,21,13),(8,32,22,46),(9,33,56,51),(10,61,53,20),(11,35,54,49),(12,63,55,18),(14,40,57,27),(16,38,59,25),(26,45,39,31),(28,47,37,29)], [(1,60,55,31),(2,46,56,14),(3,58,53,29),(4,48,54,16),(5,18,37,50),(6,33,38,64),(7,20,39,52),(8,35,40,62),(9,57,43,32),(10,47,44,15),(11,59,41,30),(12,45,42,13),(17,22,49,27),(19,24,51,25),(21,61,26,34),(23,63,28,36)])

C23.84C23 is a maximal subgroup of
C24.577C23  C23.410C24  C24.315C23  C23.428C24  C24.340C23  C23.494C24  C23.543C24  C23.553C24  C23.555C24  C23.645C24  C24.432C23  C23.659C24  C23.660C24  C23.662C24  C24.443C23  C23.666C24  C23.676C24  C23.678C24  C23.687C24  C23.689C24  C23.701C24  C23.725C24  C23.733C24  C23.734C24  C23.738C24  C23.19(C2×A4)  C23.F8
 C2p.(C422C2): C23.301C24  C24.304C23  C23.414C24  C23.545C24  (C22×C4).30D6  (C22×C4).D10  (C22×C4).D14 ...
C23.84C23 is a maximal quotient of
C24.4C23
 (C22×C4).D2p: C24.633C23  (C22×C4).30D6  (C22×C4).D10  (C22×C4).D14 ...

Matrix representation of C23.84C23 in GL6(𝔽5)

400000
040000
001000
000100
000040
000004
,
400000
040000
004000
000400
000040
000004
,
100000
010000
004000
000400
000040
000004
,
010000
100000
000200
003000
000030
000042
,
400000
010000
000100
004000
000011
000004
,
300000
020000
001000
000400
000011
000034

G:=sub<GL(6,GF(5))| [4,0,0,0,0,0,0,4,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,4,0,0,0,0,0,0,4],[4,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,4],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,4],[0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,3,0,0,0,0,2,0,0,0,0,0,0,0,3,4,0,0,0,0,0,2],[4,0,0,0,0,0,0,1,0,0,0,0,0,0,0,4,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,1,4],[3,0,0,0,0,0,0,2,0,0,0,0,0,0,1,0,0,0,0,0,0,4,0,0,0,0,0,0,1,3,0,0,0,0,1,4] >;

C23.84C23 in GAP, Magma, Sage, TeX

C_2^3._{84}C_2^3
% in TeX

G:=Group("C2^3.84C2^3");
// GroupNames label

G:=SmallGroup(64,82);
// by ID

G=gap.SmallGroup(64,82);
# by ID

G:=PCGroup([6,-2,2,2,-2,2,2,336,121,151,362,332,50]);
// Polycyclic

G:=Group<a,b,c,d,e,f|a^2=b^2=c^2=1,d^2=a*b*c,e^2=b*a=a*b,f^2=a,a*c=c*a,e*d*e^-1=a*d=d*a,a*e=e*a,a*f=f*a,b*c=c*b,f*d*f^-1=b*d=d*b,b*e=e*b,b*f=f*b,c*d=d*c,f*e*f^-1=c*e=e*c,c*f=f*c>;
// generators/relations

Export

Subgroup lattice of C23.84C23 in TeX
Character table of C23.84C23 in TeX

׿
×
𝔽