Copied to
clipboard

G = C23.34D4order 64 = 26

5th non-split extension by C23 of D4 acting via D4/C22=C2

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

Aliases: C23.34D4, C24.25C22, C23.57C23, (C22×C4)⋊7C4, (C23×C4).3C2, C23.26(C2×C4), C22.31(C2×D4), C2.C422C2, C2.6(C42⋊C2), C22.16(C4○D4), (C22×C4).88C22, C22.30(C22×C4), C22.15(C22⋊C4), C2.1(C22.D4), (C2×C4).52(C2×C4), C2.6(C2×C22⋊C4), (C2×C22⋊C4).4C2, SmallGroup(64,62)

Series: Derived Chief Lower central Upper central Jennings

C1C22 — C23.34D4
C1C2C22C23C22×C4C23×C4 — C23.34D4
C1C22 — C23.34D4
C1C23 — C23.34D4
C1C23 — C23.34D4

Generators and relations for C23.34D4
 G = < a,b,c,d,e | a2=b2=c2=d4=1, e2=cb=bc, dad-1=eae-1=ab=ba, ac=ca, bd=db, be=eb, cd=dc, ce=ec, ede-1=cd-1 >

Subgroups: 177 in 109 conjugacy classes, 49 normal (7 characteristic)
C1, C2, C2 [×6], C2 [×4], C4 [×8], C22, C22 [×10], C22 [×12], C2×C4 [×4], C2×C4 [×24], C23, C23 [×6], C23 [×4], C22⋊C4 [×4], C22×C4 [×10], C22×C4 [×4], C24, C2.C42 [×4], C2×C22⋊C4 [×2], C23×C4, C23.34D4
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C2×C4 [×6], D4 [×4], C23, C22⋊C4 [×4], C22×C4, C2×D4 [×2], C4○D4 [×4], C2×C22⋊C4, C42⋊C2 [×2], C22.D4 [×4], C23.34D4

Character table of C23.34D4

 class 12A2B2C2D2E2F2G2H2I2J2K4A4B4C4D4E4F4G4H4I4J4K4L4M4N4O4P
 size 1111111122222222222244444444
ρ11111111111111111111111111111    trivial
ρ211111111-1-1-1-111-111-1-1-11-1-1111-1-1    linear of order 2
ρ311111111-1-1-1-111-111-1-1-1-111-1-1-111    linear of order 2
ρ411111111111111111111-1-1-1-1-1-1-1-1    linear of order 2
ρ511111111-1-1-1-1-1-11-1-1111-1-1111-1-11    linear of order 2
ρ6111111111111-1-1-1-1-1-1-1-1-11-111-11-1    linear of order 2
ρ7111111111111-1-1-1-1-1-1-1-11-11-1-11-11    linear of order 2
ρ811111111-1-1-1-1-1-11-1-111111-1-1-111-1    linear of order 2
ρ91-11-11-11-11-11-11-1-1-111-11i-i-ii-i-iii    linear of order 4
ρ101-11-11-11-1-11-111-11-11-11-1iiii-i-i-i-i    linear of order 4
ρ111-11-11-11-11-11-11-1-1-111-11-iii-iii-i-i    linear of order 4
ρ121-11-11-11-1-11-111-11-11-11-1-i-i-i-iiiii    linear of order 4
ρ131-11-11-11-11-11-1-1111-1-11-1-i-iii-iii-i    linear of order 4
ρ141-11-11-11-1-11-11-11-11-11-11-ii-ii-ii-ii    linear of order 4
ρ151-11-11-11-11-11-1-1111-1-11-1ii-i-ii-i-ii    linear of order 4
ρ161-11-11-11-1-11-11-11-11-11-11i-ii-ii-ii-i    linear of order 4
ρ172-2-2-222-22-222-20000000000000000    orthogonal lifted from D4
ρ1822-222-2-2-2-2-2220000000000000000    orthogonal lifted from D4
ρ1922-222-2-2-222-2-20000000000000000    orthogonal lifted from D4
ρ202-2-2-222-222-2-220000000000000000    orthogonal lifted from D4
ρ212-2-22-222-20000002i00-2i-2i2i00000000    complex lifted from C4○D4
ρ2222-2-2-2-2220000002i002i-2i-2i00000000    complex lifted from C4○D4
ρ23222-2-22-2-200002i2i0-2i-2i00000000000    complex lifted from C4○D4
ρ24222-2-22-2-20000-2i-2i02i2i00000000000    complex lifted from C4○D4
ρ2522-2-2-2-222000000-2i00-2i2i2i00000000    complex lifted from C4○D4
ρ262-2-22-222-2000000-2i002i2i-2i00000000    complex lifted from C4○D4
ρ272-222-2-2-220000-2i2i0-2i2i00000000000    complex lifted from C4○D4
ρ282-222-2-2-2200002i-2i02i-2i00000000000    complex lifted from C4○D4

Smallest permutation representation of C23.34D4
On 32 points
Generators in S32
(2 12)(4 10)(5 22)(7 24)(14 26)(16 28)(18 32)(20 30)
(1 11)(2 12)(3 9)(4 10)(5 22)(6 23)(7 24)(8 21)(13 25)(14 26)(15 27)(16 28)(17 31)(18 32)(19 29)(20 30)
(1 25)(2 26)(3 27)(4 28)(5 18)(6 19)(7 20)(8 17)(9 15)(10 16)(11 13)(12 14)(21 31)(22 32)(23 29)(24 30)
(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 18 13 22)(2 8 14 31)(3 20 15 24)(4 6 16 29)(5 11 32 25)(7 9 30 27)(10 23 28 19)(12 21 26 17)

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

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

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

C23.34D4 is a maximal subgroup of
C23.165C24  C23.195C24  C24.547C23  C23.201C24  C23.214C24  C23.215C24  C24.549C23  C23.224C24  C23.225C24  C23.226C24  C23.235C24  C24.212C23  C23.259C24  C24.227C23  C23.304C24  C24.563C23  C24.254C23  C24.567C23  C23.344C24  C24.271C23  C24.278C23  C23.359C24  C24.286C23  C23.368C24  C23.388C24  C24.577C23  C24.309C23  C23.416C24  C23.426C24  C24.315C23  C23.434C24  C23.472C24  C23.473C24  C24.339C23  C24.340C23  C23.500C24  C24.355C23  C23.508C24  C249D4  C23.535C24  C24.379C23  C23.567C24  C24.393C23  C24.394C23  C24.395C23  C23.589C24  C24.405C23  C24.406C23  C23.600C24  C24.407C23  C23.637C24  C24.426C23  C24.427C23  C23.640C24  C23.643C24  C24.430C23  C23.645C24  C24.432C23  C23.649C24  C24.435C23  C23.651C24  C23.652C24  C23.660C24  C24.440C23  C23.664C24  C24.443C23  C23.671C24  C23.715C24  C23.741C24  (C22×C4)⋊7F5
 C24.D2p: C24.46D4  C24.56D4  C24.57D4  C24.59D4  C24.26D4  C24.31D4  C24.90D4  C24.95D4 ...
 (C22×C4).D2p: C23.8C42  C23.15M4(2)  C24.165C23  C24.169C23  C24.174C23  C24.180C23  C25.85C22  C4×C22.D4 ...
C23.34D4 is a maximal quotient of
(C2×C42)⋊C4  C24.C23  C24.6(C2×C4)  (C2×Q8).211D4  (C22×C4)⋊7F5
 C24.D2p: C24.17Q8  C24.52D4  C24.68D4  C23.36D8  C24.157D4  C24.69D4  C24.70D4  C24.56D6 ...
 (C22×C4).D2p: C24.624C23  C24.635C23  C23.32M4(2)  C24.53(C2×C4)  C24.169C23  (C22×C4).275D4  (C22×C4).276D4  C22.58(S3×D4) ...

Matrix representation of C23.34D4 in GL5(𝔽5)

40000
01000
00100
00010
00014
,
10000
01000
00100
00040
00004
,
40000
04000
00400
00040
00004
,
20000
01000
00400
00013
00004
,
20000
00400
01000
00021
00023

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

C23.34D4 in GAP, Magma, Sage, TeX

C_2^3._{34}D_4
% in TeX

G:=Group("C2^3.34D4");
// GroupNames label

G:=SmallGroup(64,62);
// by ID

G=gap.SmallGroup(64,62);
# by ID

G:=PCGroup([6,-2,2,2,-2,2,2,192,121,362,50]);
// Polycyclic

G:=Group<a,b,c,d,e|a^2=b^2=c^2=d^4=1,e^2=c*b=b*c,d*a*d^-1=e*a*e^-1=a*b=b*a,a*c=c*a,b*d=d*b,b*e=e*b,c*d=d*c,c*e=e*c,e*d*e^-1=c*d^-1>;
// generators/relations

Export

Character table of C23.34D4 in TeX

׿
×
𝔽