Copied to
clipboard

G = C23.65C23order 64 = 26

15th central extension by C23 of C23

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

Aliases: C23.65C23, C4⋊C46C4, C42(C4⋊C4), C2.8(C4×D4), C2.4(C4×Q8), C2.2(C4⋊Q8), (C2×C4).15Q8, (C2×C4).116D4, C2.4(C4⋊D4), (C2×C42).10C2, C22.38(C2×D4), C2.4(C22⋊Q8), C22.14(C2×Q8), C2.3(C42.C2), C22.23(C4○D4), C2.C42.6C2, (C22×C4).24C22, C22.38(C22×C4), C2.8(C2×C4⋊C4), (C2×C4⋊C4).7C2, (C2×C4).18(C2×C4), SmallGroup(64,70)

Series: Derived Chief Lower central Upper central Jennings

C1C22 — C23.65C23
C1C2C22C23C22×C4C2×C42 — C23.65C23
C1C22 — C23.65C23
C1C23 — C23.65C23
C1C23 — C23.65C23

Generators and relations for C23.65C23
 G = < a,b,c,d,e,f | a2=b2=c2=1, d2=e2=c, f2=b, ab=ba, ac=ca, ede-1=ad=da, ae=ea, af=fa, bc=cb, fdf-1=bd=db, be=eb, bf=fb, cd=dc, ce=ec, cf=fc, ef=fe >

Subgroups: 121 in 85 conjugacy classes, 53 normal (23 characteristic)
C1, C2 [×7], C4 [×4], C4 [×10], C22 [×7], C2×C4 [×14], C2×C4 [×14], C23, C42 [×2], C4⋊C4 [×4], C4⋊C4 [×6], C22×C4 [×3], C22×C4 [×4], C2.C42 [×2], C2×C42, C2×C4⋊C4 [×2], C2×C4⋊C4 [×2], C23.65C23
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C2×C4 [×6], D4 [×4], Q8 [×4], C23, C4⋊C4 [×4], C22×C4, C2×D4 [×2], C2×Q8 [×2], C4○D4 [×2], C2×C4⋊C4, C4×D4, C4×Q8, C4⋊D4, C22⋊Q8, C42.C2, C4⋊Q8, C23.65C23

Character table of C23.65C23

 class 12A2B2C2D2E2F2G4A4B4C4D4E4F4G4H4I4J4K4L4M4N4O4P4Q4R4S4T
 size 1111111122222222222244444444
ρ11111111111111111111111111111    trivial
ρ211111111-1-11-1-1-1-1-1-1111-1-1111-1-11    linear of order 2
ρ311111111-11-1-1-1-1111-1-1-1-11-111-11-1    linear of order 2
ρ4111111111-1-1111-1-1-1-1-1-11-1-1111-1-1    linear of order 2
ρ511111111-11-1-1-1-1111-1-1-11-11-1-11-11    linear of order 2
ρ611111111-1-11-1-1-1-1-1-111111-1-1-111-1    linear of order 2
ρ7111111111-1-1111-1-1-1-1-1-1-111-1-1-111    linear of order 2
ρ811111111111111111111-1-1-1-1-1-1-1-1    linear of order 2
ρ91-11-11-11-1-1i-i1-11-ii-ii-ii-i-11i-ii1-1    linear of order 4
ρ101-11-11-11-11-i-i-11-1i-iii-iii11i-i-i-1-1    linear of order 4
ρ111-11-11-11-1-1i-i1-11-ii-ii-iii1-1-ii-i-11    linear of order 4
ρ121-11-11-11-11-i-i-11-1i-iii-ii-i-1-1-iii11    linear of order 4
ρ131-11-11-11-11ii-11-1-ii-i-ii-ii-1-1i-i-i11    linear of order 4
ρ141-11-11-11-1-1-ii1-11i-ii-ii-i-i1-1i-ii-11    linear of order 4
ρ151-11-11-11-11ii-11-1-ii-i-ii-i-i11-iii-1-1    linear of order 4
ρ161-11-11-11-1-1-ii1-11i-ii-ii-ii-11-ii-i1-1    linear of order 4
ρ172-222-2-2-222002-2-200000000000000    orthogonal lifted from D4
ρ182-222-2-2-22-200-22200000000000000    orthogonal lifted from D4
ρ192-2-2-222-220-20000-22200000000000    orthogonal lifted from D4
ρ202-2-2-222-220200002-2-200000000000    orthogonal lifted from D4
ρ21222-2-22-2-2200-2-2200000000000000    symplectic lifted from Q8, Schur index 2
ρ22222-2-22-2-2-20022-200000000000000    symplectic lifted from Q8, Schur index 2
ρ232-2-22-222-2002000000-2-2200000000    symplectic lifted from Q8, Schur index 2
ρ242-2-22-222-200-200000022-200000000    symplectic lifted from Q8, Schur index 2
ρ2522-222-2-2-202i0000-2i-2i2i00000000000    complex lifted from C4○D4
ρ2622-222-2-2-20-2i00002i2i-2i00000000000    complex lifted from C4○D4
ρ2722-2-2-2-222002i0000002i-2i-2i00000000    complex lifted from C4○D4
ρ2822-2-2-2-22200-2i000000-2i2i2i00000000    complex lifted from C4○D4

Smallest permutation representation of C23.65C23
Regular action on 64 points
Generators in S64
(1 11)(2 12)(3 9)(4 10)(5 38)(6 39)(7 40)(8 37)(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 53)(26 54)(27 55)(28 56)(29 57)(30 58)(31 59)(32 60)(33 63)(34 64)(35 61)(36 62)
(1 51)(2 52)(3 49)(4 50)(5 36)(6 33)(7 34)(8 35)(9 21)(10 22)(11 23)(12 24)(13 25)(14 26)(15 27)(16 28)(17 29)(18 30)(19 31)(20 32)(37 61)(38 62)(39 63)(40 64)(41 53)(42 54)(43 55)(44 56)(45 57)(46 58)(47 59)(48 60)
(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 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 45 3 47)(2 18 4 20)(5 42 7 44)(6 15 8 13)(9 19 11 17)(10 48 12 46)(14 40 16 38)(21 31 23 29)(22 60 24 58)(25 33 27 35)(26 64 28 62)(30 50 32 52)(34 56 36 54)(37 41 39 43)(49 59 51 57)(53 63 55 61)
(1 55 51 43)(2 44 52 56)(3 53 49 41)(4 42 50 54)(5 30 36 18)(6 19 33 31)(7 32 34 20)(8 17 35 29)(9 25 21 13)(10 14 22 26)(11 27 23 15)(12 16 24 28)(37 45 61 57)(38 58 62 46)(39 47 63 59)(40 60 64 48)

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

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

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

C23.65C23 is a maximal subgroup of
C4⋊C4⋊C8  C2.D84C4  C4.Q89C4  C4.Q810C4  C2.D85C4  C4⋊C47D4  C4⋊C4.94D4  C4⋊C4.95D4  C4⋊C4⋊Q8  (C2×C8)⋊Q8  C2.(C8⋊Q8)  C4⋊C4.106D4  (C2×Q8).8Q8  (C2×C4).23D8  (C2×C8).52D4  (C2×C8).1Q8  C2.(C83Q8)  (C2×C8).24Q8  (C2×C4).26D8  (C2×C4).21Q16  C4.(C4⋊Q8)  (C2×C4).28D8  (C2×C4).23Q16  C4⋊C4.Q8  C4214Q8  C23.178C24  C4×C4⋊D4  C4×C42.C2  C24.545C23  C23.199C24  C23.201C24  C23.202C24  C24.195C23  C23.211C24  C42.33Q8  C424Q8  C24.203C23  C24.204C23  C23.218C24  C23.226C24  C23.227C24  C24.208C23  C23.229C24  D4×C4⋊C4  C23.231C24  Q8×C4⋊C4  C23.233C24  C23.241C24  C24.215C23  C23.251C24  C23.252C24  C23.255C24  C24.225C23  C24.227C23  C23.264C24  C24.230C23  C23.313C24  C24.249C23  C23.315C24  C23.316C24  C24.252C23  C24.254C23  C23.321C24  C23.322C24  C23.323C24  C24.258C23  C24.259C23  C23.327C24  C23.328C24  C23.329C24  C24.567C23  C24.267C23  C24.568C23  C24.268C23  C24.569C23  C23.345C24  C23.346C24  C24.271C23  C23.348C24  C23.349C24  C23.351C24  C23.352C24  C23.353C24  C23.354C24  C24.276C23  C23.362C24  C24.285C23  C24.286C23  C23.369C24  C24.289C23  C24.572C23  C23.375C24  C24.295C23  C23.379C24  C23.385C24  C24.299C23  C24.300C23  C24.304C23  C23.395C24  C23.396C24  C23.397C24  C23.405C24  C23.406C24  C23.407C24  C23.408C24  C23.409C24  C23.411C24  C23.412C24  C23.413C24  C23.414C24  C23.418C24  C23.419C24  C23.422C24  C23.425C24  C23.426C24  C23.428C24  C23.429C24  C23.430C24  C23.433C24  C42.166D4  C42.169D4  C23.449C24  C426Q8  C427Q8  C42.35Q8  C24.326C23  C23.456C24  C23.458C24  C24.331C23  C42.176D4  C42.36Q8  C42.37Q8  C23.473C24  C42.178D4  C42.179D4  C42.180D4  C23.483C24  C42.181D4  C23.485C24  C23.486C24  C24.345C23  C23.488C24  C24.346C23  C23.490C24  C23.494C24  C23.496C24  C4223D4  C23.502C24  C42.184D4  C42.38Q8  C23.508C24  C42.185D4  C429Q8  C23.524C24  C23.525C24  C42.187D4  C42.188D4  C23.544C24  C23.545C24  C42.39Q8  C24.375C23  C24.376C23  C23.554C24  C23.555C24  C24.378C23  C42.198D4  C4211Q8  C23.567C24  C23.572C24  C23.574C24  C23.581C24  C24.389C23  C23.583C24  C23.589C24  C24.401C23  C23.595C24  C24.405C23  C24.407C23  C23.602C24  C24.408C23  C23.605C24  C23.607C24  C24.412C23  C23.613C24  C23.615C24  C23.618C24  C23.619C24  C23.620C24  C23.621C24  C23.622C24  C24.418C23  C23.624C24  C23.625C24  C23.626C24  C23.627C24  C24.421C23  C23.632C24  C23.634C24  C24.426C23  C24.427C23  C24.428C23  C23.655C24  C24.438C23  C23.658C24  C24.443C23  C23.666C24  C23.667C24  C23.668C24  C23.669C24  C24.445C23  C23.671C24  C23.672C24  C23.673C24  C23.674C24  C23.675C24  C23.676C24  C23.677C24  C23.679C24  C24.448C23  C23.683C24  C23.687C24  C23.688C24  C24.454C23  C23.691C24  C23.692C24  C23.696C24  C23.698C24  C23.700C24  C23.701C24  C23.702C24  C23.703C24  C23.705C24  C23.706C24  C23.707C24  C23.708C24  C23.709C24  C23.710C24  C23.736C24  C23.737C24  C23.738C24  C23.739C24  C42.439D4  C24.598C23  C24.599C23  C42.440D4  C43.15C2  C4313C2  C4215Q8  C43.18C2
 C4p⋊(C4⋊C4): C87(C4⋊C4)  C85(C4⋊C4)  C4.(C4×Q8)  C8⋊(C4⋊C4)  C124(C4⋊C4)  C12⋊(C4⋊C4)  C4⋊C46Dic3  C207(C4⋊C4) ...
 C2p.(C4×D4): C2.(C4×D8)  Q8⋊(C4⋊C4)  D4⋊(C4⋊C4)  Q8⋊C4⋊C4  C2.(C88D4)  C2.(C87D4)  C2.(C8⋊D4)  C2.(C82D4) ...
C23.65C23 is a maximal quotient of
C24.625C23  C24.626C23  C24.631C23  C24.632C23  C24.634C23  C42.61Q8  C42.27Q8  M4(2).5Q8  M4(2).6Q8  M4(2).27D4
 C4p⋊(C4⋊C4): C87(C4⋊C4)  C85(C4⋊C4)  C4.(C4×Q8)  C8⋊(C4⋊C4)  C124(C4⋊C4)  C12⋊(C4⋊C4)  C4⋊C46Dic3  C207(C4⋊C4) ...
 C2p.(C4×Q8): C42.62Q8  C42.28Q8  C42.29Q8  C42.30Q8  C42.31Q8  C42.430D4  C6.(C4×Q8)  Dic3⋊(C4⋊C4) ...

Matrix representation of C23.65C23 in GL6(𝔽5)

400000
040000
004000
000400
000010
000001
,
100000
010000
001000
000100
000040
000004
,
100000
010000
004000
000400
000040
000004
,
030000
200000
000200
002000
000013
000014
,
010000
100000
000100
004000
000020
000002
,
100000
010000
004000
000400
000030
000032

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

C23.65C23 in GAP, Magma, Sage, TeX

C_2^3._{65}C_2^3
% in TeX

G:=Group("C2^3.65C2^3");
// GroupNames label

G:=SmallGroup(64,70);
// by ID

G=gap.SmallGroup(64,70);
# by ID

G:=PCGroup([6,-2,2,2,-2,2,2,192,121,199,362,86]);
// Polycyclic

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

Export

Character table of C23.65C23 in TeX

׿
×
𝔽