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G = C2×F13  order 312 = 23·3·13

Direct product of C2 and F13

direct product, metacyclic, supersoluble, monomial, A-group

Aliases: C2×F13, C26⋊C12, D13⋊C12, D26.C6, C13⋊C4⋊C6, C13⋊C6⋊C4, C13⋊(C2×C12), D13.(C2×C6), C13⋊C6.C22, (C2×C13⋊C4)⋊C3, C13⋊C3⋊(C2×C4), (C2×C13⋊C3)⋊C4, (C2×C13⋊C6).C2, Aut(D26), Hol(C26), SmallGroup(312,45)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C13 — C2×F13
Chief series
C1 — C13 — D13 — C13⋊C6 — F13 — C2×F13
Lower central
C13 — C2×F13
Upper central
C1 — C2

Generators and relations for C2×F13
 G = < a,b,c | a2=b13=c12=1, ab=ba, ac=ca, cbc-1=b6 >

C1
C2
13C2
13C2
13C3
C13
13C4
13C22
13C4
13C6
13C6
13C6
D13
C26
D13
C13⋊C3
13C2×C4
13C12
13C12
13C2×C6
C13⋊C4
D26
C13⋊C4
C13⋊C6
C13⋊C6
C2×C13⋊C3
13C2×C12
C2×C13⋊C4
F13
F13
C2×C13⋊C6
C2×F13

Character table of C2×F13

 class 12A2B2C3A3B4A4B4C4D6A6B6C6D6E6F12A12B12C12D12E12F12G12H1326
 size 11131313131313131313131313131313131313131313131212
ρ111111111111111111111111111    trivial
ρ21-1-1111-111-1-111-1-1-11111-1-1-1-11-1    linear of order 2
ρ3111111-1-1-1-1111111-1-1-1-1-1-1-1-111    linear of order 2
ρ41-1-11111-1-11-111-1-1-1-1-1-1-111111-1    linear of order 2
ρ51111ζ32ζ31111ζ3ζ32ζ3ζ32ζ32ζ3ζ32ζ32ζ3ζ3ζ32ζ32ζ3ζ311    linear of order 3
ρ61-1-11ζ3ζ32-111-1ζ6ζ3ζ32ζ65ζ65ζ6ζ3ζ3ζ32ζ32ζ65ζ65ζ6ζ61-1    linear of order 6
ρ71111ζ3ζ32-1-1-1-1ζ32ζ3ζ32ζ3ζ3ζ32ζ65ζ65ζ6ζ6ζ65ζ65ζ6ζ611    linear of order 6
ρ81-1-11ζ3ζ321-1-11ζ6ζ3ζ32ζ65ζ65ζ6ζ65ζ65ζ6ζ6ζ3ζ3ζ32ζ321-1    linear of order 6
ρ91111ζ3ζ321111ζ32ζ3ζ32ζ3ζ3ζ32ζ3ζ3ζ32ζ32ζ3ζ3ζ32ζ3211    linear of order 3
ρ101-1-11ζ32ζ3-111-1ζ65ζ32ζ3ζ6ζ6ζ65ζ32ζ32ζ3ζ3ζ6ζ6ζ65ζ651-1    linear of order 6
ρ111111ζ32ζ3-1-1-1-1ζ3ζ32ζ3ζ32ζ32ζ3ζ6ζ6ζ65ζ65ζ6ζ6ζ65ζ6511    linear of order 6
ρ121-1-11ζ32ζ31-1-11ζ65ζ32ζ3ζ6ζ6ζ65ζ6ζ6ζ65ζ65ζ32ζ32ζ3ζ31-1    linear of order 6
ρ1311-1-111i-ii-i-1-1-11-11-ii-ii-ii-ii11    linear of order 4
ρ141-11-111-i-iii1-1-1-11-1-ii-iii-ii-i1-1    linear of order 4
ρ151-11-111ii-i-i1-1-1-11-1i-ii-i-ii-ii1-1    linear of order 4
ρ1611-1-111-ii-ii-1-1-11-11i-ii-ii-ii-i11    linear of order 4
ρ1711-1-1ζ3ζ32i-ii-iζ6ζ65ζ6ζ3ζ65ζ32ζ43ζ3ζ4ζ3ζ43ζ32ζ4ζ32ζ43ζ3ζ4ζ3ζ43ζ32ζ4ζ3211    linear of order 12
ρ181-11-1ζ32ζ3ii-i-iζ3ζ6ζ65ζ6ζ32ζ65ζ4ζ32ζ43ζ32ζ4ζ3ζ43ζ3ζ43ζ32ζ4ζ32ζ43ζ3ζ4ζ31-1    linear of order 12
ρ191-11-1ζ3ζ32-i-iiiζ32ζ65ζ6ζ65ζ3ζ6ζ43ζ3ζ4ζ3ζ43ζ32ζ4ζ32ζ4ζ3ζ43ζ3ζ4ζ32ζ43ζ321-1    linear of order 12
ρ2011-1-1ζ32ζ3i-ii-iζ65ζ6ζ65ζ32ζ6ζ3ζ43ζ32ζ4ζ32ζ43ζ3ζ4ζ3ζ43ζ32ζ4ζ32ζ43ζ3ζ4ζ311    linear of order 12
ρ211-11-1ζ3ζ32ii-i-iζ32ζ65ζ6ζ65ζ3ζ6ζ4ζ3ζ43ζ3ζ4ζ32ζ43ζ32ζ43ζ3ζ4ζ3ζ43ζ32ζ4ζ321-1    linear of order 12
ρ221-11-1ζ32ζ3-i-iiiζ3ζ6ζ65ζ6ζ32ζ65ζ43ζ32ζ4ζ32ζ43ζ3ζ4ζ3ζ4ζ32ζ43ζ32ζ4ζ3ζ43ζ31-1    linear of order 12
ρ2311-1-1ζ3ζ32-ii-iiζ6ζ65ζ6ζ3ζ65ζ32ζ4ζ3ζ43ζ3ζ4ζ32ζ43ζ32ζ4ζ3ζ43ζ3ζ4ζ32ζ43ζ3211    linear of order 12
ρ2411-1-1ζ32ζ3-ii-iiζ65ζ6ζ65ζ32ζ6ζ3ζ4ζ32ζ43ζ32ζ4ζ3ζ43ζ3ζ4ζ32ζ43ζ32ζ4ζ3ζ43ζ311    linear of order 12
ρ2512-120000000000000000000000-11    orthogonal faithful
ρ2612120000000000000000000000-1-1    orthogonal lifted from F13

Permutation representations of C2×F13
►On 26 points - transitive group 26T10
Generators in S26
(1 14)(2 15)(3 16)(4 17)(5 18)(6 19)(7 20)(8 21)(9 22)(10 23)(11 24)(12 25)(13 26)

(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)

(2 12 5 6 4 8 13 3 10 9 11 7)(15 25 18 19 17 21 26 16 23 22 24 20)
 
G:=sub<Sym(26)| (1,14)(2,15)(3,16)(4,17)(5,18)(6,19)(7,20)(8,21)(9,22)(10,23)(11,24)(12,25)(13,26), (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), (2,12,5,6,4,8,13,3,10,9,11,7)(15,25,18,19,17,21,26,16,23,22,24,20)>;
 
G:=Group( (1,14)(2,15)(3,16)(4,17)(5,18)(6,19)(7,20)(8,21)(9,22)(10,23)(11,24)(12,25)(13,26), (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), (2,12,5,6,4,8,13,3,10,9,11,7)(15,25,18,19,17,21,26,16,23,22,24,20) );
 
G=PermutationGroup([[(1,14),(2,15),(3,16),(4,17),(5,18),(6,19),(7,20),(8,21),(9,22),(10,23),(11,24),(12,25),(13,26)], [(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)], [(2,12,5,6,4,8,13,3,10,9,11,7),(15,25,18,19,17,21,26,16,23,22,24,20)]])
 
G:=TransitiveGroup(26,10);
 

Matrix representation of C2×F13 ►in GL12(ℤ)

-100000000000
0-10000000000
00-1000000000
000-100000000
0000-10000000
00000-1000000
000000-100000
0000000-10000
00000000-1000
000000000-100
0000000000-10
00000000000-1
,
-1-1-1-1-1-1-1-1-1-1-1-1
100000000000
010000000000
001000000000
000100000000
000010000000
000001000000
000000100000
000000010000
000000001000
000000000100
000000000010
,
-100000000000
000000-100000
111111111111
00000-1000000
00000000000-1
0000-10000000
0000000000-10
000-100000000
000000000-100
00-1000000000
00000000-1000
0-10000000000

G:=sub<GL(12,Integers())| [-1,0,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,-1],[-1,1,0,0,0,0,0,0,0,0,0,0,-1,0,1,0,0,0,0,0,0,0,0,0,-1,0,0,1,0,0,0,0,0,0,0,0,-1,0,0,0,1,0,0,0,0,0,0,0,-1,0,0,0,0,1,0,0,0,0,0,0,-1,0,0,0,0,0,1,0,0,0,0,0,-1,0,0,0,0,0,0,1,0,0,0,0,-1,0,0,0,0,0,0,0,1,0,0,0,-1,0,0,0,0,0,0,0,0,1,0,0,-1,0,0,0,0,0,0,0,0,0,1,0,-1,0,0,0,0,0,0,0,0,0,0,1,-1,0,0,0,0,0,0,0,0,0,0,0],[-1,0,1,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,-1,0,0,1,0,0,0,0,0,0,-1,0,0,0,0,1,0,0,0,0,-1,0,0,0,0,0,0,1,0,0,-1,0,0,0,0,0,0,0,0,1,-1,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,-1,0,0,0,1,0,0,0,0,0,-1,0,0,0,0,0,1,0,0,0,-1,0,0,0,0,0,0,0,1,0,-1,0,0,0,0,0,0,0] >;
 

C2×F13 in GAP, Magma, Sage, TeX 

C_2\times F_{13}
 
% in TeX
 
G:=Group("C2xF13");
 
// GroupNames label
 
G:=SmallGroup(312,45);
 
// by ID
 
G=gap.SmallGroup(312,45);
 
# by ID
 
G:=PCGroup([5,-2,-2,-3,-2,-13,60,4804,464,619]);
 
// Polycyclic
 
G:=Group<a,b,c|a^2=b^13=c^12=1,a*b=b*a,a*c=c*a,c*b*c^-1=b^6>;
 
// generators/relations
 

Export

Subgroup lattice of C2×F13 in TeX
Character table of C2×F13 in TeX

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