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G = C4×C13⋊C4order 208 = 24·13

Direct product of C4 and C13⋊C4

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

Aliases: C4×C13⋊C4, C13⋊C42, C522C4, Dic132C4, D26.4C22, D13.(C2×C4), C26.3(C2×C4), (C4×D13).6C2, C2.2(C2×C13⋊C4), (C2×C13⋊C4).2C2, SmallGroup(208,30)

Series: Derived Chief Lower central Upper central

C1C13 — C4×C13⋊C4
C1C13D13D26C2×C13⋊C4 — C4×C13⋊C4
C13 — C4×C13⋊C4
C1C4

Generators and relations for C4×C13⋊C4
 G = < a,b,c | a4=b13=c4=1, ab=ba, ac=ca, cbc-1=b5 >

13C2
13C2
13C4
13C4
13C22
13C4
13C4
13C4
13C2×C4
13C2×C4
13C2×C4
13C42

Character table of C4×C13⋊C4

 class 12A2B2C4A4B4C4D4E4F4G4H4I4J4K4L13A13B13C26A26B26C52A52B52C52D52E52F
 size 1113131113131313131313131313444444444444
ρ11111111111111111111111111111    trivial
ρ21111-1-11-1-1-1-1-1-1111111111-1-1-1-1-1-1    linear of order 2
ρ3111111-111-1-1-1-1-1-1-1111111111111    linear of order 2
ρ41111-1-1-1-1-11111-1-1-1111111-1-1-1-1-1-1    linear of order 2
ρ51-1-11-ii-1-iii-i-ii-111111-1-1-1-i-iiii-i    linear of order 4
ρ61-11-1-iiii-i-11-11-ii-i111-1-1-1-i-iiii-i    linear of order 4
ρ711-1-1-1-1i11ii-i-i-i-ii111111-1-1-1-1-1-1    linear of order 4
ρ81-11-1i-ii-ii1-11-1-ii-i111-1-1-1ii-i-i-ii    linear of order 4
ρ91-1-11i-i-1i-i-iii-i-111111-1-1-1ii-i-i-ii    linear of order 4
ρ1011-1-111i-1-1-i-iii-i-ii111111111111    linear of order 4
ρ111-11-1-ii-ii-i1-11-1i-ii111-1-1-1-i-iiii-i    linear of order 4
ρ1211-1-1-1-1-i11-i-iiiii-i111111-1-1-1-1-1-1    linear of order 4
ρ131-1-11-ii1-ii-iii-i1-1-1111-1-1-1-i-iiii-i    linear of order 4
ρ1411-1-111-i-1-1ii-i-iii-i111111111111    linear of order 4
ρ151-11-1i-i-i-ii-11-11i-ii111-1-1-1ii-i-i-ii    linear of order 4
ρ161-1-11i-i1i-ii-i-ii1-1-1111-1-1-1ii-i-i-ii    linear of order 4
ρ174400-4-40000000000ζ139137136134ζ131213813513ζ13111310133132ζ139137136134ζ131213813513ζ131113101331321391371361341312138135131391371361341312138135131311131013313213111310133132    orthogonal lifted from C2×C13⋊C4
ρ184400440000000000ζ131213813513ζ13111310133132ζ139137136134ζ131213813513ζ13111310133132ζ139137136134ζ131213813513ζ13111310133132ζ131213813513ζ13111310133132ζ139137136134ζ139137136134    orthogonal lifted from C13⋊C4
ρ194400-4-40000000000ζ13111310133132ζ139137136134ζ131213813513ζ13111310133132ζ139137136134ζ1312138135131311131013313213913713613413111310133132139137136134131213813513131213813513    orthogonal lifted from C2×C13⋊C4
ρ204400440000000000ζ139137136134ζ131213813513ζ13111310133132ζ139137136134ζ131213813513ζ13111310133132ζ139137136134ζ131213813513ζ139137136134ζ131213813513ζ13111310133132ζ13111310133132    orthogonal lifted from C13⋊C4
ρ214400440000000000ζ13111310133132ζ139137136134ζ131213813513ζ13111310133132ζ139137136134ζ131213813513ζ13111310133132ζ139137136134ζ13111310133132ζ139137136134ζ131213813513ζ131213813513    orthogonal lifted from C13⋊C4
ρ224400-4-40000000000ζ131213813513ζ13111310133132ζ139137136134ζ131213813513ζ13111310133132ζ1391371361341312138135131311131013313213121381351313111310133132139137136134139137136134    orthogonal lifted from C2×C13⋊C4
ρ234-400-4i4i0000000000ζ13111310133132ζ139137136134ζ13121381351313111310133132139137136134131213813513ζ43ζ131143ζ131043ζ13343ζ132ζ43ζ13943ζ13743ζ13643ζ134ζ4ζ13114ζ13104ζ1334ζ132ζ4ζ1394ζ1374ζ1364ζ134ζ4ζ13124ζ1384ζ1354ζ13ζ43ζ131243ζ13843ζ13543ζ13    complex faithful
ρ244-4004i-4i0000000000ζ139137136134ζ131213813513ζ1311131013313213913713613413121381351313111310133132ζ4ζ1394ζ1374ζ1364ζ134ζ4ζ13124ζ1384ζ1354ζ13ζ43ζ13943ζ13743ζ13643ζ134ζ43ζ131243ζ13843ζ13543ζ13ζ43ζ131143ζ131043ζ13343ζ132ζ4ζ13114ζ13104ζ1334ζ132    complex faithful
ρ254-400-4i4i0000000000ζ131213813513ζ13111310133132ζ13913713613413121381351313111310133132139137136134ζ43ζ131243ζ13843ζ13543ζ13ζ43ζ131143ζ131043ζ13343ζ132ζ4ζ13124ζ1384ζ1354ζ13ζ4ζ13114ζ13104ζ1334ζ132ζ4ζ1394ζ1374ζ1364ζ134ζ43ζ13943ζ13743ζ13643ζ134    complex faithful
ρ264-4004i-4i0000000000ζ13111310133132ζ139137136134ζ13121381351313111310133132139137136134131213813513ζ4ζ13114ζ13104ζ1334ζ132ζ4ζ1394ζ1374ζ1364ζ134ζ43ζ131143ζ131043ζ13343ζ132ζ43ζ13943ζ13743ζ13643ζ134ζ43ζ131243ζ13843ζ13543ζ13ζ4ζ13124ζ1384ζ1354ζ13    complex faithful
ρ274-400-4i4i0000000000ζ139137136134ζ131213813513ζ1311131013313213913713613413121381351313111310133132ζ43ζ13943ζ13743ζ13643ζ134ζ43ζ131243ζ13843ζ13543ζ13ζ4ζ1394ζ1374ζ1364ζ134ζ4ζ13124ζ1384ζ1354ζ13ζ4ζ13114ζ13104ζ1334ζ132ζ43ζ131143ζ131043ζ13343ζ132    complex faithful
ρ284-4004i-4i0000000000ζ131213813513ζ13111310133132ζ13913713613413121381351313111310133132139137136134ζ4ζ13124ζ1384ζ1354ζ13ζ4ζ13114ζ13104ζ1334ζ132ζ43ζ131243ζ13843ζ13543ζ13ζ43ζ131143ζ131043ζ13343ζ132ζ43ζ13943ζ13743ζ13643ζ134ζ4ζ1394ζ1374ζ1364ζ134    complex faithful

Smallest permutation representation of C4×C13⋊C4
On 52 points
Generators in S52
(1 40 14 27)(2 41 15 28)(3 42 16 29)(4 43 17 30)(5 44 18 31)(6 45 19 32)(7 46 20 33)(8 47 21 34)(9 48 22 35)(10 49 23 36)(11 50 24 37)(12 51 25 38)(13 52 26 39)
(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 33 34 35 36 37 38 39)(40 41 42 43 44 45 46 47 48 49 50 51 52)
(1 40 14 27)(2 48 26 32)(3 43 25 37)(4 51 24 29)(5 46 23 34)(6 41 22 39)(7 49 21 31)(8 44 20 36)(9 52 19 28)(10 47 18 33)(11 42 17 38)(12 50 16 30)(13 45 15 35)

G:=sub<Sym(52)| (1,40,14,27)(2,41,15,28)(3,42,16,29)(4,43,17,30)(5,44,18,31)(6,45,19,32)(7,46,20,33)(8,47,21,34)(9,48,22,35)(10,49,23,36)(11,50,24,37)(12,51,25,38)(13,52,26,39), (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)(27,28,29,30,31,32,33,34,35,36,37,38,39)(40,41,42,43,44,45,46,47,48,49,50,51,52), (1,40,14,27)(2,48,26,32)(3,43,25,37)(4,51,24,29)(5,46,23,34)(6,41,22,39)(7,49,21,31)(8,44,20,36)(9,52,19,28)(10,47,18,33)(11,42,17,38)(12,50,16,30)(13,45,15,35)>;

G:=Group( (1,40,14,27)(2,41,15,28)(3,42,16,29)(4,43,17,30)(5,44,18,31)(6,45,19,32)(7,46,20,33)(8,47,21,34)(9,48,22,35)(10,49,23,36)(11,50,24,37)(12,51,25,38)(13,52,26,39), (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)(27,28,29,30,31,32,33,34,35,36,37,38,39)(40,41,42,43,44,45,46,47,48,49,50,51,52), (1,40,14,27)(2,48,26,32)(3,43,25,37)(4,51,24,29)(5,46,23,34)(6,41,22,39)(7,49,21,31)(8,44,20,36)(9,52,19,28)(10,47,18,33)(11,42,17,38)(12,50,16,30)(13,45,15,35) );

G=PermutationGroup([(1,40,14,27),(2,41,15,28),(3,42,16,29),(4,43,17,30),(5,44,18,31),(6,45,19,32),(7,46,20,33),(8,47,21,34),(9,48,22,35),(10,49,23,36),(11,50,24,37),(12,51,25,38),(13,52,26,39)], [(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),(27,28,29,30,31,32,33,34,35,36,37,38,39),(40,41,42,43,44,45,46,47,48,49,50,51,52)], [(1,40,14,27),(2,48,26,32),(3,43,25,37),(4,51,24,29),(5,46,23,34),(6,41,22,39),(7,49,21,31),(8,44,20,36),(9,52,19,28),(10,47,18,33),(11,42,17,38),(12,50,16,30),(13,45,15,35)])

C4×C13⋊C4 is a maximal subgroup of   C104⋊C4  Dic26⋊C4  D52⋊C4  D26.C23
C4×C13⋊C4 is a maximal quotient of   C104⋊C4  C26.C42  D26.Q8

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

2000
0200
0020
0002
,
2003
2030
0033
1112
,
4120
0204
0440
0340
G:=sub<GL(4,GF(5))| [2,0,0,0,0,2,0,0,0,0,2,0,0,0,0,2],[2,2,0,1,0,0,0,1,0,3,3,1,3,0,3,2],[4,0,0,0,1,2,4,3,2,0,4,4,0,4,0,0] >;

C4×C13⋊C4 in GAP, Magma, Sage, TeX

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

G:=Group("C4xC13:C4");
// GroupNames label

G:=SmallGroup(208,30);
// by ID

G=gap.SmallGroup(208,30);
# by ID

G:=PCGroup([5,-2,-2,-2,-2,-13,20,46,3204,1214]);
// Polycyclic

G:=Group<a,b,c|a^4=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 C4×C13⋊C4 in TeX
Character table of C4×C13⋊C4 in TeX

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