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G = C23.81C23order 64 = 26

7th central stem extension by C23 of C23

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

Aliases: C23.81C23, (C2×C4).4Q8, (C2×C4).18D4, C2.5(C4⋊Q8), C22.74(C2×D4), C2.9(C22⋊Q8), C22.23(C2×Q8), C2.10(C4⋊D4), C2.4(C42.C2), C22.41(C4○D4), C2.C42.9C2, (C22×C4).10C22, C2.8(C22.D4), (C2×C4⋊C4).10C2, SmallGroup(64,79)

Series: Derived Chief Lower central Upper central Jennings

C1C23 — C23.81C23
C1C2C22C23C22×C4C2×C4⋊C4 — C23.81C23
C1C23 — C23.81C23
C1C23 — C23.81C23
C1C23 — C23.81C23

Generators and relations for C23.81C23
 G = < a,b,c,d,e,f | a2=b2=c2=1, d2=b, e2=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, fef-1=ce=ec, cf=fc >

Subgroups: 117 in 75 conjugacy classes, 39 normal (15 characteristic)
C1, C2 [×3], C2 [×4], C4 [×11], C22 [×3], C22 [×4], C2×C4 [×8], C2×C4 [×17], C23, C4⋊C4 [×8], C22×C4 [×3], C22×C4 [×4], C2.C42, C2.C42 [×2], C2×C4⋊C4 [×2], C2×C4⋊C4 [×2], C23.81C23
Quotients: C1, C2 [×7], C22 [×7], D4 [×4], Q8 [×4], C23, C2×D4 [×2], C2×Q8 [×2], C4○D4 [×3], C4⋊D4, C22⋊Q8 [×2], C22.D4, C42.C2 [×2], C4⋊Q8, C23.81C23

Character table of C23.81C23

 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
ρ9222-2-2-22-2000-20000002000    orthogonal lifted from D4
ρ1022-2-222-2-202000000-200000    orthogonal lifted from D4
ρ11222-2-2-22-20002000000-2000    orthogonal lifted from D4
ρ1222-2-222-2-20-2000000200000    orthogonal lifted from D4
ρ132-2-2-22-222002000000-20000    symplectic lifted from Q8, Schur index 2
ρ142-2-22-222-20000002000000-2    symplectic lifted from Q8, Schur index 2
ρ152-2-22-222-2000000-20000002    symplectic lifted from Q8, Schur index 2
ρ162-2-2-22-22200-200000020000    symplectic lifted from Q8, Schur index 2
ρ172-2222-2-2-2-2i0000002i000000    complex lifted from C4○D4
ρ182-22-2-22-220000-2i0000002i00    complex lifted from C4○D4
ρ192-2222-2-2-22i000000-2i000000    complex lifted from C4○D4
ρ2022-22-2-2-22000002i000000-2i0    complex lifted from C4○D4
ρ2122-22-2-2-2200000-2i0000002i0    complex lifted from C4○D4
ρ222-22-2-22-2200002i000000-2i00    complex lifted from C4○D4

Smallest permutation representation of C23.81C23
Regular action on 64 points
Generators in S64
(1 39)(2 40)(3 37)(4 38)(5 17)(6 18)(7 19)(8 20)(9 41)(10 42)(11 43)(12 44)(13 45)(14 46)(15 47)(16 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 62)(34 63)(35 64)(36 61)
(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 47 39 15)(2 16 40 48)(3 45 37 13)(4 14 38 46)(5 41 17 9)(6 10 18 42)(7 43 19 11)(8 12 20 44)(21 57 49 29)(22 30 50 58)(23 59 51 31)(24 32 52 60)(25 61 53 36)(26 33 54 62)(27 63 55 34)(28 35 56 64)
(1 43 39 11)(2 42 40 10)(3 41 37 9)(4 44 38 12)(5 29 17 57)(6 32 18 60)(7 31 19 59)(8 30 20 58)(13 36 45 61)(14 35 46 64)(15 34 47 63)(16 33 48 62)(21 53 49 25)(22 56 50 28)(23 55 51 27)(24 54 52 26)

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

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

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

C23.81C23 is a maximal subgroup of
C4⋊C4.12D4  (C2×C4).SD16  C4215D4  C23.295C24  C42.162D4  C42.34Q8  C23.313C24  C24.252C23  C24.563C23  C23.323C24  C24.258C23  C23.329C24  C24.567C23  C24.568C23  C24.569C23  C24.269C23  C23.346C24  C23.349C24  C23.350C24  C23.351C24  C23.353C24  C24.279C23  C23.362C24  C23.364C24  C23.368C24  C24.572C23  C23.375C24  C24.293C23  C24.295C23  C23.379C24  C24.576C23  C24.299C23  C23.398C24  C23.402C24  C23.405C24  C23.406C24  C23.407C24  C23.408C24  C24.309C23  C23.419C24  C23.420C24  C23.422C24  C23.424C24  C23.425C24  C23.428C24  C23.429C24  C23.431C24  C23.432C24  C42.165D4  C23.443C24  C42.169D4  C23.449C24  C24.326C23  C24.327C23  C23.456C24  C24.332C23  C42.174D4  C24.584C23  C42.36Q8  C24.338C23  C24.341C23  C23.479C24  C42.180D4  C23.483C24  C23.485C24  C23.486C24  C24.345C23  C23.488C24  C24.346C23  C23.490C24  C23.493C24  C23.494C24  C42.183D4  C4223D4  C4224D4  C42.38Q8  C4225D4  C42.185D4  C429Q8  C24.587C23  C42.186D4  C24.589C23  C23.525C24  C23.527C24  C42.188D4  C23.530C24  C42.190D4  C42.191D4  C42.192D4  C24.374C23  C23.546C24  C42.39Q8  C23.551C24  C24.376C23  C23.553C24  C23.554C24  C23.555C24  C23.559C24  C24.377C23  C42.198D4  C4211Q8  C23.567C24  C23.571C24  C24.385C23  C23.580C24  C23.581C24  C24.394C23  C23.589C24  C23.590C24  C23.591C24  C23.592C24  C24.401C23  C23.595C24  C24.405C23  C24.406C23  C24.407C23  C23.602C24  C24.408C23  C23.606C24  C23.607C24  C23.608C24  C23.613C24  C23.616C24  C23.618C24  C23.619C24  C23.620C24  C23.621C24  C23.622C24  C23.624C24  C23.625C24  C23.626C24  C24.420C23  C24.421C23  C23.632C24  C23.634C24  C24.426C23  C24.427C23  C23.640C24  C23.641C24  C24.430C23  C24.432C23  C23.647C24  C24.434C23  C23.654C24  C23.655C24  C23.656C24  C24.438C23  C24.440C23  C23.662C24  C23.664C24  C24.443C23  C23.666C24  C23.667C24  C23.668C24  C23.669C24  C24.445C23  C23.672C24  C23.673C24  C23.674C24  C23.676C24  C23.677C24  C24.448C23  C23.681C24  C23.683C24  C23.686C24  C23.687C24  C23.688C24  C23.689C24  C24.454C23  C23.691C24  C23.692C24  C23.693C24  C23.694C24  C23.699C24  C23.702C24  C23.706C24  C23.707C24  C23.709C24  C23.714C24  C23.716C24  C42.200D4  C42.201D4  C4235D4  C23.727C24  C23.729C24  C23.730C24  C23.731C24  C23.733C24  C23.734C24  C23.736C24  C23.737C24  C23.738C24  C23.739C24  C23.741C24  C4212Q8  C42.40Q8
 (C2×C4).D4p: (C2×C4).D8  (C2×C4).5D8  (C22×C4).85D6  (C2×C4).44D12  C10.(C4⋊Q8)  (C2×C20).53D4  C14.(C4⋊Q8)  (C2×C4).44D28 ...
 C2p.(C4⋊Q8): C425Q8  C426Q8  C42.35Q8  C42.181D4  C4210Q8  C6.(C4⋊Q8)  (C2×C12).54D4  (C2×Dic3).Q8 ...
C23.81C23 is a maximal quotient of
C24.631C23  C24.632C23  C24.634C23  C24.635C23  (C2×C8).1Q8  C2.(C83Q8)  (C2×C8).24Q8
 C2p.(C4⋊Q8): (C2×C4).26D8  (C2×C4).21Q16  C4.(C4⋊Q8)  M4(2).Q8  M4(2).2Q8  C6.(C4⋊Q8)  (C22×C4).85D6  (C2×C4).44D12 ...

Matrix representation of C23.81C23 in GL6(𝔽5)

400000
040000
001000
000100
000040
000004
,
400000
040000
001000
000100
000010
000001
,
400000
040000
004000
000400
000040
000004
,
200000
030000
004000
000400
000040
000011
,
030000
300000
004000
000100
000043
000011
,
010000
400000
000100
001000
000020
000033

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],[4,0,0,0,0,0,0,4,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[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],[2,0,0,0,0,0,0,3,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,4,1,0,0,0,0,0,1],[0,3,0,0,0,0,3,0,0,0,0,0,0,0,4,0,0,0,0,0,0,1,0,0,0,0,0,0,4,1,0,0,0,0,3,1],[0,4,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,2,3,0,0,0,0,0,3] >;

C23.81C23 in GAP, Magma, Sage, TeX

C_2^3._{81}C_2^3
% in TeX

G:=Group("C2^3.81C2^3");
// GroupNames label

G:=SmallGroup(64,79);
// by ID

G=gap.SmallGroup(64,79);
# by ID

G:=PCGroup([6,-2,2,2,-2,2,2,96,121,55,362,332,50]);
// Polycyclic

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

Export

Character table of C23.81C23 in TeX

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