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G = C23.63C23order 64 = 26

13rd central extension by C23 of C23

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

Aliases: C23.63C23, C4⋊C45C4, C2.6(C4×D4), C2.3(C4×Q8), (C2×C4).99D4, (C2×C4).10Q8, (C2×C42).4C2, C22.36(C2×D4), C2.3(C22⋊Q8), C22.13(C2×Q8), C2.2(C42.C2), C2.9(C42⋊C2), C2.2(C422C2), C22.21(C4○D4), C2.C42.5C2, C22.36(C22×C4), (C22×C4).22C22, C2.4(C22.D4), (C2×C4⋊C4).6C2, (C2×C4).16(C2×C4), SmallGroup(64,68)

Series: Derived Chief Lower central Upper central Jennings

C1C22 — C23.63C23
C1C2C22C23C22×C4C2×C42 — C23.63C23
C1C22 — C23.63C23
C1C23 — C23.63C23
C1C23 — C23.63C23

Generators and relations for C23.63C23
 G = < a,b,c,d,e,f | a2=b2=c2=1, d2=e2=c, f2=a, 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: 113 in 77 conjugacy classes, 45 normal (31 characteristic)
C1, C2 [×7], C4 [×12], C22 [×7], C2×C4 [×10], C2×C4 [×16], C23, C42 [×2], C4⋊C4 [×4], C4⋊C4 [×2], C22×C4 [×7], C2.C42 [×4], C2×C42, C2×C4⋊C4 [×2], C23.63C23
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C2×C4 [×6], D4 [×2], Q8 [×2], C23, C22×C4, C2×D4, C2×Q8, C4○D4 [×4], C42⋊C2, C4×D4, C4×Q8, C22⋊Q8, C22.D4, C42.C2, C422C2, C23.63C23

Character table of C23.63C23

 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
ρ172-2-2-222-220200002-2-200000000000    orthogonal lifted from D4
ρ182-2-2-222-220-20000-22200000000000    orthogonal lifted from D4
ρ192-2-22-222-200-200000022-200000000    symplectic lifted from Q8, Schur index 2
ρ202-2-22-222-2002000000-2-2200000000    symplectic lifted from Q8, Schur index 2
ρ2122-222-2-2-202i0000-2i-2i2i00000000000    complex lifted from C4○D4
ρ2222-222-2-2-20-2i00002i2i-2i00000000000    complex lifted from C4○D4
ρ23222-2-22-2-22i00-2i-2i2i00000000000000    complex lifted from C4○D4
ρ2422-2-2-2-222002i0000002i-2i-2i00000000    complex lifted from C4○D4
ρ252-222-2-2-222i002i-2i-2i00000000000000    complex lifted from C4○D4
ρ2622-2-2-2-22200-2i000000-2i2i2i00000000    complex lifted from C4○D4
ρ272-222-2-2-22-2i00-2i2i2i00000000000000    complex lifted from C4○D4
ρ28222-2-22-2-2-2i002i2i-2i00000000000000    complex lifted from C4○D4

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

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

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

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

C23.63C23 is a maximal subgroup of
C4214Q8  C432C2  C4×C22.D4  C4×C422C2  C4×C42.C2  C23.195C24  C24.192C23  C24.547C23  C23.202C24  C24.195C23  C24.198C23  C23.211C24  C42.33Q8  C424Q8  C23.214C24  C23.215C24  C24.203C23  C24.204C23  C23.218C24  C23.225C24  C23.226C24  C23.227C24  C23.235C24  C23.237C24  C23.238C24  C23.241C24  C24.218C23  C23.250C24  C23.252C24  C23.253C24  C23.255C24  C24.223C23  C24.225C23  C23.259C24  C24.227C23  C23.263C24  C23.264C24  C24.230C23  C24.563C23  C23.321C24  C23.323C24  C24.567C23  C24.267C23  C24.268C23  C24.269C23  C23.344C24  C23.346C24  C23.349C24  C23.350C24  C23.351C24  C23.353C24  C23.356C24  C24.278C23  C24.279C23  C23.359C24  C23.360C24  C23.362C24  C24.283C23  C24.285C23  C24.286C23  C23.367C24  C23.368C24  C23.369C24  C24.289C23  C24.290C23  C24.572C23  C23.374C24  C23.375C24  C24.293C23  C23.377C24  C24.295C23  C23.379C24  C24.573C23  C24.576C23  C23.385C24  C24.577C23  C24.304C23  C23.396C24  C23.397C24  C23.398C24  C24.308C23  C23.405C24  C23.406C24  C23.407C24  C23.408C24  C23.409C24  C23.410C24  C23.411C24  C23.414C24  C24.309C23  C23.416C24  C23.417C24  C23.418C24  C23.419C24  C23.420C24  C24.311C23  C23.422C24  C24.313C23  C23.424C24  C23.425C24  C23.426C24  C24.315C23  C23.428C24  C23.429C24  C23.430C24  C23.431C24  C23.432C24  C23.433C24  C24.326C23  C23.456C24  C23.457C24  C24.332C23  C24.583C23  C42.174D4  C24.584C23  C42.36Q8  C23.473C24  C24.338C23  C24.339C23  C24.340C23  C24.341C23  C23.478C24  C23.479C24  C42.179D4  C23.485C24  C23.486C24  C24.345C23  C23.488C24  C24.346C23  C23.490C24  C23.494C24  C24.347C23  C23.496C24  C24.348C23  C42.183D4  C23.500C24  C4223D4  C23.502C24  C42.184D4  C428Q8  C42.38Q8  C24.355C23  C23.508C24  C4225D4  C42.185D4  C429Q8  C23.530C24  C42.191D4  C42.192D4  C24.374C23  C23.548C24  C24.375C23  C23.550C24  C23.551C24  C24.376C23  C23.553C24  C23.554C24  C23.555C24  C24.379C23  C4211Q8  C23.567C24  C23.573C24  C23.574C24  C23.576C24  C24.385C23  C24.393C23  C23.589C24  C23.590C24  C23.591C24  C23.592C24  C24.401C23  C24.403C23  C24.405C23  C23.600C24  C23.605C24  C23.607C24  C24.411C23  C23.611C24  C23.613C24  C24.413C23  C23.615C24  C23.616C24  C23.617C24  C23.619C24  C23.620C24  C23.621C24  C23.622C24  C24.418C23  C23.625C24  C23.626C24  C24.421C23  C23.630C24  C23.631C24  C23.634C24  C23.637C24  C24.426C23  C24.427C23  C23.640C24  C23.641C24  C24.428C23  C23.643C24  C24.430C23  C23.645C24  C24.432C23  C23.647C24  C24.434C23  C24.435C23  C23.651C24  C23.652C24  C23.654C24  C23.655C24  C24.438C23  C23.658C24  C23.659C24  C24.440C23  C23.662C24  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  C23.681C24  C23.683C24  C24.450C23  C23.686C24  C23.687C24  C23.688C24  C23.689C24  C24.454C23  C23.691C24  C23.692C24  C23.693C24  C23.694C24  C23.695C24  C23.696C24  C23.697C24  C23.698C24  C23.699C24  C23.700C24  C23.702C24  C23.703C24  C23.705C24  C23.706C24  C23.708C24  C23.709C24  C23.710C24  C23.729C24  C23.730C24  C23.731C24  C23.733C24  C23.736C24  C23.738C24  C23.739C24  C42.439D4  C4243D4  C23.753C24  C24.599C23  C4215Q8  C43.18C2  C434C2  C435C2
 C2p.(C4×D4): C4242D4  C4×C22⋊Q8  C42.159D4  C42.161D4  C23.234C24  C24.558C23  C23.244C24  C23.247C24 ...
C23.63C23 is a maximal quotient of
C24.624C23  C24.626C23  C24.631C23  C24.632C23  C24.633C23  C24.635C23  C4⋊C43C8  (C2×C8).Q8
 C2.(C4×D4p): C2.D84C4  C2.(C4×D12)  C2.(C4×D20)  C4⋊Dic77C4 ...
 C2p.(C4×Q8): C4.Q89C4  C4.Q810C4  C2.D85C4  M4(2).3Q8  C6.(C4×D4)  C2.(C4×Dic6)  Dic3⋊C4⋊C4  (C2×C42).6S3 ...

Matrix representation of C23.63C23 in GL5(𝔽5)

10000
04000
00400
00040
00004
,
10000
01000
00100
00040
00004
,
40000
01000
00100
00040
00004
,
20000
01000
00400
00030
00042
,
30000
00100
01000
00022
00003
,
10000
03000
00300
00033
00002

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

C23.63C23 in GAP, Magma, Sage, TeX

C_2^3._{63}C_2^3
% in TeX

G:=Group("C2^3.63C2^3");
// GroupNames label

G:=SmallGroup(64,68);
// by ID

G=gap.SmallGroup(64,68);
# by ID

G:=PCGroup([6,-2,2,2,-2,2,2,192,121,199,362,50]);
// Polycyclic

G:=Group<a,b,c,d,e,f|a^2=b^2=c^2=1,d^2=e^2=c,f^2=a,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 C23.63C23 in TeX

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