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G = C16⋊C4order 64 = 26

2nd semidirect product of C16 and C4 acting faithfully

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

Aliases: C162C4, C42.1C4, C4.11C42, C4.6M4(2), M5(2).2C2, C22.4M4(2), (C2×C8).2C4, C8.20(C2×C4), C8⋊C4.4C2, C2.3(C8⋊C4), (C2×C8).41C22, (C2×C4).66(C2×C4), SmallGroup(64,28)

Series: Derived Chief Lower central Upper central Jennings

C1C4 — C16⋊C4
C1C2C4C2×C4C2×C8C8⋊C4 — C16⋊C4
C1C4 — C16⋊C4
C1C4 — C16⋊C4
C1C2C2C2C2C4C4C2×C8 — C16⋊C4

Generators and relations for C16⋊C4
 G = < a,b | a16=b4=1, bab-1=a13 >

2C2
4C4
2C2×C4
2C8

Character table of C16⋊C4

 class 12A2B4A4B4C4D4E8A8B8C8D8E8F16A16B16C16D16E16F16G16H
 size 1121124422224444444444
ρ11111111111111111111111    trivial
ρ211111111111111-1-1-1-1-1-1-1-1    linear of order 2
ρ3111111-1-11111-1-1-11-111-11-1    linear of order 2
ρ4111111-1-11111-1-11-11-1-11-11    linear of order 2
ρ511-111-1-ii-1-111i-ii-1i11-i-1-i    linear of order 4
ρ6111111-1-1-1-1-1-111i-i-ii-i-iii    linear of order 4
ρ711-111-1i-i-1-111-ii-i-1-i11i-1i    linear of order 4
ρ811-111-1-ii-1-111i-i-i1-i-1-1i1i    linear of order 4
ρ9111111-1-1-1-1-1-111-iii-iii-i-i    linear of order 4
ρ1011-111-1i-i-1-111-iii1i-1-1-i1-i    linear of order 4
ρ1111-111-1i-i11-1-1i-i1-i-1-ii1i-1    linear of order 4
ρ1211111111-1-1-1-1-1-1-i-iii-iii-i    linear of order 4
ρ1311-111-1-ii11-1-1-ii-1-i1-ii-1i1    linear of order 4
ρ1411-111-1i-i11-1-1i-i-1i1i-i-1-i1    linear of order 4
ρ1511-111-1-ii11-1-1-ii1i-1i-i1-i-1    linear of order 4
ρ1611111111-1-1-1-1-1-1ii-i-ii-i-ii    linear of order 4
ρ1722-2-2-22002i-2i2i-2i0000000000    complex lifted from M4(2)
ρ1822-2-2-2200-2i2i-2i2i0000000000    complex lifted from M4(2)
ρ19222-2-2-2002i-2i-2i2i0000000000    complex lifted from M4(2)
ρ20222-2-2-200-2i2i2i-2i0000000000    complex lifted from M4(2)
ρ214-404i-4i00000000000000000    complex faithful
ρ224-40-4i4i00000000000000000    complex faithful

Permutation representations of C16⋊C4
On 16 points - transitive group 16T125
Generators in S16
(1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16)
(2 6 10 14)(3 11)(4 16 12 8)(7 15)

G:=sub<Sym(16)| (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16), (2,6,10,14)(3,11)(4,16,12,8)(7,15)>;

G:=Group( (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16), (2,6,10,14)(3,11)(4,16,12,8)(7,15) );

G=PermutationGroup([(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16)], [(2,6,10,14),(3,11),(4,16,12,8),(7,15)])

G:=TransitiveGroup(16,125);

C16⋊C4 is a maximal subgroup of
C4.C4≀C2  C42.(C2×C4)  C8.5M4(2)  C8.19M4(2)  Q32⋊C4  D16⋊C4
 C4p.C42: C8.23C42  C12.15C42  C48⋊C4  C20.45C42  C80⋊C4  C16⋊F5  C164F5  C42.3F5 ...
C16⋊C4 is a maximal quotient of
C42.2C8  C16⋊F5  C164F5
 C4p.M4(2): C16⋊C8  C12.15C42  C48⋊C4  C20.45C42  C80⋊C4  C42.3F5  C20.23C42  C28.15C42 ...

Matrix representation of C16⋊C4 in GL4(𝔽5) generated by

0003
0040
1000
0100
,
2000
0300
0040
0001
G:=sub<GL(4,GF(5))| [0,0,1,0,0,0,0,1,0,4,0,0,3,0,0,0],[2,0,0,0,0,3,0,0,0,0,4,0,0,0,0,1] >;

C16⋊C4 in GAP, Magma, Sage, TeX

C_{16}\rtimes C_4
% in TeX

G:=Group("C16:C4");
// GroupNames label

G:=SmallGroup(64,28);
// by ID

G=gap.SmallGroup(64,28);
# by ID

G:=PCGroup([6,-2,2,-2,2,-2,-2,24,217,55,332,86,963,88]);
// Polycyclic

G:=Group<a,b|a^16=b^4=1,b*a*b^-1=a^13>;
// generators/relations

Export

Subgroup lattice of C16⋊C4 in TeX
Character table of C16⋊C4 in TeX

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