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G = C23.37C23order 64 = 26

10th non-split extension by C23 of C23 acting via C23/C22=C2

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

Aliases: C23.37C23, C42.36C22, C22.27C24, (C2×C4)⋊5Q8, C42(C4⋊Q8), C4⋊Q820C2, (C4×Q8)⋊8C2, C4.25(C2×Q8), C43(C22⋊Q8), C4.19(C4○D4), C22.3(C2×Q8), C2.6(C22×Q8), C4⋊C4.73C22, (C2×C4).15C23, (C2×C42).20C2, C42(C42.C2), C42.C214C2, C22⋊Q8.10C2, (C2×Q8).57C22, C42⋊C2.11C2, C22⋊C4.15C22, (C22×C4).126C22, (C2×C4)(C4⋊Q8), C2.14(C2×C4○D4), SmallGroup(64,214)

Series: Derived Chief Lower central Upper central Jennings

C1C22 — C23.37C23
C1C2C22C2×C4C42C2×C42 — C23.37C23
C1C22 — C23.37C23
C1C2×C4 — C23.37C23
C1C22 — C23.37C23

Generators and relations for C23.37C23
 G = < a,b,c,d,e,f | a2=b2=c2=1, d2=e2=c, f2=b, dad-1=ab=ba, ac=ca, ae=ea, af=fa, bc=cb, bd=db, be=eb, bf=fb, ede-1=cd=dc, ce=ec, cf=fc, df=fd, ef=fe >

Subgroups: 137 in 111 conjugacy classes, 85 normal (11 characteristic)
C1, C2, C2 [×2], C2 [×2], C4 [×8], C4 [×10], C22, C22 [×2], C22 [×2], C2×C4 [×2], C2×C4 [×16], C2×C4 [×4], Q8 [×8], C23, C42 [×2], C42 [×6], C22⋊C4 [×4], C4⋊C4 [×16], C22×C4, C22×C4 [×2], C2×Q8 [×4], C2×C42, C42⋊C2 [×2], C4×Q8 [×4], C22⋊Q8 [×4], C42.C2 [×2], C4⋊Q8 [×2], C23.37C23
Quotients: C1, C2 [×15], C22 [×35], Q8 [×4], C23 [×15], C2×Q8 [×6], C4○D4 [×4], C24, C22×Q8, C2×C4○D4 [×2], C23.37C23

Character table of C23.37C23

 class 12A2B2C2D2E4A4B4C4D4E4F4G4H4I4J4K4L4M4N4O4P4Q4R4S4T4U4V
 size 1111221111222222222244444444
ρ11111111111111111111111111111    trivial
ρ21111-1-11111-1-1-11111-1-1-1-1-1-1-11111    linear of order 2
ρ3111111-1-1-1-1-11-1-11-11-11-11-11-11-11-1    linear of order 2
ρ41111-1-1-1-1-1-11-11-11-111-11-11-111-11-1    linear of order 2
ρ51111-1-1-1-1-1-11-11-11-111-111-11-1-11-11    linear of order 2
ρ6111111-1-1-1-1-11-1-11-11-11-1-11-11-11-11    linear of order 2
ρ71111-1-11111-1-1-11111-1-1-11111-1-1-1-1    linear of order 2
ρ811111111111111111111-1-1-1-1-1-1-1-1    linear of order 2
ρ91111-1-1-1-1-1-1-11-11-11-1111-111-11-1-11    linear of order 2
ρ10111111-1-1-1-11-111-11-1-1-1-11-1-111-1-11    linear of order 2
ρ111111-1-11111111-1-1-1-1-11-1-1-11111-1-1    linear of order 2
ρ121111111111-1-1-1-1-1-1-11-1111-1-111-1-1    linear of order 2
ρ131111111111-1-1-1-1-1-1-11-11-1-111-1-111    linear of order 2
ρ141111-1-11111111-1-1-1-1-11-111-1-1-1-111    linear of order 2
ρ15111111-1-1-1-11-111-11-1-1-1-1-111-1-111-1    linear of order 2
ρ161111-1-1-1-1-1-1-11-11-11-11111-1-11-111-1    linear of order 2
ρ172-22-2-222-22-20000000-20200000000    symplectic lifted from Q8, Schur index 2
ρ182-22-2-22-22-22000000020-200000000    symplectic lifted from Q8, Schur index 2
ρ192-22-22-2-22-220000000-20200000000    symplectic lifted from Q8, Schur index 2
ρ202-22-22-22-22-2000000020-200000000    symplectic lifted from Q8, Schur index 2
ρ212-2-22002i-2i-2i2i0002i-2-2i200000000000    complex lifted from C4○D4
ρ222-2-2200-2i2i2i-2i0002i2-2i-200000000000    complex lifted from C4○D4
ρ2322-2-2002i2i-2i-2i-22i200000-2i000000000    complex lifted from C4○D4
ρ2422-2-200-2i-2i2i2i-2-2i2000002i000000000    complex lifted from C4○D4
ρ252-2-2200-2i2i2i-2i000-2i-22i200000000000    complex lifted from C4○D4
ρ2622-2-200-2i-2i2i2i22i-200000-2i000000000    complex lifted from C4○D4
ρ2722-2-2002i2i-2i-2i2-2i-2000002i000000000    complex lifted from C4○D4
ρ282-2-22002i-2i-2i2i000-2i22i-200000000000    complex lifted from C4○D4

Smallest permutation representation of C23.37C23
On 32 points
Generators in S32
(2 28)(4 26)(5 20)(7 18)(10 14)(12 16)(22 30)(24 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)
(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 21 3 23)(2 24 4 22)(5 12 7 10)(6 11 8 9)(13 17 15 19)(14 20 16 18)(25 31 27 29)(26 30 28 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)| (2,28)(4,26)(5,20)(7,18)(10,14)(12,16)(22,30)(24,32), (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), (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), (1,21,3,23)(2,24,4,22)(5,12,7,10)(6,11,8,9)(13,17,15,19)(14,20,16,18)(25,31,27,29)(26,30,28,32), (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( (2,28)(4,26)(5,20)(7,18)(10,14)(12,16)(22,30)(24,32), (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), (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), (1,21,3,23)(2,24,4,22)(5,12,7,10)(6,11,8,9)(13,17,15,19)(14,20,16,18)(25,31,27,29)(26,30,28,32), (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([(2,28),(4,26),(5,20),(7,18),(10,14),(12,16),(22,30),(24,32)], [(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)], [(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)], [(1,21,3,23),(2,24,4,22),(5,12,7,10),(6,11,8,9),(13,17,15,19),(14,20,16,18),(25,31,27,29),(26,30,28,32)], [(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)])

C23.37C23 is a maximal subgroup of
C22.33C25  C22.44C25  C22.47C25  C22.50C25  C22.64C25  Q8×C4○D4  C22.71C25  C22.84C25  C22.90C25  C22.91C25  C22.92C25  C22.93C25  C22.95C25  C22.96C25  C22.97C25  C22.98C25  C22.99C25  C22.100C25  C22.104C25  C22.107C25  C23.144C24  C23.146C24  C22.120C25  C22.136C25  C22.137C25  C22.139C25  C22.143C25  C22.144C25  C22.145C25  C22.146C25  C22.150C25  C22.152C25  C22.153C25  C22.154C25
 C42.D2p: C42.46D4  C42.401D4  C42.316D4  C42.54D4  C42.404D4  C42.56D4  C42.60D4  C42.62D4 ...
 (C2×C4p).C23: C42.286C23  C42.287C23  M4(2)⋊9Q8  C42.696C23  C42.304C23  C42.305C23  (Q8×Dic3)⋊C2  (Q8×Dic5)⋊C2 ...
C23.37C23 is a maximal quotient of
C23.167C24  C4214Q8  C23.178C24  C4×C42.C2  C42.34Q8  C23.323C24  C24.567C23  C23.346C24  C23.397C24  C23.407C24  C23.411C24  C23.420C24  C23.422C24  C23.449C24  C24.584C23  C42.36Q8  C24.338C23  C23.485C24  C23.486C24  C24.345C23  C23.488C24  C24.346C23  C23.490C24  C428Q8  C42.38Q8  C24.355C23  C23.508C24  C429Q8  C24.379C23  C4211Q8  C23.567C24  C24.599C23  C4215Q8  C43.18C2
 C42.D2p: C4×C22⋊Q8  C4×C4⋊Q8  C42.162D4  C425Q8  C42.439D4  C42.440D4  C43.15C2  C4218Q8 ...
 C4⋊C4.D2p: C23.329C24  C24.267C23  C24.268C23  C23.351C24  C23.362C24  C24.285C23  C23.392C24  C426Q8 ...

Matrix representation of C23.37C23 in GL4(𝔽5) generated by

1000
0400
0010
0004
,
4000
0400
0040
0004
,
1000
0100
0040
0004
,
0100
1000
0001
0040
,
4000
0400
0020
0003
,
3000
0300
0020
0002
G:=sub<GL(4,GF(5))| [1,0,0,0,0,4,0,0,0,0,1,0,0,0,0,4],[4,0,0,0,0,4,0,0,0,0,4,0,0,0,0,4],[1,0,0,0,0,1,0,0,0,0,4,0,0,0,0,4],[0,1,0,0,1,0,0,0,0,0,0,4,0,0,1,0],[4,0,0,0,0,4,0,0,0,0,2,0,0,0,0,3],[3,0,0,0,0,3,0,0,0,0,2,0,0,0,0,2] >;

C23.37C23 in GAP, Magma, Sage, TeX

C_2^3._{37}C_2^3
% in TeX

G:=Group("C2^3.37C2^3");
// GroupNames label

G:=SmallGroup(64,214);
// by ID

G=gap.SmallGroup(64,214);
# by ID

G:=PCGroup([6,-2,2,2,2,-2,2,192,217,650,158,69]);
// Polycyclic

G:=Group<a,b,c,d,e,f|a^2=b^2=c^2=1,d^2=e^2=c,f^2=b,d*a*d^-1=a*b=b*a,a*c=c*a,a*e=e*a,a*f=f*a,b*c=c*b,b*d=d*b,b*e=e*b,b*f=f*b,e*d*e^-1=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 C23.37C23 in TeX

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