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G = C23.32D8order 128 = 27

3rd non-split extension by C23 of D8 acting via D8/D4=C2

p-group, metabelian, nilpotent (class 4), monomial

Aliases: C23.32D8, C22.2Q32, C22.2SD32, C2.D82C4, C4.19C4≀C2, (C2×Q16)⋊1C4, (C2×C8).301D4, C22⋊C16.3C2, C4.6(C23⋊C4), (C2×C4).14SD16, C2.5(D82C4), C8.18D4.1C2, (C22×C4).186D4, C2.3(C2.Q32), (C22×C8).98C22, C22.4Q16.33C2, C22.56(D4⋊C4), C2.13(C22.SD16), (C2×C8).18(C2×C4), (C2×C4).218(C22⋊C4), SmallGroup(128,80)

Series: Derived Chief Lower central Upper central Jennings

C1C2×C8 — C23.32D8
C1C2C4C2×C4C22×C4C22×C8C8.18D4 — C23.32D8
C1C2C2×C4C2×C8 — C23.32D8
C1C22C22×C4C22×C8 — C23.32D8
C1C2C2C2C2C4C2×C4C22×C8 — C23.32D8

Generators and relations for C23.32D8
 G = < a,b,c,d,e | a2=b2=c2=1, d8=c, e2=a, ab=ba, ac=ca, dad-1=abc, ae=ea, bc=cb, bd=db, be=eb, cd=dc, ce=ec, ede-1=ad7 >

2C2
2C2
2C22
2C22
2C4
8C4
8C4
8C4
8C4
2C2×C4
2C8
2C2×C4
2C8
4Q8
4C2×C4
4Q8
4C2×C4
4C2×C4
4C2×C4
8C2×C4
8C2×C4
2C4⋊C4
2C2×Q8
2C4⋊C4
2C4⋊C4
4C16
4C22⋊C4
4Q16
4C2×C8
4C4⋊C4
4C4⋊C4
4C22×C4
2Q8⋊C4
2C2×C16
2C2×C4⋊C4
2C22⋊Q8

Character table of C23.32D8

 class 12A2B2C2D2E4A4B4C4D4E4F4G4H4I8A8B8C8D8E8F16A16B16C16D16E16F16G16H
 size 1111222248888161622224444444444
ρ111111111111111111111111111111    trivial
ρ21111111111111-1-1111111-1-1-1-1-1-1-1-1    linear of order 2
ρ3111111111-1-1-1-111111111-1-1-1-1-1-1-1-1    linear of order 2
ρ4111111111-1-1-1-1-1-111111111111111    linear of order 2
ρ51111-1-111-1-ii-ii-11-1-1-1-111i-iii-i-i-ii    linear of order 4
ρ61111-1-111-1-ii-ii1-1-1-1-1-111-ii-i-iiii-i    linear of order 4
ρ71111-1-111-1i-ii-i1-1-1-1-1-111i-iii-i-i-ii    linear of order 4
ρ81111-1-111-1i-ii-i-11-1-1-1-111-ii-i-iiii-i    linear of order 4
ρ92222-2-222-20000002222-2-200000000    orthogonal lifted from D4
ρ10222222222000000-2-2-2-2-2-200000000    orthogonal lifted from D4
ρ11222222-2-2-2000000000000-2-2-222-222    orthogonal lifted from D8
ρ12222222-2-2-2000000000000222-2-22-2-2    orthogonal lifted from D8
ρ132-2-22-220000000002-22-2-22ζ165163ζ165163165163ζ1615169ζ161516916516316151691615169    symplectic lifted from Q32, Schur index 2
ρ142-2-22-220000000002-22-2-22165163165163ζ16516316151691615169ζ165163ζ1615169ζ1615169    symplectic lifted from Q32, Schur index 2
ρ152-2-22-22000000000-22-222-2ζ1615169ζ161516916151691651631651631615169ζ165163ζ165163    symplectic lifted from Q32, Schur index 2
ρ162-2-22-22000000000-22-222-216151691615169ζ1615169ζ165163ζ165163ζ1615169165163165163    symplectic lifted from Q32, Schur index 2
ρ172-22-200-220-1+i1+i1-i-1-i002i-2i-2i2i0000000000    complex lifted from C4≀C2
ρ182-22-200-2201+i-1+i-1-i1-i00-2i2i2i-2i0000000000    complex lifted from C4≀C2
ρ192222-2-2-2-22000000000000-2--2-2--2-2--2-2--2    complex lifted from SD16
ρ202222-2-2-2-22000000000000--2-2--2-2--2-2--2-2    complex lifted from SD16
ρ212-22-200-2201-i-1-i-1+i1+i002i-2i-2i2i0000000000    complex lifted from C4≀C2
ρ222-22-200-220-1-i1-i1+i-1+i00-2i2i2i-2i0000000000    complex lifted from C4≀C2
ρ232-2-222-2000000000-22-22-22ζ1615169ζ16716ζ16716ζ165163ζ16131611ζ1615169ζ165163ζ16131611    complex lifted from SD32
ρ242-2-222-2000000000-22-22-22ζ16716ζ1615169ζ1615169ζ16131611ζ165163ζ16716ζ16131611ζ165163    complex lifted from SD32
ρ252-2-222-20000000002-22-22-2ζ16131611ζ165163ζ165163ζ1615169ζ16716ζ16131611ζ1615169ζ16716    complex lifted from SD32
ρ262-2-222-20000000002-22-22-2ζ165163ζ16131611ζ16131611ζ16716ζ1615169ζ165163ζ16716ζ1615169    complex lifted from SD32
ρ274-44-4004-4000000000000000000000    orthogonal lifted from C23⋊C4
ρ2844-4-400000000000-2-2-2-22-22-20000000000    complex lifted from D82C4
ρ2944-4-4000000000002-22-2-2-2-2-20000000000    complex lifted from D82C4

Smallest permutation representation of C23.32D8
On 32 points
Generators in S32
(2 25)(4 27)(6 29)(8 31)(10 17)(12 19)(14 21)(16 23)
(1 32)(2 17)(3 18)(4 19)(5 20)(6 21)(7 22)(8 23)(9 24)(10 25)(11 26)(12 27)(13 28)(14 29)(15 30)(16 31)
(1 9)(2 10)(3 11)(4 12)(5 13)(6 14)(7 15)(8 16)(17 25)(18 26)(19 27)(20 28)(21 29)(22 30)(23 31)(24 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 13)(2 27 25 4)(3 18)(5 9)(6 23 29 16)(7 30)(8 14 31 21)(10 19 17 12)(11 26)(15 22)(20 24)(28 32)

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

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

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

Matrix representation of C23.32D8 in GL4(𝔽17) generated by

1000
0100
0010
00016
,
16000
01600
00160
00016
,
16000
01600
0010
0001
,
11200
16900
0004
00160
,
161500
0100
00160
0004
G:=sub<GL(4,GF(17))| [1,0,0,0,0,1,0,0,0,0,1,0,0,0,0,16],[16,0,0,0,0,16,0,0,0,0,16,0,0,0,0,16],[16,0,0,0,0,16,0,0,0,0,1,0,0,0,0,1],[11,16,0,0,2,9,0,0,0,0,0,16,0,0,4,0],[16,0,0,0,15,1,0,0,0,0,16,0,0,0,0,4] >;

C23.32D8 in GAP, Magma, Sage, TeX

C_2^3._{32}D_8
% in TeX

G:=Group("C2^3.32D8");
// GroupNames label

G:=SmallGroup(128,80);
// by ID

G=gap.SmallGroup(128,80);
# by ID

G:=PCGroup([7,-2,2,-2,2,-2,2,-2,56,85,456,422,387,520,1690,2804,1411,172,4037,2028,124]);
// Polycyclic

G:=Group<a,b,c,d,e|a^2=b^2=c^2=1,d^8=c,e^2=a,a*b=b*a,a*c=c*a,d*a*d^-1=a*b*c,a*e=e*a,b*c=c*b,b*d=d*b,b*e=e*b,c*d=d*c,c*e=e*c,e*d*e^-1=a*d^7>;
// generators/relations

Export

Subgroup lattice of C23.32D8 in TeX
Character table of C23.32D8 in TeX

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