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G = C23.19D4order 64 = 26

12nd non-split extension by C23 of D4 acting via D4/C2=C22

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

Aliases: C23.19D4, C2.D87C2, C22⋊C87C2, C4.Q810C2, (C2×C4).106D4, C4⋊D4.7C2, D4⋊C413C2, C42⋊C25C2, C4.31(C4○D4), C2.15(C4○D8), C4⋊C4.65C22, (C2×C8).38C22, C2.18(C8⋊C22), (C2×C4).107C23, (C2×D4).23C22, C22.103(C2×D4), (C22×C4).53C22, C2.13(C22.D4), SmallGroup(64,163)

Series: Derived Chief Lower central Upper central Jennings

C1C2×C4 — C23.19D4
C1C2C4C2×C4C4⋊C4C42⋊C2 — C23.19D4
C1C2C2×C4 — C23.19D4
C1C22C22×C4 — C23.19D4
C1C2C2C2×C4 — C23.19D4

Generators and relations for C23.19D4
 G = < a,b,c,d,e | a2=b2=c2=e2=1, d4=c, dad-1=ab=ba, ac=ca, eae=abc, bc=cb, bd=db, be=eb, cd=dc, ce=ec, ede=bcd3 >

4C2
8C2
2C22
2C22
2C22
2C4
4C4
4C4
4C22
4C22
4C22
4C4
2C2×C4
2C2×C4
2C23
2C2×C4
2C2×C4
2C8
2C8
2D4
2C2×C4
2D4
4D4
4D4
2C2×D4
2C42
2C22⋊C4
2C22⋊C4

Character table of C23.19D4

 class 12A2B2C2D2E4A4B4C4D4E4F4G4H4I8A8B8C8D
 size 1111482222444484444
ρ11111111111111111111    trivial
ρ211111-11111-1-1-1-1-11111    linear of order 2
ρ31111-1-111-1-1-111-11-1-111    linear of order 2
ρ41111-1111-1-11-1-11-1-1-111    linear of order 2
ρ51111111111-1-1-1-11-1-1-1-1    linear of order 2
ρ611111-111111111-1-1-1-1-1    linear of order 2
ρ71111-1-111-1-11-1-11111-1-1    linear of order 2
ρ81111-1111-1-1-111-1-111-1-1    linear of order 2
ρ9222220-2-2-2-2000000000    orthogonal lifted from D4
ρ102222-20-2-222000000000    orthogonal lifted from D4
ρ112-22-200-220002i-2i000000    complex lifted from C4○D4
ρ122-22-200-22000-2i2i000000    complex lifted from C4○D4
ρ132-22-2002-2002i00-2i00000    complex lifted from C4○D4
ρ142-22-2002-200-2i002i00000    complex lifted from C4○D4
ρ152-2-2200002i-2i00000-22-2--2    complex lifted from C4○D8
ρ162-2-220000-2i2i00000-22--2-2    complex lifted from C4○D8
ρ172-2-2200002i-2i000002-2--2-2    complex lifted from C4○D8
ρ182-2-220000-2i2i000002-2-2--2    complex lifted from C4○D8
ρ1944-4-4000000000000000    orthogonal lifted from C8⋊C22

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

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

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

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

C23.19D4 is a maximal subgroup of
C24.115D4  C24.116D4  C24.117D4  C42.229D4  C42.233D4  C24.121D4  C24.124D4  C24.127D4  C24.130D4  C42.287D4  C42.292D4
 (C2×D4).D2p: (C2×D4).301D4  (C2×D4).303D4  (C2×D4).304D4  C4.2+ 1+4  C4.142+ 1+4  C4.152+ 1+4  C4.162+ 1+4  C42.461C23 ...
 C4⋊C4.D2p: C42.352C23  C42.355C23  C42.356C23  C42.360C23  C42.423C23  C42.425C23  C42.426C23  C4.Q8⋊S3 ...
 C2p.(C4○D8): C42.384D4  C42.450D4  C42.280D4  C42.285D4  C23.18D12  C23.13D20  C23.13D28 ...
C23.19D4 is a maximal quotient of
C24.69D4  C24.74D4  C4.Q810C4  C2.D85C4  C428C4⋊C2  C4⋊C4.Q8
 C4⋊C4.D2p: C24.71D4  D4⋊C4⋊C4  C4.67(C4×D4)  C24.83D4  C24.84D4  (C2×C8).24Q8  D6⋊C811C2  C241C4⋊C2 ...
 (C2×C8).D2p: C24.88D4  C24.89D4  (C2×C8).168D4  C23.18D12  C23.13D20  C23.13D28 ...

Matrix representation of C23.19D4 in GL4(𝔽17) generated by

11500
01600
00138
00134
,
16000
01600
0010
0001
,
1000
0100
00160
00016
,
13000
13400
00011
00311
,
1000
11600
0010
00116
G:=sub<GL(4,GF(17))| [1,0,0,0,15,16,0,0,0,0,13,13,0,0,8,4],[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],[13,13,0,0,0,4,0,0,0,0,0,3,0,0,11,11],[1,1,0,0,0,16,0,0,0,0,1,1,0,0,0,16] >;

C23.19D4 in GAP, Magma, Sage, TeX

C_2^3._{19}D_4
% in TeX

G:=Group("C2^3.19D4");
// GroupNames label

G:=SmallGroup(64,163);
// by ID

G=gap.SmallGroup(64,163);
# by ID

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

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

Export

Subgroup lattice of C23.19D4 in TeX
Character table of C23.19D4 in TeX

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