Copied to
clipboard

G = C23.36C23order 64 = 26

9th non-split extension by C23 of C23 acting via C23/C22=C2

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

Aliases: C23.36C23, C42.61C22, C22.23C24, (C4×D4)⋊8C2, (C4×Q8)⋊7C2, C42(C4⋊D4), (C2×C42)⋊10C2, C4(C4.4D4), C42(C22⋊Q8), C22⋊Q822C2, C4(C42.C2), C4(C422C2), C42⋊C29C2, C4.56(C4○D4), C4.4D417C2, C4⋊D4.10C2, C4⋊C4.70C22, (C2×C4).54C23, C42.C213C2, C422C210C2, (C2×D4).62C22, C22.3(C4○D4), (C2×Q8).54C22, C42(C22.D4), C22.D416C2, C22⋊C4.13C22, (C22×C4).100C22, C2.12(C2×C4○D4), (C2×C4)(C4.4D4), (C2×C4)(C42.C2), (C2×C4)(C422C2), (C2×C4)(C22.D4), SmallGroup(64,210)

Series: Derived Chief Lower central Upper central Jennings

C1C22 — C23.36C23
C1C2C22C2×C4C22×C4C2×C42 — C23.36C23
C1C22 — C23.36C23
C1C2×C4 — C23.36C23
C1C22 — C23.36C23

Generators and relations for C23.36C23
 G = < a,b,c,d,e,f | a2=b2=c2=d2=1, e2=c, f2=b, ab=ba, dad=ac=ca, ae=ea, af=fa, bc=cb, ede-1=bd=db, be=eb, bf=fb, cd=dc, ce=ec, cf=fc, df=fd, ef=fe >

Subgroups: 161 in 117 conjugacy classes, 77 normal (31 characteristic)
C1, C2 [×3], C2 [×4], C4 [×4], C4 [×10], C22, C22 [×2], C22 [×8], C2×C4 [×6], C2×C4 [×6], C2×C4 [×10], D4 [×6], Q8 [×2], C23, C23 [×2], C42 [×4], C42 [×2], C22⋊C4 [×10], C4⋊C4 [×2], C4⋊C4 [×8], C22×C4 [×3], C22×C4 [×2], C2×D4, C2×D4 [×2], C2×Q8, C2×C42, C42⋊C2 [×2], C4×D4, C4×D4 [×2], C4×Q8, C4⋊D4, C22⋊Q8, C22.D4 [×2], C4.4D4, C42.C2, C422C2 [×2], C23.36C23
Quotients: C1, C2 [×15], C22 [×35], C23 [×15], C4○D4 [×6], C24, C2×C4○D4 [×3], C23.36C23

Character table of C23.36C23

 class 12A2B2C2D2E2F2G4A4B4C4D4E4F4G4H4I4J4K4L4M4N4O4P4Q4R4S4T
 size 1111224411112222222222444444
ρ11111111111111111111111111111    trivial
ρ21111-1-1-1-1-1-1-1-1111-1-111-1-111-1-1111    linear of order 2
ρ31111-1-111-1-1-1-1111-1-111-1-11-111-1-1-1    linear of order 2
ρ4111111-1-111111111111111-1-1-1-1-1-1    linear of order 2
ρ51111-1-1-111111-1-111-1-111-1-1-1-111-11    linear of order 2
ρ61111111-1-1-1-1-1-1-11-11-11-11-1-11-11-11    linear of order 2
ρ7111111-11-1-1-1-1-1-11-11-11-11-11-11-11-1    linear of order 2
ρ81111-1-11-11111-1-111-1-111-1-111-1-11-1    linear of order 2
ρ9111111-111111-11-1-1-11-1-1-1-111-1-1-11    linear of order 2
ρ101111-1-11-1-1-1-1-1-11-1111-111-11-11-1-11    linear of order 2
ρ111111-1-1-11-1-1-1-1-11-1111-111-1-11-111-1    linear of order 2
ρ121111111-11111-11-1-1-11-1-1-1-1-1-1111-1    linear of order 2
ρ131111-1-11111111-1-1-11-1-1-111-1-1-1-111    linear of order 2
ρ14111111-1-1-1-1-1-11-1-11-1-1-11-11-111-111    linear of order 2
ρ1511111111-1-1-1-11-1-11-1-1-11-111-1-11-1-1    linear of order 2
ρ161111-1-1-1-111111-1-1-11-1-1-1111111-1-1    linear of order 2
ρ172-22-22-200-2i2i2i-2i02i000-2i0000000000    complex lifted from C4○D4
ρ182-2-2200002i2i-2i-2i2000-2i0002i-2000000    complex lifted from C4○D4
ρ1922-2-20000-22-2200-2i-2i002i2i00000000    complex lifted from C4○D4
ρ202-22-2-2200-2i2i2i-2i0-2i0002i0000000000    complex lifted from C4○D4
ρ2122-2-200002-22-2002i-2i00-2i2i00000000    complex lifted from C4○D4
ρ2222-2-200002-22-200-2i2i002i-2i00000000    complex lifted from C4○D4
ρ2322-2-20000-22-22002i2i00-2i-2i00000000    complex lifted from C4○D4
ρ242-2-220000-2i-2i2i2i20002i000-2i-2000000    complex lifted from C4○D4
ρ252-2-2200002i2i-2i-2i-20002i000-2i2000000    complex lifted from C4○D4
ρ262-22-2-22002i-2i-2i2i02i000-2i0000000000    complex lifted from C4○D4
ρ272-22-22-2002i-2i-2i2i0-2i0002i0000000000    complex lifted from C4○D4
ρ282-2-220000-2i-2i2i2i-2000-2i0002i2000000    complex lifted from C4○D4

Smallest permutation representation of C23.36C23
On 32 points
Generators in S32
(1 23)(2 24)(3 21)(4 22)(5 10)(6 11)(7 12)(8 9)(13 19)(14 20)(15 17)(16 18)(25 29)(26 30)(27 31)(28 32)
(1 27)(2 28)(3 25)(4 26)(5 20)(6 17)(7 18)(8 19)(9 13)(10 14)(11 15)(12 16)(21 29)(22 30)(23 31)(24 32)
(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)
(2 28)(4 26)(5 18)(6 8)(7 20)(10 14)(12 16)(17 19)(21 23)(22 32)(24 30)(29 31)
(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)
(1 15 27 11)(2 16 28 12)(3 13 25 9)(4 14 26 10)(5 22 20 30)(6 23 17 31)(7 24 18 32)(8 21 19 29)

G:=sub<Sym(32)| (1,23)(2,24)(3,21)(4,22)(5,10)(6,11)(7,12)(8,9)(13,19)(14,20)(15,17)(16,18)(25,29)(26,30)(27,31)(28,32), (1,27)(2,28)(3,25)(4,26)(5,20)(6,17)(7,18)(8,19)(9,13)(10,14)(11,15)(12,16)(21,29)(22,30)(23,31)(24,32), (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), (2,28)(4,26)(5,18)(6,8)(7,20)(10,14)(12,16)(17,19)(21,23)(22,32)(24,30)(29,31), (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), (1,15,27,11)(2,16,28,12)(3,13,25,9)(4,14,26,10)(5,22,20,30)(6,23,17,31)(7,24,18,32)(8,21,19,29)>;

G:=Group( (1,23)(2,24)(3,21)(4,22)(5,10)(6,11)(7,12)(8,9)(13,19)(14,20)(15,17)(16,18)(25,29)(26,30)(27,31)(28,32), (1,27)(2,28)(3,25)(4,26)(5,20)(6,17)(7,18)(8,19)(9,13)(10,14)(11,15)(12,16)(21,29)(22,30)(23,31)(24,32), (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), (2,28)(4,26)(5,18)(6,8)(7,20)(10,14)(12,16)(17,19)(21,23)(22,32)(24,30)(29,31), (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), (1,15,27,11)(2,16,28,12)(3,13,25,9)(4,14,26,10)(5,22,20,30)(6,23,17,31)(7,24,18,32)(8,21,19,29) );

G=PermutationGroup([(1,23),(2,24),(3,21),(4,22),(5,10),(6,11),(7,12),(8,9),(13,19),(14,20),(15,17),(16,18),(25,29),(26,30),(27,31),(28,32)], [(1,27),(2,28),(3,25),(4,26),(5,20),(6,17),(7,18),(8,19),(9,13),(10,14),(11,15),(12,16),(21,29),(22,30),(23,31),(24,32)], [(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)], [(2,28),(4,26),(5,18),(6,8),(7,20),(10,14),(12,16),(17,19),(21,23),(22,32),(24,30),(29,31)], [(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)], [(1,15,27,11),(2,16,28,12),(3,13,25,9),(4,14,26,10),(5,22,20,30),(6,23,17,31),(7,24,18,32),(8,21,19,29)])

C23.36C23 is a maximal subgroup of
C22.33C25  C22.44C25  C22.48C25  C22.49C25  C22.50C25  C22.64C25  C22.69C25  C22.70C25  C22.71C25  C22.80C25  C22.82C25  C22.83C25  C22.84C25  C22.94C25  C22.96C25  C22.99C25  C22.101C25  C22.102C25  C22.103C25  C22.104C25  C22.105C25  C22.106C25  C22.107C25  C22.108C25  C23.144C24  C22.110C25  C22.111C25  C23.146C24  C22.113C25  C22.118C25  C22.120C25  C22.132C25  C22.133C25  C22.134C25  C22.135C25  C22.136C25  C22.140C25  C22.141C25  C22.142C25  C22.143C25  C22.144C25  C22.146C25  C22.147C25  C22.148C25  C22.149C25  C22.150C25  C22.151C25  C22.152C25  C22.153C25  C22.154C25  C22.155C25  C22.156C25  C22.157C25
 C42.D2p: C42.373D4  C42.47D4  C42.305D4  C42.52D4  C42.375D4  C42.57D4  C42.58D4  C42.63D4 ...
 (C2×C4p).C23: C42.291C23  C42.292C23  C42.293C23  C42.294C23  C42.307C23  C42.308C23  C42.309C23  C42.310C23 ...
C23.36C23 is a maximal quotient of
C23.165C24  C4242D4  C439C2  C4214Q8  C432C2  C4×C4⋊D4  C4×C22.D4  C4×C422C2  C4×C42.C2  C4215D4  C23.295C24  C23.301C24  C42.34Q8  C24.563C23  C24.254C23  C23.321C24  C24.258C23  C23.327C24  C23.344C24  C24.271C23  C23.348C24  C23.350C24  C24.278C23  C23.359C24  C24.286C23  C23.367C24  C23.368C24  C24.289C23  C24.290C23  C23.374C24  C24.293C23  C23.377C24  C24.295C23  C23.379C24  C23.388C24  C24.577C23  C24.304C23  C23.395C24  C23.405C24  C23.408C24  C23.409C24  C23.410C24  C23.413C24  C23.414C24  C24.309C23  C23.416C24  C23.417C24  C23.418C24  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  C24.327C23  C23.457C24  C24.331C23  C24.332C23  C23.472C24  C23.473C24  C24.338C23  C24.339C23  C24.340C23  C24.341C23  C23.478C24  C23.485C24  C24.345C23  C23.488C24  C24.346C23  C23.490C24  C23.493C24  C23.494C24  C24.347C23  C23.496C24  C24.348C23  C23.548C24  C24.375C23  C23.550C24  C23.551C24  C24.376C23  C23.553C24  C23.554C24  C23.555C24  C4246D4  C4243D4  C23.753C24  C24.598C23  C24.599C23  C4313C2  C4215Q8  C43.18C2  C434C2  C435C2
 C42.D2p: C4×C22⋊Q8  C4×C4.4D4  C42.162D4  C42.163D4  C42.439D4  C4314C2  C42.277D6  C42.93D6 ...
 C4⋊C4.D2p: C24.259C23  C23.353C24  C23.354C24  C24.279C23  C23.360C24  C23.369C24  C23.375C24  C24.301C23 ...

Matrix representation of C23.36C23 in GL4(𝔽5) generated by

0100
1000
0001
0010
,
4000
0400
0010
0001
,
4000
0400
0040
0004
,
1000
0400
0010
0004
,
0200
2000
0030
0003
,
2000
0200
0040
0004
G:=sub<GL(4,GF(5))| [0,1,0,0,1,0,0,0,0,0,0,1,0,0,1,0],[4,0,0,0,0,4,0,0,0,0,1,0,0,0,0,1],[4,0,0,0,0,4,0,0,0,0,4,0,0,0,0,4],[1,0,0,0,0,4,0,0,0,0,1,0,0,0,0,4],[0,2,0,0,2,0,0,0,0,0,3,0,0,0,0,3],[2,0,0,0,0,2,0,0,0,0,4,0,0,0,0,4] >;

C23.36C23 in GAP, Magma, Sage, TeX

C_2^3._{36}C_2^3
% in TeX

G:=Group("C2^3.36C2^3");
// GroupNames label

G:=SmallGroup(64,210);
// by ID

G=gap.SmallGroup(64,210);
# by ID

G:=PCGroup([6,-2,2,2,2,-2,2,217,199,650,69]);
// Polycyclic

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

Export

Character table of C23.36C23 in TeX

׿
×
𝔽