Copied to
clipboard

G = C23.78C23order 64 = 26

4th central stem extension by C23 of C23

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

Aliases: C23.78C23, (C2×C4)⋊1Q8, (C2×C4).15D4, C2.4(C4⋊Q8), C2.7C22≀C2, C22.71(C2×D4), C2.7(C22⋊Q8), (C22×Q8).2C2, C22.21(C2×Q8), C22.38(C4○D4), C2.C42.8C2, (C22×C4).27C22, (C2×C4⋊C4).9C2, SmallGroup(64,76)

Series: Derived Chief Lower central Upper central Jennings

C1C23 — C23.78C23
C1C2C22C23C22×C4C2×C4⋊C4 — C23.78C23
C1C23 — C23.78C23
C1C23 — C23.78C23
C1C23 — C23.78C23

Generators and relations for C23.78C23
 G = < a,b,c,d,e,f | a2=b2=c2=1, d2=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: 141 in 91 conjugacy classes, 43 normal (7 characteristic)
C1, C2, C2 [×6], C4 [×13], C22, C22 [×6], C2×C4 [×12], C2×C4 [×15], Q8 [×8], C23, C4⋊C4 [×6], C22×C4, C22×C4 [×6], C2×Q8 [×6], C2.C42 [×3], C2×C4⋊C4 [×3], C22×Q8, C23.78C23
Quotients: C1, C2 [×7], C22 [×7], D4 [×6], Q8 [×6], C23, C2×D4 [×3], C2×Q8 [×3], C4○D4, C22≀C2, C22⋊Q8 [×3], C4⋊Q8 [×3], C23.78C23

Character table of C23.78C23

 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
ρ922-2-222-2-20-2000000200000    orthogonal lifted from D4
ρ1022-22-2-2-22000002000000-20    orthogonal lifted from D4
ρ1122-22-2-2-2200000-200000020    orthogonal lifted from D4
ρ1222-2-222-2-202000000-200000    orthogonal lifted from D4
ρ13222-2-2-22-20002000000-2000    orthogonal lifted from D4
ρ14222-2-2-22-2000-20000002000    orthogonal lifted from D4
ρ152-2-22-222-20000002000000-2    symplectic lifted from Q8, Schur index 2
ρ162-2-2-22-222002000000-20000    symplectic lifted from Q8, Schur index 2
ρ172-2-22-222-2000000-20000002    symplectic lifted from Q8, Schur index 2
ρ182-22-2-22-2200002000000-200    symplectic lifted from Q8, Schur index 2
ρ192-22-2-22-220000-2000000200    symplectic lifted from Q8, Schur index 2
ρ202-2-2-22-22200-200000020000    symplectic lifted from Q8, Schur index 2
ρ212-2222-2-2-2-2i0000002i000000    complex lifted from C4○D4
ρ222-2222-2-2-22i000000-2i000000    complex lifted from C4○D4

Smallest permutation representation of C23.78C23
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 63 3 61)(2 62 4 64)(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)(45 55 47 53)(46 54 48 56)
(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), (1,63,3,61)(2,62,4,64)(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)(45,55,47,53)(46,54,48,56), (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), (1,63,3,61)(2,62,4,64)(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)(45,55,47,53)(46,54,48,56), (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)], [(1,63,3,61),(2,62,4,64),(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),(45,55,47,53),(46,54,48,56)], [(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)])

C23.78C23 is a maximal subgroup of
C2.7C2≀C4  C4⋊C4.6D4  Q8⋊D4⋊C2  (C2×C4)⋊Q16  C42.162D4  C425Q8  C24.243C23  C24.262C23  C24.567C23  C24.568C23  C23.349C24  C23.351C24  C24.282C23  C23.362C24  C24.285C23  C23.369C24  C23.374C24  C23.420C24  C24.311C23  C23.449C24  C426Q8  C427Q8  C23.455C24  C23.456C24  C24.583C23  C42.176D4  C24.338C23  C42.179D4  C23.483C24  C42.181D4  C23.486C24  C23.488C24  C42.183D4  C42.184D4  C428Q8  C4226D4  C429Q8  C23.514C24  C23.527C24  C42.187D4  C42.189D4  C42.191D4  C42.192D4  C24.374C23  C23.550C24  C23.559C24  C4210Q8  C4232D4  C24.378C23  C24.379C23  C4211Q8  C23.572C24  C23.574C24  C24.385C23  C23.580C24  C23.583C24  C24.393C23  C23.589C24  C23.590C24  C23.592C24  C24.405C23  C23.600C24  C24.408C23  C23.613C24  C23.615C24  C23.617C24  C23.620C24  C24.418C23  C24.421C23  C23.634C24  C23.637C24  C24.428C23  C23.645C24  C23.655C24  C23.658C24  C23.659C24  C23.662C24  C23.663C24  C23.674C24  C23.675C24  C24.450C23  C23.685C24  C23.688C24  C23.689C24  C23.692C24  C23.699C24  C23.705C24  C23.706C24  C23.711C24  C23.714C24  C24.462C23  C42.199D4  C42.200D4  C23.730C24  C23.731C24  C23.732C24  C23.733C24  C23.735C24  C23.738C24  C23.741C24  C4212Q8  C4213Q8  (C22×C4).A4
 C2p.C22≀C2: C23.288C24  C23.309C24  C23.334C24  C24.361C23  (C2×C4)⋊Dic6  (C2×Dic3)⋊Q8  C22.52(S3×Q8)  (C2×Dic5)⋊Q8 ...
C23.78C23 is a maximal quotient of
C24.631C23  C24.634C23  C24.636C23  C24.182C23
 (C2×C4p)⋊Q8: C4⋊C4⋊Q8  (C2×C8)⋊Q8  (C2×C4)⋊Dic6  (C2×Dic3)⋊Q8  (C2×Dic5)⋊Q8  (C2×C4)⋊Dic10  (C2×Dic7)⋊Q8  (C2×C4)⋊Dic14 ...
 C2p.(C4⋊Q8): C2.(C8⋊Q8)  M4(2)⋊Q8  C423Q8  C22.52(S3×Q8)  C10.C22≀C2  C14.C22≀C2 ...

Matrix representation of C23.78C23 in GL6(𝔽5)

400000
040000
001000
000100
000040
000004
,
100000
010000
004000
000400
000010
000001
,
400000
040000
004000
000400
000010
000001
,
300000
020000
000400
004000
000020
000013
,
010000
400000
000100
001000
000044
000021
,
200000
030000
001000
000400
000020
000002

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

C23.78C23 in GAP, Magma, Sage, TeX

C_2^3._{78}C_2^3
% in TeX

G:=Group("C2^3.78C2^3");
// GroupNames label

G:=SmallGroup(64,76);
// by ID

G=gap.SmallGroup(64,76);
# by ID

G:=PCGroup([6,-2,2,2,-2,2,2,48,121,55,362,332,50]);
// Polycyclic

G:=Group<a,b,c,d,e,f|a^2=b^2=c^2=1,d^2=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 C23.78C23 in TeX

׿
×
𝔽