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G = C23.67C23order 64 = 26

17th central extension by C23 of C23

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

Aliases: C23.67C23, (C2×C4)⋊3Q8, (C2×Q8)⋊3C4, C2.5(C4×Q8), (C2×C4).68D4, C2.3(C4⋊Q8), C4.7(C22⋊C4), (C2×C42).11C2, C22.40(C2×D4), C2.5(C22⋊Q8), (C22×Q8).1C2, C22.15(C2×Q8), C2.4(C4.4D4), (C22×C4).6C22, C22.25(C4○D4), C2.C42.7C2, C22.40(C22×C4), (C2×C4⋊C4).8C2, (C2×C4).19(C2×C4), C2.9(C2×C22⋊C4), SmallGroup(64,72)

Series: Derived Chief Lower central Upper central Jennings

C1C22 — C23.67C23
C1C2C22C23C22×C4C2×C42 — C23.67C23
C1C22 — C23.67C23
C1C23 — C23.67C23
C1C23 — C23.67C23

Generators and relations for C23.67C23
 G = < a,b,c,d,e,f | a2=b2=c2=1, d2=c, e2=abc, 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: 137 in 93 conjugacy classes, 53 normal (13 characteristic)
C1, C2 [×3], C2 [×4], C4 [×4], C4 [×10], C22 [×3], C22 [×4], C2×C4 [×14], C2×C4 [×14], Q8 [×8], C23, C42 [×2], C4⋊C4 [×2], C22×C4, C22×C4 [×6], C2×Q8 [×4], C2×Q8 [×4], C2.C42 [×4], C2×C42, C2×C4⋊C4, C22×Q8, C23.67C23
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C2×C4 [×6], D4 [×4], Q8 [×4], C23, C22⋊C4 [×4], C22×C4, C2×D4 [×2], C2×Q8 [×2], C4○D4 [×2], C2×C22⋊C4, C4×Q8 [×2], C22⋊Q8 [×2], C4.4D4, C4⋊Q8, C23.67C23

Character table of C23.67C23

 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
ρ17222-2-22-2-2200-2-2200000000000000    orthogonal lifted from D4
ρ182-222-2-2-222002-2-200000000000000    orthogonal lifted from D4
ρ192-222-2-2-22-200-22200000000000000    orthogonal lifted from D4
ρ20222-2-22-2-2-20022-200000000000000    orthogonal lifted from D4
ρ2122-222-2-2-2020000-2-2200000000000    symplectic lifted from Q8, Schur index 2
ρ222-2-22-222-200-200000022-200000000    symplectic lifted from Q8, Schur index 2
ρ2322-222-2-2-20-2000022-200000000000    symplectic lifted from Q8, Schur index 2
ρ242-2-22-222-2002000000-2-2200000000    symplectic lifted from Q8, Schur index 2
ρ2522-2-2-2-222002i0000002i-2i-2i00000000    complex lifted from C4○D4
ρ262-2-2-222-2202i00002i-2i-2i00000000000    complex lifted from C4○D4
ρ2722-2-2-2-22200-2i000000-2i2i2i00000000    complex lifted from C4○D4
ρ282-2-2-222-220-2i0000-2i2i2i00000000000    complex lifted from C4○D4

Smallest permutation representation of C23.67C23
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 29 21 47)(2 58 22 20)(3 31 23 45)(4 60 24 18)(5 42 64 28)(6 15 61 53)(7 44 62 26)(8 13 63 55)(9 59 51 17)(10 32 52 46)(11 57 49 19)(12 30 50 48)(14 34 56 38)(16 36 54 40)(25 39 43 35)(27 37 41 33)
(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), (1,29,21,47)(2,58,22,20)(3,31,23,45)(4,60,24,18)(5,42,64,28)(6,15,61,53)(7,44,62,26)(8,13,63,55)(9,59,51,17)(10,32,52,46)(11,57,49,19)(12,30,50,48)(14,34,56,38)(16,36,54,40)(25,39,43,35)(27,37,41,33), (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), (1,29,21,47)(2,58,22,20)(3,31,23,45)(4,60,24,18)(5,42,64,28)(6,15,61,53)(7,44,62,26)(8,13,63,55)(9,59,51,17)(10,32,52,46)(11,57,49,19)(12,30,50,48)(14,34,56,38)(16,36,54,40)(25,39,43,35)(27,37,41,33), (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)], [(1,29,21,47),(2,58,22,20),(3,31,23,45),(4,60,24,18),(5,42,64,28),(6,15,61,53),(7,44,62,26),(8,13,63,55),(9,59,51,17),(10,32,52,46),(11,57,49,19),(12,30,50,48),(14,34,56,38),(16,36,54,40),(25,39,43,35),(27,37,41,33)], [(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)])

C23.67C23 is a maximal subgroup of
(C2×Q8)⋊C8  (C2×C42).C4  (C2×Q8).Q8  2- 1+42C4  C4.10D42C4  (C2×SD16)⋊14C4  (C2×C4)⋊9Q16  C8.C22⋊C4  C4.68(C4×D4)  C2.(C4×Q16)  C2.(C88D4)  C2.(C8⋊D4)  (C2×C4)⋊9SD16  (C2×C4)⋊6Q16  (C2×Q16)⋊10C4  C8⋊(C22⋊C4)  M4(2)⋊4D4  (C2×C8)⋊20D4  (C2×C8).41D4  (C2×C4)⋊2Q16  (C2×Q8)⋊Q8  C4⋊C4.85D4  (C2×Q8)⋊2Q8  M4(2).7D4  (C2×C4)⋊5SD16  (C2×C4)⋊3Q16  (C2×Q8).8Q8  (C2×C8).52D4  (C2×C4).19Q16  (C2×Q8).109D4  (C2×C8).60D4  (C2×C8).170D4  (C2×C8).171D4  C23.179C24  C4×C22⋊Q8  C4×C4.4D4  C4×C4⋊Q8  C23.192C24  C24.542C23  C23.202C24  C42.159D4  C42.160D4  C42.161D4  C23.211C24  C23.214C24  C24.205C23  C24.549C23  Q8×C22⋊C4  C23.238C24  C24.558C23  C23.244C24  C23.247C24  C24.220C23  C23.250C24  C24.221C23  C24.227C23  C23.261C24  C23.263C24  C24.244C23  C23.309C24  C23.315C24  C24.252C23  C23.321C24  C23.323C24  C23.327C24  C23.329C24  C24.263C23  C24.264C23  C23.334C24  C24.565C23  C24.567C23  C24.267C23  C23.346C24  C24.271C23  C23.348C24  C23.350C24  C23.351C24  C23.353C24  C23.359C24  C24.285C23  C23.374C24  C23.377C24  C23.388C24  C24.301C23  C23.392C24  C24.308C23  C23.402C24  C24.579C23  C23.411C24  C23.414C24  C23.420C24  C24.311C23  C24.313C23  C24.315C23  C23.432C24  C4221D4  C42.168D4  C42.169D4  C42.170D4  C23.449C24  C426Q8  C23.456C24  C23.457C24  C24.332C23  C42.174D4  C42.176D4  C42.177D4  C23.472C24  C24.338C23  C42.179D4  C42.180D4  C23.486C24  C23.488C24  C24.346C23  C42.182D4  C428Q8  C24.355C23  C429Q8  C4228D4  C42.186D4  C23.525C24  C42.193D4  C42.195D4  C23.545C24  C23.550C24  C23.553C24  C24.379C23  C4211Q8  C24.394C23  C23.589C24  C23.590C24  C23.592C24  C24.405C23  C23.600C24  C23.602C24  C24.408C23  C24.412C23  C23.612C24  C23.613C24  C23.615C24  C23.616C24  C24.420C23  C24.421C23  C23.630C24  C23.631C24  C23.632C24  C23.633C24  C23.634C24  C23.645C24  C23.651C24  C23.654C24  C23.655C24  C23.658C24  C23.659C24  C23.662C24  C23.663C24  C23.664C24  C23.674C24  C23.675C24  C23.688C24  C23.689C24  C23.698C24  C23.699C24  C24.456C23  C23.705C24  C23.706C24  C23.708C24  C23.709C24  C23.711C24  C23.731C24  C23.732C24  C23.733C24  C4246D4  C24.599C23  C42.440D4  C4314C2  C4218Q8  C4215Q8
 C2p.(C4×Q8): C4214Q8  C424Q8  C23.237C24  C23.251C24  (C2×C12)⋊Q8  (C2×Dic6)⋊7C4  C4.(D6⋊C4)  (C6×Q8)⋊7C4 ...
C23.67C23 is a maximal quotient of
C24.625C23  C24.626C23  C24.631C23  C24.635C23  C24.636C23  C24.176C23  M4(2)⋊8Q8  C42.128D4
 (C2×C4p)⋊Q8: C42.327D4  C42.120D4  C42.436D4  C42.125D4  (C2×C12)⋊Q8  (C2×Dic6)⋊7C4  (C2×C20)⋊Q8  (C2×C20)⋊10Q8 ...
 C2p.(C4×Q8): M4(2)⋊7Q8  C42.121D4  C42.122D4  C42.123D4  C42.437D4  C42.124D4  C4216Q8  C42⋊Q8 ...

Matrix representation of C23.67C23 in GL6(𝔽5)

400000
040000
004000
000400
000040
000004
,
100000
010000
001000
000100
000040
000004
,
100000
010000
004000
000400
000010
000001
,
420000
010000
000400
001000
000001
000010
,
130000
140000
000100
001000
000010
000004
,
100000
010000
004000
000400
000030
000002

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

C23.67C23 in GAP, Magma, Sage, TeX

C_2^3._{67}C_2^3
% in TeX

G:=Group("C2^3.67C2^3");
// GroupNames label

G:=SmallGroup(64,72);
// by ID

G=gap.SmallGroup(64,72);
# by ID

G:=PCGroup([6,-2,2,2,-2,2,2,192,121,343,362,86]);
// Polycyclic

G:=Group<a,b,c,d,e,f|a^2=b^2=c^2=1,d^2=c,e^2=a*b*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 C23.67C23 in TeX

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