Copied to
clipboard

G = C23.716C24order 128 = 27

433rd central stem extension by C23 of C24

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

Aliases: C23.716C24, C24.461C23, C22.3732- 1+4, C22.4892+ 1+4, C23⋊Q861C2, C23.226(C2×D4), (C22×C4).411D4, C23.4Q865C2, C23.34D462C2, C2.63(C233D4), (C23×C4).501C22, (C22×C4).227C23, C22.448(C22×D4), C23.10D4108C2, (C22×D4).294C22, (C22×Q8).232C22, C2.69(C22.29C24), C23.81C23133C2, C2.C42.419C22, C2.50(C22.56C24), C2.67(C23.38C23), (C2×C4).433(C2×D4), (C2×C22⋊Q8)⋊48C2, (C2×C4⋊D4).49C2, (C2×C4⋊C4).525C22, (C2×C22⋊C4).335C22, SmallGroup(128,1548)

Series: Derived Chief Lower central Upper central Jennings

C1C23 — C23.716C24
C1C2C22C23C22×C4C22×D4C23.10D4 — C23.716C24
C1C23 — C23.716C24
C1C23 — C23.716C24
C1C23 — C23.716C24

Generators and relations for C23.716C24
 G = < a,b,c,d,e,f,g | a2=b2=c2=f2=1, d2=e2=g2=a, ab=ba, ac=ca, ede-1=ad=da, ae=ea, gfg-1=af=fa, ag=ga, bc=cb, fdf=bd=db, be=eb, bf=fb, bg=gb, cd=dc, fef=ce=ec, cf=fc, cg=gc, dg=gd, geg-1=abe >

Subgroups: 596 in 264 conjugacy classes, 92 normal (18 characteristic)
C1, C2 [×3], C2 [×4], C2 [×4], C4 [×14], C22, C22 [×6], C22 [×24], C2×C4 [×4], C2×C4 [×38], D4 [×12], Q8 [×4], C23, C23 [×2], C23 [×20], C22⋊C4 [×18], C4⋊C4 [×12], C22×C4 [×2], C22×C4 [×14], C22×C4 [×2], C2×D4 [×12], C2×Q8 [×4], C24, C24 [×2], C2.C42 [×8], C2×C22⋊C4 [×14], C2×C4⋊C4 [×2], C2×C4⋊C4 [×6], C4⋊D4 [×4], C22⋊Q8 [×4], C23×C4, C22×D4, C22×D4 [×2], C22×Q8, C23.34D4, C23⋊Q8 [×2], C23.10D4 [×6], C23.81C23 [×2], C23.4Q8 [×2], C2×C4⋊D4, C2×C22⋊Q8, C23.716C24
Quotients: C1, C2 [×15], C22 [×35], D4 [×4], C23 [×15], C2×D4 [×6], C24, C22×D4, 2+ 1+4 [×4], 2- 1+4 [×2], C233D4, C22.29C24, C23.38C23, C22.56C24 [×4], C23.716C24

Character table of C23.716C24

 class 12A2B2C2D2E2F2G2H2I2J2K4A4B4C4D4E4F4G4H4I4J4K4L4M4N
 size 11111111448844448888888888
ρ111111111111111111111111111    trivial
ρ211111111-1-11-1-11-11-1-1-111-111-11    linear of order 2
ρ31111111111-11-1-1-1-11-1-1-11-11-111    linear of order 2
ρ411111111-1-1-1-11-11-1-111-1111-1-11    linear of order 2
ρ511111111111-1-1-1-1-11-11-1-1111-1-1    linear of order 2
ρ611111111-1-1111-11-1-11-1-1-1-1111-1    linear of order 2
ρ71111111111-1-1111111-11-1-11-1-1-1    linear of order 2
ρ811111111-1-1-11-11-11-1-111-111-11-1    linear of order 2
ρ91111111111-11-1-1-1-1-11-11-11-11-11    linear of order 2
ρ1011111111-1-1-1-11-11-11-111-1-1-1111    linear of order 2
ρ111111111111111111-1-11-1-1-1-1-1-11    linear of order 2
ρ1211111111-1-11-1-11-1111-1-1-11-1-111    linear of order 2
ρ131111111111-1-11111-1-1-1-111-111-1    linear of order 2
ρ1411111111-1-1-11-11-11111-11-1-11-1-1    linear of order 2
ρ1511111111111-1-1-1-1-1-11111-1-1-11-1    linear of order 2
ρ1611111111-1-1111-11-11-1-1111-1-1-1-1    linear of order 2
ρ172-22-22-22-2-2200-2-2220000000000    orthogonal lifted from D4
ρ182-22-22-22-2-220022-2-20000000000    orthogonal lifted from D4
ρ192-22-22-22-22-200-222-20000000000    orthogonal lifted from D4
ρ202-22-22-22-22-2002-2-220000000000    orthogonal lifted from D4
ρ214-4-44-444-4000000000000000000    orthogonal lifted from 2+ 1+4
ρ2244-444-4-4-4000000000000000000    orthogonal lifted from 2+ 1+4
ρ234-4-4-444-44000000000000000000    orthogonal lifted from 2+ 1+4
ρ2444-4-4-4-444000000000000000000    orthogonal lifted from 2+ 1+4
ρ254-444-4-4-44000000000000000000    symplectic lifted from 2- 1+4, Schur index 2
ρ26444-4-44-4-4000000000000000000    symplectic lifted from 2- 1+4, Schur index 2

Smallest permutation representation of C23.716C24
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 9)(2 10)(3 11)(4 12)(5 39)(6 40)(7 37)(8 38)(13 41)(14 42)(15 43)(16 44)(17 45)(18 46)(19 47)(20 48)(21 49)(22 50)(23 51)(24 52)(25 55)(26 56)(27 53)(28 54)(29 59)(30 60)(31 57)(32 58)(33 62)(34 63)(35 64)(36 61)
(1 53)(2 54)(3 55)(4 56)(5 52)(6 49)(7 50)(8 51)(9 27)(10 28)(11 25)(12 26)(13 31)(14 32)(15 29)(16 30)(17 36)(18 33)(19 34)(20 35)(21 40)(22 37)(23 38)(24 39)(41 57)(42 58)(43 59)(44 60)(45 61)(46 62)(47 63)(48 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 47 3 45)(2 46 4 48)(5 58 7 60)(6 57 8 59)(9 19 11 17)(10 18 12 20)(13 23 15 21)(14 22 16 24)(25 36 27 34)(26 35 28 33)(29 40 31 38)(30 39 32 37)(41 51 43 49)(42 50 44 52)(53 63 55 61)(54 62 56 64)
(1 41)(2 14)(3 43)(4 16)(5 20)(6 45)(7 18)(8 47)(9 13)(10 42)(11 15)(12 44)(17 40)(19 38)(21 36)(22 62)(23 34)(24 64)(25 29)(26 60)(27 31)(28 58)(30 56)(32 54)(33 50)(35 52)(37 46)(39 48)(49 61)(51 63)(53 57)(55 59)
(1 43 3 41)(2 44 4 42)(5 33 7 35)(6 34 8 36)(9 15 11 13)(10 16 12 14)(17 49 19 51)(18 50 20 52)(21 47 23 45)(22 48 24 46)(25 31 27 29)(26 32 28 30)(37 64 39 62)(38 61 40 63)(53 59 55 57)(54 60 56 58)

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

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

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

Matrix representation of C23.716C24 in GL12(𝔽5)

400000000000
040000000000
004000000000
000400000000
000040000000
000004000000
000000400000
000000040000
000000004000
000000000400
000000000040
000000000004
,
100000000000
010000000000
001000000000
000100000000
000010000000
000001000000
000000100000
000000010000
000000004000
000000000400
000000000040
000000000004
,
100000000000
010000000000
001000000000
000100000000
000040000000
000004000000
000000400000
000000040000
000000001000
000000000100
000000000010
000000000001
,
200000000000
020000000000
003000000000
000300000000
000000300000
000000020000
000030000000
000002000000
000000000400
000000001000
000000000001
000000000040
,
001000000000
000100000000
400000000000
040000000000
000000100000
000000010000
000040000000
000004000000
000000000010
000000000001
000000004000
000000000400
,
420000000000
010000000000
004200000000
000100000000
000001000000
000010000000
000000040000
000000400000
000000000100
000000001000
000000000001
000000000010
,
420000000000
410000000000
001300000000
001400000000
000001000000
000040000000
000000040000
000000100000
000000000100
000000004000
000000000001
000000000040

G:=sub<GL(12,GF(5))| [4,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,0,0,0,4],[1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,0,0,0,4],[1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1],[2,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,0,0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,0,1,0],[0,0,4,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0],[4,0,0,0,0,0,0,0,0,0,0,0,2,1,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,0,0,2,1,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0],[4,4,0,0,0,0,0,0,0,0,0,0,2,1,0,0,0,0,0,0,0,0,0,0,0,0,1,1,0,0,0,0,0,0,0,0,0,0,3,4,0,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,0,1,0] >;

C23.716C24 in GAP, Magma, Sage, TeX

C_2^3._{716}C_2^4
% in TeX

G:=Group("C2^3.716C2^4");
// GroupNames label

G:=SmallGroup(128,1548);
// by ID

G=gap.SmallGroup(128,1548);
# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,2,112,253,120,758,723,794,185,80]);
// Polycyclic

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

Export

Character table of C23.716C24 in TeX

׿
×
𝔽