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G = C23.83C23order 64 = 26

9th central stem extension by C23 of C23

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

Aliases: C23.83C23, (C2×C4).5Q8, (C2×C4).20D4, C22.76(C2×D4), C22.25(C2×Q8), C2.8(C4.4D4), C2.11(C22⋊Q8), C2.5(C42.C2), C2.6(C422C2), C22.43(C4○D4), (C22×C4).28C22, C2.C42.10C2, C2.10(C22.D4), (C2×C4⋊C4).11C2, SmallGroup(64,81)

Series: Derived Chief Lower central Upper central Jennings

C1C23 — C23.83C23
C1C2C22C23C22×C4C2.C42 — C23.83C23
C1C23 — C23.83C23
C1C23 — C23.83C23
C1C23 — C23.83C23

Generators and relations for C23.83C23
 G = < a,b,c,d,e,f | a2=b2=c2=1, d2=f2=a, e2=ba=ab, 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: 109 in 67 conjugacy classes, 35 normal (15 characteristic)
C1, C2 [×3], C2 [×4], C4 [×9], C22 [×3], C22 [×4], C2×C4 [×4], C2×C4 [×19], C23, C4⋊C4 [×4], C22×C4 [×3], C22×C4 [×4], C2.C42, C2.C42 [×4], C2×C4⋊C4 [×2], C23.83C23
Quotients: C1, C2 [×7], C22 [×7], D4 [×2], Q8 [×2], C23, C2×D4, C2×Q8, C4○D4 [×5], C22⋊Q8, C22.D4, C4.4D4, C42.C2 [×2], C422C2 [×2], C23.83C23

Character table of C23.83C23

 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
ρ10222-2-2-22-20002000000-2000    orthogonal lifted from D4
ρ112-22-2-22-220000-2000000200    symplectic lifted from Q8, Schur index 2
ρ122-22-2-22-2200002000000-200    symplectic lifted from Q8, Schur index 2
ρ132-2-22-222-20000002i000000-2i    complex lifted from C4○D4
ρ142-2222-2-2-2-2i0000002i000000    complex lifted from C4○D4
ρ1522-2-222-2-202i000000-2i00000    complex lifted from C4○D4
ρ162-2-2-22-222002i000000-2i0000    complex lifted from C4○D4
ρ172-2222-2-2-22i000000-2i000000    complex lifted from C4○D4
ρ1822-22-2-2-22000002i000000-2i0    complex lifted from C4○D4
ρ192-2-22-222-2000000-2i0000002i    complex lifted from C4○D4
ρ2022-22-2-2-2200000-2i0000002i0    complex lifted from C4○D4
ρ2122-2-222-2-20-2i0000002i00000    complex lifted from C4○D4
ρ222-2-2-22-22200-2i0000002i0000    complex lifted from C4○D4

Smallest permutation representation of C23.83C23
Regular action 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 41)(2 42)(3 43)(4 44)(5 20)(6 17)(7 18)(8 19)(9 57)(10 58)(11 59)(12 60)(13 25)(14 26)(15 27)(16 28)(21 34)(22 35)(23 36)(24 33)(29 45)(30 46)(31 47)(32 48)(37 56)(38 53)(39 54)(40 55)(49 61)(50 62)(51 63)(52 64)
(1 11)(2 12)(3 9)(4 10)(5 50)(6 51)(7 52)(8 49)(13 31)(14 32)(15 29)(16 30)(17 63)(18 64)(19 61)(20 62)(21 40)(22 37)(23 38)(24 39)(25 47)(26 48)(27 45)(28 46)(33 54)(34 55)(35 56)(36 53)(41 59)(42 60)(43 57)(44 58)
(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 51 43 61)(2 50 44 64)(3 49 41 63)(4 52 42 62)(5 58 18 12)(6 57 19 11)(7 60 20 10)(8 59 17 9)(13 36 27 21)(14 35 28 24)(15 34 25 23)(16 33 26 22)(29 55 47 38)(30 54 48 37)(31 53 45 40)(32 56 46 39)
(1 27 3 25)(2 16 4 14)(5 33 7 35)(6 21 8 23)(9 47 11 45)(10 32 12 30)(13 41 15 43)(17 34 19 36)(18 22 20 24)(26 42 28 44)(29 57 31 59)(37 62 39 64)(38 51 40 49)(46 58 48 60)(50 54 52 56)(53 63 55 61)

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

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

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

C23.83C23 is a maximal subgroup of
C42.162D4  C42.163D4  C23.301C24  C42.34Q8  C24.563C23  C23.321C24  C24.567C23  C24.569C23  C23.344C24  C23.346C24  C23.350C24  C23.353C24  C24.278C23  C23.369C24  C24.289C23  C24.572C23  C23.375C24  C23.377C24  C24.573C23  C23.388C24  C24.301C23  C23.392C24  C24.577C23  C23.395C24  C23.396C24  C23.397C24  C23.405C24  C23.406C24  C23.408C24  C23.409C24  C23.410C24  C23.411C24  C23.414C24  C24.309C23  C23.416C24  C23.420C24  C23.424C24  C23.425C24  C23.426C24  C24.315C23  C23.428C24  C23.429C24  C23.430C24  C23.432C24  C23.433C24  C23.458C24  C24.331C23  C42.172D4  C42.177D4  C24.584C23  C42.36Q8  C42.37Q8  C23.472C24  C23.473C24  C24.339C23  C24.341C23  C23.478C24  C42.179D4  C23.485C24  C24.345C23  C23.488C24  C24.346C23  C23.490C24  C23.493C24  C24.347C23  C23.496C24  C24.348C23  C4222D4  C42.184D4  C428Q8  C42.185D4  C429Q8  C42.187D4  C42.191D4  C23.535C24  C4230D4  C42.195D4  C23.543C24  C23.545C24  C23.546C24  C42.39Q8  C23.548C24  C24.375C23  C23.550C24  C24.376C23  C23.554C24  C23.555C24  C42.198D4  C24.379C23  C4211Q8  C24.394C23  C24.395C23  C23.589C24  C23.590C24  C23.593C24  C23.595C24  C24.403C23  C24.405C23  C23.602C24  C23.603C24  C23.608C24  C23.613C24  C23.616C24  C23.618C24  C23.619C24  C23.620C24  C23.625C24  C24.420C23  C24.421C23  C24.426C23  C24.427C23  C23.641C24  C24.428C23  C23.643C24  C24.430C23  C23.645C24  C24.432C23  C23.647C24  C23.649C24  C24.435C23  C23.651C24  C24.437C23  C23.654C24  C23.655C24  C23.658C24  C23.662C24  C23.663C24  C23.664C24  C24.443C23  C23.666C24  C23.667C24  C23.669C24  C24.445C23  C23.671C24  C23.672C24  C23.673C24  C23.674C24  C23.675C24  C23.676C24  C23.677C24  C23.678C24  C23.679C24  C24.448C23  C23.681C24  C23.682C24  C23.683C24  C24.450C23  C23.687C24  C23.688C24  C23.689C24  C24.454C23  C23.691C24  C23.693C24  C23.694C24  C23.695C24  C23.696C24  C23.698C24  C23.699C24  C23.702C24  C24.456C23  C23.705C24  C23.707C24  C23.709C24  C23.710C24  C24.459C23  C42.200D4  C42.201D4  C23.724C24  C23.726C24  C23.727C24  C23.728C24  C23.731C24  C23.732C24  C23.733C24  C23.735C24  C23.736C24  C23.737C24  C23.738C24  C23.739C24  C23.741C24  C4212Q8  C4213Q8
 (C22×C4).D2p: (C2×C4).Q16  C4⋊C4.18D4  C4⋊C4.19D4  C4⋊C4.20D4  C23.295C24  C24.576C23  C24.300C23  C23.398C24 ...
C23.83C23 is a maximal quotient of
 (C2×C4p).D4: C42.32Q8  C22⋊C4.Q8  (C2×C4).17D12  (C2×C12).288D4  (C2×C12).55D4  (C2×C20).28D4  (C2×C20).288D4  (C2×C20).55D4 ...
 (C22×C4).D2p: C24.631C23  C24.632C23  C24.633C23  C24.635C23  (C2×C4).28D8  (C2×C4).23Q16  C4⋊C4.Q8  (C2×Dic3).9D4 ...

Matrix representation of C23.83C23 in GL6(𝔽5)

400000
040000
001000
000100
000010
000001
,
400000
040000
001000
000100
000040
000004
,
100000
010000
004000
000400
000040
000004
,
330000
020000
001000
000100
000044
000001
,
100000
340000
000300
002000
000033
000002
,
200000
130000
000100
001000
000010
000034

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

C23.83C23 in GAP, Magma, Sage, TeX

C_2^3._{83}C_2^3
% in TeX

G:=Group("C2^3.83C2^3");
// GroupNames label

G:=SmallGroup(64,81);
// by ID

G=gap.SmallGroup(64,81);
# by ID

G:=PCGroup([6,-2,2,2,-2,2,2,48,121,151,362,332,50]);
// Polycyclic

G:=Group<a,b,c,d,e,f|a^2=b^2=c^2=1,d^2=f^2=a,e^2=b*a=a*b,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 C23.83C23 in TeX

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