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G = C32⋊C22order 128 = 27

The semidirect product of C32 and C22 acting faithfully

p-group, metabelian, nilpotent (class 5), monomial

Aliases: C32⋊C22, C8.8D8, D322C2, C16.3D4, SD641C2, C4.14D16, D162C22, M6(2)⋊1C2, Q322C22, C22.5D16, C16.10C23, C4○D162C2, C4.17(C2×D8), C8.49(C2×D4), (C2×C4).51D8, (C2×D16)⋊11C2, C2.16(C2×D16), (C2×C8).141D4, (C2×C16).32C22, 2-Sylow(PGammaL(2,289)), SmallGroup(128,995)

Series: Derived Chief Lower central Upper central Jennings

C1C16 — C32⋊C22
C1C2C4C8C16C2×C16C2×D16 — C32⋊C22
C1C2C4C8C16 — C32⋊C22
C1C2C2×C4C2×C8C2×C16 — C32⋊C22
C1C2C2C2C2C2C2C2C2C4C4C4C4C8C8C16 — C32⋊C22

Generators and relations for C32⋊C22
 G = < a,b,c | a32=b2=c2=1, bab=a15, cac=a17, bc=cb >

2C2
16C2
16C2
16C2
8C4
8C22
8C22
8C22
16C22
16C22
4D4
4D4
4D4
4Q8
8D4
8D4
8C2×C4
8C23
2D8
2D8
2Q16
2D8
4D8
4SD16
4C2×D4
4C4○D4
2C4○D8
2SD32
2C2×D8
2D16

Character table of C32⋊C22

 class 12A2B2C2D2E4A4B4C8A8B8C16A16B16C16D16E16F32A32B32C32D32E32F32G32H
 size 112161616221622422224444444444
ρ111111111111111111111111111    trivial
ρ211-1-1111-1-111-11111-1-1-11-11-11-11    linear of order 2
ρ311-1-1-111-1111-11111-1-11-11-11-11-1    linear of order 2
ρ41111-1111-1111111111-1-1-1-1-1-1-1-1    linear of order 2
ρ5111-11-1111111111111-1-1-1-1-1-1-1-1    linear of order 2
ρ611-111-11-1-111-11111-1-11-11-11-11-1    linear of order 2
ρ711-11-1-11-1111-11111-1-1-11-11-11-11    linear of order 2
ρ8111-1-1-111-111111111111111111    linear of order 2
ρ9222000220222-2-2-2-2-2-200000000    orthogonal lifted from D4
ρ1022-20002-2022-2-2-2-2-22200000000    orthogonal lifted from D4
ρ1122-20002-20-2-22000000-222-22-2-22    orthogonal lifted from D8
ρ1222-20002-20-2-220000002-2-22-222-2    orthogonal lifted from D8
ρ13222000220-2-2-200000022-2-2-2-222    orthogonal lifted from D8
ρ14222000220-2-2-2000000-2-22222-2-2    orthogonal lifted from D8
ρ1522-2000-220000-22-22-22ζ1671616716165163ζ165163ζ16516316516316716ζ16716    orthogonal lifted from D16
ρ1622-2000-220000-22-22-2216716ζ16716ζ165163165163165163ζ165163ζ1671616716    orthogonal lifted from D16
ρ1722-2000-2200002-22-22-2ζ165163165163ζ167161671616716ζ16716165163ζ165163    orthogonal lifted from D16
ρ1822-2000-2200002-22-22-2165163ζ16516316716ζ16716ζ1671616716ζ165163165163    orthogonal lifted from D16
ρ19222000-2-200002-22-2-221651631651631671616716ζ16716ζ16716ζ165163ζ165163    orthogonal lifted from D16
ρ20222000-2-20000-22-222-21671616716ζ165163ζ165163165163165163ζ16716ζ16716    orthogonal lifted from D16
ρ21222000-2-200002-22-2-22ζ165163ζ165163ζ16716ζ167161671616716165163165163    orthogonal lifted from D16
ρ22222000-2-20000-22-222-2ζ16716ζ16716165163165163ζ165163ζ1651631671616716    orthogonal lifted from D16
ρ234-40000000-22220-2ζ165+2ζ163-2ζ1615+2ζ169165-2ζ1631615-2ζ1690000000000    orthogonal faithful
ρ244-4000000022-2201615-2ζ169-2ζ165+2ζ163-2ζ1615+2ζ169165-2ζ1630000000000    orthogonal faithful
ρ254-40000000-22220165-2ζ1631615-2ζ169-2ζ165+2ζ163-2ζ1615+2ζ1690000000000    orthogonal faithful
ρ264-4000000022-220-2ζ1615+2ζ169165-2ζ1631615-2ζ169-2ζ165+2ζ1630000000000    orthogonal faithful

Smallest permutation representation of C32⋊C22
On 32 points
Generators in S32
(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 16)(3 31)(4 14)(5 29)(6 12)(7 27)(8 10)(9 25)(11 23)(13 21)(15 19)(18 32)(20 30)(22 28)(24 26)
(2 18)(4 20)(6 22)(8 24)(10 26)(12 28)(14 30)(16 32)

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

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

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

Matrix representation of C32⋊C22 in GL4(𝔽97) generated by

4057195
831410
622069
1511043
,
0100
1000
7621694
66317128
,
1000
0100
2869960
5443096
G:=sub<GL(4,GF(97))| [40,83,6,15,57,14,22,11,1,1,0,0,95,0,69,43],[0,1,76,66,1,0,21,31,0,0,69,71,0,0,4,28],[1,0,28,54,0,1,69,43,0,0,96,0,0,0,0,96] >;

C32⋊C22 in GAP, Magma, Sage, TeX

C_{32}\rtimes C_2^2
% in TeX

G:=Group("C32:C2^2");
// GroupNames label

G:=SmallGroup(128,995);
// by ID

G=gap.SmallGroup(128,995);
# by ID

G:=PCGroup([7,-2,2,2,-2,-2,-2,-2,141,1430,675,346,192,1684,851,242,4037,2028,124]);
// Polycyclic

G:=Group<a,b,c|a^32=b^2=c^2=1,b*a*b=a^15,c*a*c=a^17,b*c=c*b>;
// generators/relations

Export

Subgroup lattice of C32⋊C22 in TeX
Character table of C32⋊C22 in TeX

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