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G = C52.C4order 208 = 24·13

1st non-split extension by C52 of C4 acting faithfully

metacyclic, supersoluble, monomial, 2-hyperelementary

Aliases: C52.1C4, D26.3C4, C131M4(2), Dic13.5C22, C13⋊C81C2, C4.(C13⋊C4), C26.2(C2×C4), (C4×D13).3C2, C2.4(C2×C13⋊C4), SmallGroup(208,29)

Series: Derived Chief Lower central Upper central

C1C26 — C52.C4
C1C13C26Dic13C13⋊C8 — C52.C4
C13C26 — C52.C4
C1C2C4

Generators and relations for C52.C4
 G = < a,b | a52=1, b4=a26, bab-1=a31 >

26C2
13C4
13C22
2D13
13C8
13C2×C4
13C8
13M4(2)

Character table of C52.C4

 class 12A2B4A4B4C8A8B8C8D13A13B13C26A26B26C52A52B52C52D52E52F
 size 11262131326262626444444444444
ρ11111111111111111111111    trivial
ρ211-1-1111-11-1111111-1-1-1-1-1-1    linear of order 2
ρ3111111-1-1-1-1111111111111    linear of order 2
ρ411-1-111-11-11111111-1-1-1-1-1-1    linear of order 2
ρ5111-1-1-1-iii-i111111-1-1-1-1-1-1    linear of order 4
ρ611-11-1-1-i-iii111111111111    linear of order 4
ρ7111-1-1-1i-i-ii111111-1-1-1-1-1-1    linear of order 4
ρ811-11-1-1ii-i-i111111111111    linear of order 4
ρ92-2002i-2i0000222-2-2-2000000    complex lifted from M4(2)
ρ102-200-2i2i0000222-2-2-2000000    complex lifted from M4(2)
ρ11440-4000000ζ13111310133132ζ139137136134ζ131213813513ζ131213813513ζ13111310133132ζ1391371361341391371361341312138135131311131013313213121381351313111310133132139137136134    orthogonal lifted from C2×C13⋊C4
ρ124404000000ζ131213813513ζ13111310133132ζ139137136134ζ139137136134ζ131213813513ζ13111310133132ζ13111310133132ζ139137136134ζ131213813513ζ139137136134ζ131213813513ζ13111310133132    orthogonal lifted from C13⋊C4
ρ134404000000ζ139137136134ζ131213813513ζ13111310133132ζ13111310133132ζ139137136134ζ131213813513ζ131213813513ζ13111310133132ζ139137136134ζ13111310133132ζ139137136134ζ131213813513    orthogonal lifted from C13⋊C4
ρ14440-4000000ζ139137136134ζ131213813513ζ13111310133132ζ13111310133132ζ139137136134ζ1312138135131312138135131311131013313213913713613413111310133132139137136134131213813513    orthogonal lifted from C2×C13⋊C4
ρ154404000000ζ13111310133132ζ139137136134ζ131213813513ζ131213813513ζ13111310133132ζ139137136134ζ139137136134ζ131213813513ζ13111310133132ζ131213813513ζ13111310133132ζ139137136134    orthogonal lifted from C13⋊C4
ρ16440-4000000ζ131213813513ζ13111310133132ζ139137136134ζ139137136134ζ131213813513ζ131113101331321311131013313213913713613413121381351313913713613413121381351313111310133132    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 104 points
Generators in S104
(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)(53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104)
(1 67 40 80 27 93 14 54)(2 62 13 59 28 88 39 85)(3 57 38 90 29 83 12 64)(4 104 11 69 30 78 37 95)(5 99 36 100 31 73 10 74)(6 94 9 79 32 68 35 53)(7 89 34 58 33 63 8 84)(15 101 26 98 41 75 52 72)(16 96 51 77 42 70 25 103)(17 91 24 56 43 65 50 82)(18 86 49 87 44 60 23 61)(19 81 22 66 45 55 48 92)(20 76 47 97 46 102 21 71)

G:=sub<Sym(104)| (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)(53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104), (1,67,40,80,27,93,14,54)(2,62,13,59,28,88,39,85)(3,57,38,90,29,83,12,64)(4,104,11,69,30,78,37,95)(5,99,36,100,31,73,10,74)(6,94,9,79,32,68,35,53)(7,89,34,58,33,63,8,84)(15,101,26,98,41,75,52,72)(16,96,51,77,42,70,25,103)(17,91,24,56,43,65,50,82)(18,86,49,87,44,60,23,61)(19,81,22,66,45,55,48,92)(20,76,47,97,46,102,21,71)>;

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)(53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104), (1,67,40,80,27,93,14,54)(2,62,13,59,28,88,39,85)(3,57,38,90,29,83,12,64)(4,104,11,69,30,78,37,95)(5,99,36,100,31,73,10,74)(6,94,9,79,32,68,35,53)(7,89,34,58,33,63,8,84)(15,101,26,98,41,75,52,72)(16,96,51,77,42,70,25,103)(17,91,24,56,43,65,50,82)(18,86,49,87,44,60,23,61)(19,81,22,66,45,55,48,92)(20,76,47,97,46,102,21,71) );

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),(53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104)], [(1,67,40,80,27,93,14,54),(2,62,13,59,28,88,39,85),(3,57,38,90,29,83,12,64),(4,104,11,69,30,78,37,95),(5,99,36,100,31,73,10,74),(6,94,9,79,32,68,35,53),(7,89,34,58,33,63,8,84),(15,101,26,98,41,75,52,72),(16,96,51,77,42,70,25,103),(17,91,24,56,43,65,50,82),(18,86,49,87,44,60,23,61),(19,81,22,66,45,55,48,92),(20,76,47,97,46,102,21,71)])

C52.C4 is a maximal subgroup of   C104.C4  C104.1C4  Dic26⋊C4  D52⋊C4  D13⋊M4(2)  Dic26.C4  D52.C4
C52.C4 is a maximal quotient of   C52⋊C8  C26.C42  D26⋊C8

Matrix representation of C52.C4 in GL6(𝔽313)

28800000
139250000
0010314517673
00240168138241
0072103721
00312000
,
205850000
31080000
007213411332
0042135113103
00209168138210
000281272281

G:=sub<GL(6,GF(313))| [288,139,0,0,0,0,0,25,0,0,0,0,0,0,103,240,72,312,0,0,145,168,103,0,0,0,176,138,72,0,0,0,73,241,1,0],[205,3,0,0,0,0,85,108,0,0,0,0,0,0,72,42,209,0,0,0,134,135,168,281,0,0,113,113,138,272,0,0,32,103,210,281] >;

C52.C4 in GAP, Magma, Sage, TeX

C_{52}.C_4
% in TeX

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

G:=SmallGroup(208,29);
// by ID

G=gap.SmallGroup(208,29);
# by ID

G:=PCGroup([5,-2,-2,-2,-2,-13,20,101,46,42,3204,1214]);
// Polycyclic

G:=Group<a,b|a^52=1,b^4=a^26,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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