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G = C23.20D4order 64 = 26

13rd non-split extension by C23 of D4 acting via D4/C2=C22

p-group, metabelian, nilpotent (class 3), monomial

Aliases: C23.20D4, C2.D89C2, C4.Q812C2, Q8⋊C48C2, (C2×C4).107D4, C22⋊C8.5C2, C22⋊Q8.7C2, C2.16(C4○D8), C4.34(C4○D4), C4⋊C4.68C22, (C2×C8).40C22, (C2×C4).110C23, C42⋊C2.9C2, C22.106(C2×D4), (C2×Q8).19C22, C2.19(C8.C22), (C22×C4).56C22, C2.16(C22.D4), SmallGroup(64,166)

Series: Derived Chief Lower central Upper central Jennings

C1C2×C4 — C23.20D4
C1C2C4C2×C4C4⋊C4C42⋊C2 — C23.20D4
C1C2C2×C4 — C23.20D4
C1C22C22×C4 — C23.20D4
C1C2C2C2×C4 — C23.20D4

Generators and relations for C23.20D4
 G = < a,b,c,d,e | a2=b2=c2=1, d4=e2=c, dad-1=ab=ba, ac=ca, eae-1=abc, bc=cb, bd=db, be=eb, cd=dc, ce=ec, ede-1=bd3 >

4C2
2C22
2C22
2C4
2C22
4C4
4C4
4C4
4C4
2C8
2C2×C4
2C2×C4
2C2×C4
2Q8
2Q8
2C8
2C2×C4
2C2×C4
2C2×C4
2C22⋊C4
2C22⋊C4
2C42
2C4⋊C4

Character table of C23.20D4

 class 12A2B2C2D4A4B4C4D4E4F4G4H4I4J8A8B8C8D
 size 1111422224444884444
ρ11111111111111111111    trivial
ρ21111-111-1-1-1-1111-11-1-11    linear of order 2
ρ3111111111-1-1-1-111-1-1-1-1    linear of order 2
ρ41111-111-1-111-1-11-1-111-1    linear of order 2
ρ51111-111-1-1-1-111-11-111-1    linear of order 2
ρ61111111111111-1-1-1-1-1-1    linear of order 2
ρ71111-111-1-111-1-1-111-1-11    linear of order 2
ρ8111111111-1-1-1-1-1-11111    linear of order 2
ρ92222-2-2-2220000000000    orthogonal lifted from D4
ρ1022222-2-2-2-20000000000    orthogonal lifted from D4
ρ112-22-20-22002i-2i00000000    complex lifted from C4○D4
ρ122-22-202-200002i-2i000000    complex lifted from C4○D4
ρ132-22-202-20000-2i2i000000    complex lifted from C4○D4
ρ142-22-20-2200-2i2i00000000    complex lifted from C4○D4
ρ152-2-220002i-2i000000-2-22--2    complex lifted from C4○D8
ρ162-2-22000-2i2i000000--2-22-2    complex lifted from C4○D8
ρ172-2-220002i-2i000000--22-2-2    complex lifted from C4○D8
ρ182-2-22000-2i2i000000-22-2--2    complex lifted from C4○D8
ρ1944-4-4000000000000000    symplectic lifted from C8.C22, Schur index 2

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

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

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

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

C23.20D4 is a maximal subgroup of
C24.115D4  C24.116D4  C24.118D4  (C2×D4).302D4  (C2×D4).303D4  (C2×D4).304D4  C42.229D4  C42.234D4  C24.123D4  C24.124D4  C24.129D4  C24.130D4  C4.2+ 1+4  C4.152+ 1+4  C4.162+ 1+4  C4.172+ 1+4  C42.289D4  C42.292D4
 C4⋊C4.D2p: C42.352C23  C42.355C23  C42.357C23  C42.361C23  C42.424C23  C42.425C23  C42.426C23  C42.465C23 ...
 C2p.(C4○D8): C42.384D4  C42.451D4  C42.283D4  C42.285D4  C23.15D12  C23.10D20  C23.10D28 ...
C23.20D4 is a maximal quotient of
C24.69D4  C2.D84C4  C4.Q89C4  (C2×Q8).109D4  C24.89D4  (C2×C8).171D4  C4⋊C4.Q8
 C23.D4p: C23.12D8  C23.15D12  C23.10D20  C23.10D28 ...
 C4⋊C4.D2p: C24.71D4  C24.73D4  C4.68(C4×D4)  C2.(C4×Q16)  C24.85D4  C24.86D4  (C2×C8).24Q8  D6⋊C8.C2 ...

Matrix representation of C23.20D4 in GL4(𝔽17) generated by

1000
01600
0010
00016
,
16000
01600
0010
0001
,
1000
0100
00160
00016
,
0400
4000
0020
0008
,
0100
1000
0008
0020
G:=sub<GL(4,GF(17))| [1,0,0,0,0,16,0,0,0,0,1,0,0,0,0,16],[16,0,0,0,0,16,0,0,0,0,1,0,0,0,0,1],[1,0,0,0,0,1,0,0,0,0,16,0,0,0,0,16],[0,4,0,0,4,0,0,0,0,0,2,0,0,0,0,8],[0,1,0,0,1,0,0,0,0,0,0,2,0,0,8,0] >;

C23.20D4 in GAP, Magma, Sage, TeX

C_2^3._{20}D_4
% in TeX

G:=Group("C2^3.20D4");
// GroupNames label

G:=SmallGroup(64,166);
// by ID

G=gap.SmallGroup(64,166);
# by ID

G:=PCGroup([6,-2,2,2,-2,2,-2,192,121,199,362,50,1444,376,88]);
// Polycyclic

G:=Group<a,b,c,d,e|a^2=b^2=c^2=1,d^4=e^2=c,d*a*d^-1=a*b=b*a,a*c=c*a,e*a*e^-1=a*b*c,b*c=c*b,b*d=d*b,b*e=e*b,c*d=d*c,c*e=e*c,e*d*e^-1=b*d^3>;
// generators/relations

Export

Subgroup lattice of C23.20D4 in TeX
Character table of C23.20D4 in TeX

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