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G = C2×C13⋊C4order 104 = 23·13

Direct product of C2 and C13⋊C4

direct product, metacyclic, supersoluble, monomial, A-group, 2-hyperelementary

Aliases: C2×C13⋊C4, C26⋊C4, D13⋊C4, D26.C2, D13.C22, C13⋊(C2×C4), SmallGroup(104,12)

Series: Derived Chief Lower central Upper central

C1C13 — C2×C13⋊C4
C1C13D13C13⋊C4 — C2×C13⋊C4
C13 — C2×C13⋊C4
C1C2

Generators and relations for C2×C13⋊C4
 G = < a,b,c | a2=b13=c4=1, ab=ba, ac=ca, cbc-1=b5 >

13C2
13C2
13C4
13C22
13C4
13C2×C4

Character table of C2×C13⋊C4

 class 12A2B2C4A4B4C4D13A13B13C26A26B26C
 size 11131313131313444444
ρ111111111111111    trivial
ρ21-11-11-11-1111-1-1-1    linear of order 2
ρ31-11-1-11-11111-1-1-1    linear of order 2
ρ41111-1-1-1-1111111    linear of order 2
ρ511-1-1-i-iii111111    linear of order 4
ρ611-1-1ii-i-i111111    linear of order 4
ρ71-1-11i-i-ii111-1-1-1    linear of order 4
ρ81-1-11-iii-i111-1-1-1    linear of order 4
ρ944000000ζ131213813513ζ13111310133132ζ139137136134ζ13111310133132ζ139137136134ζ131213813513    orthogonal lifted from C13⋊C4
ρ104-4000000ζ13111310133132ζ139137136134ζ13121381351313913713613413121381351313111310133132    orthogonal faithful
ρ1144000000ζ13111310133132ζ139137136134ζ131213813513ζ139137136134ζ131213813513ζ13111310133132    orthogonal lifted from C13⋊C4
ρ124-4000000ζ131213813513ζ13111310133132ζ13913713613413111310133132139137136134131213813513    orthogonal faithful
ρ1344000000ζ139137136134ζ131213813513ζ13111310133132ζ131213813513ζ13111310133132ζ139137136134    orthogonal lifted from C13⋊C4
ρ144-4000000ζ139137136134ζ131213813513ζ1311131013313213121381351313111310133132139137136134    orthogonal faithful

Permutation representations of C2×C13⋊C4
On 26 points - transitive group 26T7
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)
(1 14)(2 22 13 19)(3 17 12 24)(4 25 11 16)(5 20 10 21)(6 15 9 26)(7 23 8 18)

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), (1,14)(2,22,13,19)(3,17,12,24)(4,25,11,16)(5,20,10,21)(6,15,9,26)(7,23,8,18)>;

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), (1,14)(2,22,13,19)(3,17,12,24)(4,25,11,16)(5,20,10,21)(6,15,9,26)(7,23,8,18) );

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)], [(1,14),(2,22,13,19),(3,17,12,24),(4,25,11,16),(5,20,10,21),(6,15,9,26),(7,23,8,18)])

G:=TransitiveGroup(26,7);

C2×C13⋊C4 is a maximal subgroup of   C52⋊C4  D13.D4
C2×C13⋊C4 is a maximal quotient of   D13⋊C8  C52.C4  C52⋊C4  C13⋊M4(2)  D13.D4

Matrix representation of C2×C13⋊C4 in GL4(𝔽5) generated by

4000
0400
0040
0004
,
3010
1000
4004
2100
,
1230
0221
0020
0330
G:=sub<GL(4,GF(5))| [4,0,0,0,0,4,0,0,0,0,4,0,0,0,0,4],[3,1,4,2,0,0,0,1,1,0,0,0,0,0,4,0],[1,0,0,0,2,2,0,3,3,2,2,3,0,1,0,0] >;

C2×C13⋊C4 in GAP, Magma, Sage, TeX

C_2\times C_{13}\rtimes C_4
% in TeX

G:=Group("C2xC13:C4");
// GroupNames label

G:=SmallGroup(104,12);
// by ID

G=gap.SmallGroup(104,12);
# by ID

G:=PCGroup([4,-2,-2,-2,-13,16,1027,395]);
// Polycyclic

G:=Group<a,b,c|a^2=b^13=c^4=1,a*b=b*a,a*c=c*a,c*b*c^-1=b^5>;
// generators/relations

Export

Subgroup lattice of C2×C13⋊C4 in TeX
Character table of C2×C13⋊C4 in TeX

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