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G = C52⋊C4order 208 = 24·13

1st semidirect product of C52 and C4 acting faithfully

metacyclic, supersoluble, monomial, 2-hyperelementary

Aliases: C521C4, D13.Q8, D13.1D4, Dic133C4, D26.5C22, C13⋊(C4⋊C4), C4⋊(C13⋊C4), C26.4(C2×C4), (C4×D13).4C2, (C2×C13⋊C4).C2, C2.5(C2×C13⋊C4), SmallGroup(208,31)

Series: Derived Chief Lower central Upper central

C1C26 — C52⋊C4
C1C13D13D26C2×C13⋊C4 — C52⋊C4
C13C26 — C52⋊C4
C1C2C4

Generators and relations for C52⋊C4
 G = < a,b | a52=b4=1, bab-1=a31 >

13C2
13C2
13C4
13C22
26C4
26C4
13C2×C4
13C2×C4
13C2×C4
2C13⋊C4
2C13⋊C4
13C4⋊C4

Character table of C52⋊C4

 class 12A2B2C4A4B4C4D4E4F13A13B13C26A26B26C52A52B52C52D52E52F
 size 11131322626262626444444444444
ρ11111111111111111111111    trivial
ρ211111-1-1-1-11111111111111    linear of order 2
ρ31111-11-1-11-1111111-1-1-1-1-1-1    linear of order 2
ρ41111-1-111-1-1111111-1-1-1-1-1-1    linear of order 2
ρ511-1-11i-ii-i-1111111111111    linear of order 4
ρ611-1-11-ii-ii-1111111111111    linear of order 4
ρ711-1-1-1ii-i-i1111111-1-1-1-1-1-1    linear of order 4
ρ811-1-1-1-i-iii1111111-1-1-1-1-1-1    linear of order 4
ρ92-22-2000000222-2-2-2000000    orthogonal lifted from D4
ρ102-2-22000000222-2-2-2000000    symplectic lifted from Q8, Schur index 2
ρ114400400000ζ131213813513ζ13111310133132ζ139137136134ζ139137136134ζ131213813513ζ13111310133132ζ13111310133132ζ139137136134ζ131213813513ζ139137136134ζ131213813513ζ13111310133132    orthogonal lifted from C13⋊C4
ρ124400400000ζ139137136134ζ131213813513ζ13111310133132ζ13111310133132ζ139137136134ζ131213813513ζ131213813513ζ13111310133132ζ139137136134ζ13111310133132ζ139137136134ζ131213813513    orthogonal lifted from C13⋊C4
ρ134400-400000ζ13111310133132ζ139137136134ζ131213813513ζ131213813513ζ13111310133132ζ1391371361341391371361341312138135131311131013313213121381351313111310133132139137136134    orthogonal lifted from C2×C13⋊C4
ρ144400400000ζ13111310133132ζ139137136134ζ131213813513ζ131213813513ζ13111310133132ζ139137136134ζ139137136134ζ131213813513ζ13111310133132ζ131213813513ζ13111310133132ζ139137136134    orthogonal lifted from C13⋊C4
ρ154400-400000ζ131213813513ζ13111310133132ζ139137136134ζ139137136134ζ131213813513ζ131113101331321311131013313213913713613413121381351313913713613413121381351313111310133132    orthogonal lifted from C2×C13⋊C4
ρ164400-400000ζ139137136134ζ131213813513ζ13111310133132ζ13111310133132ζ139137136134ζ1312138135131312138135131311131013313213913713613413111310133132139137136134131213813513    orthogonal lifted from C2×C13⋊C4
ρ174-400000000ζ139137136134ζ131213813513ζ1311131013313213111310133132139137136134131213813513ζ4ζ13124ζ1384ζ1354ζ134ζ13114ζ13104ζ1334ζ132ζ43ζ13943ζ13743ζ13643ζ134ζ4ζ13114ζ13104ζ1334ζ13243ζ13943ζ13743ζ13643ζ134ζ43ζ131243ζ13843ζ13543ζ13    complex faithful
ρ184-400000000ζ13111310133132ζ139137136134ζ13121381351313121381351313111310133132139137136134ζ43ζ13943ζ13743ζ13643ζ134ζ43ζ131243ζ13843ζ13543ζ134ζ13114ζ13104ζ1334ζ132ζ4ζ13124ζ1384ζ1354ζ13ζ4ζ13114ζ13104ζ1334ζ13243ζ13943ζ13743ζ13643ζ134    complex faithful
ρ194-400000000ζ13111310133132ζ139137136134ζ1312138135131312138135131311131013313213913713613443ζ13943ζ13743ζ13643ζ134ζ4ζ13124ζ1384ζ1354ζ13ζ4ζ13114ζ13104ζ1334ζ132ζ43ζ131243ζ13843ζ13543ζ134ζ13114ζ13104ζ1334ζ132ζ43ζ13943ζ13743ζ13643ζ134    complex faithful
ρ204-400000000ζ131213813513ζ13111310133132ζ13913713613413913713613413121381351313111310133132ζ4ζ13114ζ13104ζ1334ζ132ζ43ζ13943ζ13743ζ13643ζ134ζ4ζ13124ζ1384ζ1354ζ1343ζ13943ζ13743ζ13643ζ134ζ43ζ131243ζ13843ζ13543ζ134ζ13114ζ13104ζ1334ζ132    complex faithful
ρ214-400000000ζ139137136134ζ131213813513ζ1311131013313213111310133132139137136134131213813513ζ43ζ131243ζ13843ζ13543ζ13ζ4ζ13114ζ13104ζ1334ζ13243ζ13943ζ13743ζ13643ζ1344ζ13114ζ13104ζ1334ζ132ζ43ζ13943ζ13743ζ13643ζ134ζ4ζ13124ζ1384ζ1354ζ13    complex faithful
ρ224-400000000ζ131213813513ζ13111310133132ζ139137136134139137136134131213813513131113101331324ζ13114ζ13104ζ1334ζ13243ζ13943ζ13743ζ13643ζ134ζ43ζ131243ζ13843ζ13543ζ13ζ43ζ13943ζ13743ζ13643ζ134ζ4ζ13124ζ1384ζ1354ζ13ζ4ζ13114ζ13104ζ1334ζ132    complex faithful

Smallest permutation representation of C52⋊C4
On 52 points
Generators in S52
(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)(2 35 26 19)(3 30 51 50)(4 25 24 29)(5 20 49 8)(6 15 22 39)(7 10 47 18)(9 52 45 28)(11 42 43 38)(12 37 16 17)(13 32 41 48)(14 27)(21 44 33 36)(23 34 31 46)

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

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

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

C52⋊C4 is a maximal subgroup of   D26.8D4  D13.D8  D521C4  D13.Q16  D26.C23  D4×C13⋊C4  Q8×C13⋊C4
C52⋊C4 is a maximal quotient of   D26.8D4  D13.D8  C104.C4  C104.1C4  C52⋊C8  Dic13⋊C8  D26.Q8

Matrix representation of C52⋊C4 in GL4(𝔽53) generated by

2462213
146417
2745744
7314045
,
134026
4646736
129237
318045
G:=sub<GL(4,GF(53))| [2,14,27,7,46,6,45,31,22,41,7,40,13,7,44,45],[13,46,12,31,40,46,9,8,2,7,2,0,6,36,37,45] >;

C52⋊C4 in GAP, Magma, Sage, TeX

C_{52}\rtimes C_4
% in TeX

G:=Group("C52:C4");
// GroupNames label

G:=SmallGroup(208,31);
// by ID

G=gap.SmallGroup(208,31);
# by ID

G:=PCGroup([5,-2,-2,-2,-2,-13,20,101,46,3204,1214]);
// Polycyclic

G:=Group<a,b|a^52=b^4=1,b*a*b^-1=a^31>;
// generators/relations

Export

Subgroup lattice of C52⋊C4 in TeX
Character table of C52⋊C4 in TeX

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