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G = C4×C13⋊C3order 156 = 22·3·13

Direct product of C4 and C13⋊C3

direct product, metacyclic, supersoluble, monomial, Z-group, 3-hyperelementary

Aliases: C4×C13⋊C3, C52⋊C3, C134C12, C26.2C6, C2.(C2×C13⋊C3), (C2×C13⋊C3).2C2, SmallGroup(156,2)

Series: Derived Chief Lower central Upper central

C1C13 — C4×C13⋊C3
C1C13C26C2×C13⋊C3 — C4×C13⋊C3
C13 — C4×C13⋊C3
C1C4

Generators and relations for C4×C13⋊C3
 G = < a,b,c | a4=b13=c3=1, ab=ba, ac=ca, cbc-1=b9 >

13C3
13C6
13C12

Character table of C4×C13⋊C3

 class 123A3B4A4B6A6B12A12B12C12D13A13B13C13D26A26B26C26D52A52B52C52D52E52F52G52H
 size 111313111313131313133333333333333333
ρ11111111111111111111111111111    trivial
ρ21111-1-111-1-1-1-111111111-1-1-1-1-1-1-1-1    linear of order 2
ρ311ζ32ζ311ζ3ζ32ζ3ζ32ζ32ζ31111111111111111    linear of order 3
ρ411ζ3ζ32-1-1ζ32ζ3ζ6ζ65ζ65ζ611111111-1-1-1-1-1-1-1-1    linear of order 6
ρ511ζ32ζ3-1-1ζ3ζ32ζ65ζ6ζ6ζ6511111111-1-1-1-1-1-1-1-1    linear of order 6
ρ611ζ3ζ3211ζ32ζ3ζ32ζ3ζ3ζ321111111111111111    linear of order 3
ρ71-111i-i-1-1-ii-ii1111-1-1-1-1-i-i-i-iiiii    linear of order 4
ρ81-111-ii-1-1i-ii-i1111-1-1-1-1iiii-i-i-i-i    linear of order 4
ρ91-1ζ32ζ3i-iζ65ζ6ζ43ζ3ζ4ζ32ζ43ζ32ζ4ζ31111-1-1-1-1-i-i-i-iiiii    linear of order 12
ρ101-1ζ32ζ3-iiζ65ζ6ζ4ζ3ζ43ζ32ζ4ζ32ζ43ζ31111-1-1-1-1iiii-i-i-i-i    linear of order 12
ρ111-1ζ3ζ32-iiζ6ζ65ζ4ζ32ζ43ζ3ζ4ζ3ζ43ζ321111-1-1-1-1iiii-i-i-i-i    linear of order 12
ρ121-1ζ3ζ32i-iζ6ζ65ζ43ζ32ζ4ζ3ζ43ζ3ζ4ζ321111-1-1-1-1-i-i-i-iiiii    linear of order 12
ρ133300-3-3000000ζ13121310134ζ13913313ζ1311138137ζ136135132ζ13121310134ζ13913313ζ136135132ζ13111381371312131013413111381371391331313613513213111381371391331313613513213121310134    complex lifted from C2×C13⋊C3
ρ143300-3-3000000ζ136135132ζ1311138137ζ13121310134ζ13913313ζ136135132ζ1311138137ζ13913313ζ131213101341361351321312131013413111381371391331313121310134131113813713913313136135132    complex lifted from C2×C13⋊C3
ρ15330033000000ζ13913313ζ13121310134ζ136135132ζ1311138137ζ13913313ζ13121310134ζ1311138137ζ136135132ζ13913313ζ136135132ζ13121310134ζ1311138137ζ136135132ζ13121310134ζ1311138137ζ13913313    complex lifted from C13⋊C3
ρ163300-3-3000000ζ1311138137ζ136135132ζ13913313ζ13121310134ζ1311138137ζ136135132ζ13121310134ζ139133131311138137139133131361351321312131013413913313136135132131213101341311138137    complex lifted from C2×C13⋊C3
ρ173300-3-3000000ζ13913313ζ13121310134ζ136135132ζ1311138137ζ13913313ζ13121310134ζ1311138137ζ1361351321391331313613513213121310134131113813713613513213121310134131113813713913313    complex lifted from C2×C13⋊C3
ρ18330033000000ζ136135132ζ1311138137ζ13121310134ζ13913313ζ136135132ζ1311138137ζ13913313ζ13121310134ζ136135132ζ13121310134ζ1311138137ζ13913313ζ13121310134ζ1311138137ζ13913313ζ136135132    complex lifted from C13⋊C3
ρ19330033000000ζ13121310134ζ13913313ζ1311138137ζ136135132ζ13121310134ζ13913313ζ136135132ζ1311138137ζ13121310134ζ1311138137ζ13913313ζ136135132ζ1311138137ζ13913313ζ136135132ζ13121310134    complex lifted from C13⋊C3
ρ20330033000000ζ1311138137ζ136135132ζ13913313ζ13121310134ζ1311138137ζ136135132ζ13121310134ζ13913313ζ1311138137ζ13913313ζ136135132ζ13121310134ζ13913313ζ136135132ζ13121310134ζ1311138137    complex lifted from C13⋊C3
ρ213-3003i-3i000000ζ1311138137ζ136135132ζ13913313ζ1312131013413111381371361351321312131013413913313ζ43ζ131143ζ13843ζ137ζ43ζ13943ζ13343ζ13ζ43ζ13643ζ13543ζ132ζ43ζ131243ζ131043ζ134ζ4ζ1394ζ1334ζ13ζ4ζ1364ζ1354ζ132ζ4ζ13124ζ13104ζ134ζ4ζ13114ζ1384ζ137    complex faithful
ρ223-3003i-3i000000ζ13121310134ζ13913313ζ1311138137ζ13613513213121310134139133131361351321311138137ζ43ζ131243ζ131043ζ134ζ43ζ131143ζ13843ζ137ζ43ζ13943ζ13343ζ13ζ43ζ13643ζ13543ζ132ζ4ζ13114ζ1384ζ137ζ4ζ1394ζ1334ζ13ζ4ζ1364ζ1354ζ132ζ4ζ13124ζ13104ζ134    complex faithful
ρ233-300-3i3i000000ζ136135132ζ1311138137ζ13121310134ζ1391331313613513213111381371391331313121310134ζ4ζ1364ζ1354ζ132ζ4ζ13124ζ13104ζ134ζ4ζ13114ζ1384ζ137ζ4ζ1394ζ1334ζ13ζ43ζ131243ζ131043ζ134ζ43ζ131143ζ13843ζ137ζ43ζ13943ζ13343ζ13ζ43ζ13643ζ13543ζ132    complex faithful
ρ243-300-3i3i000000ζ13121310134ζ13913313ζ1311138137ζ13613513213121310134139133131361351321311138137ζ4ζ13124ζ13104ζ134ζ4ζ13114ζ1384ζ137ζ4ζ1394ζ1334ζ13ζ4ζ1364ζ1354ζ132ζ43ζ131143ζ13843ζ137ζ43ζ13943ζ13343ζ13ζ43ζ13643ζ13543ζ132ζ43ζ131243ζ131043ζ134    complex faithful
ρ253-3003i-3i000000ζ136135132ζ1311138137ζ13121310134ζ1391331313613513213111381371391331313121310134ζ43ζ13643ζ13543ζ132ζ43ζ131243ζ131043ζ134ζ43ζ131143ζ13843ζ137ζ43ζ13943ζ13343ζ13ζ4ζ13124ζ13104ζ134ζ4ζ13114ζ1384ζ137ζ4ζ1394ζ1334ζ13ζ4ζ1364ζ1354ζ132    complex faithful
ρ263-300-3i3i000000ζ13913313ζ13121310134ζ136135132ζ131113813713913313131213101341311138137136135132ζ4ζ1394ζ1334ζ13ζ4ζ1364ζ1354ζ132ζ4ζ13124ζ13104ζ134ζ4ζ13114ζ1384ζ137ζ43ζ13643ζ13543ζ132ζ43ζ131243ζ131043ζ134ζ43ζ131143ζ13843ζ137ζ43ζ13943ζ13343ζ13    complex faithful
ρ273-3003i-3i000000ζ13913313ζ13121310134ζ136135132ζ131113813713913313131213101341311138137136135132ζ43ζ13943ζ13343ζ13ζ43ζ13643ζ13543ζ132ζ43ζ131243ζ131043ζ134ζ43ζ131143ζ13843ζ137ζ4ζ1364ζ1354ζ132ζ4ζ13124ζ13104ζ134ζ4ζ13114ζ1384ζ137ζ4ζ1394ζ1334ζ13    complex faithful
ρ283-300-3i3i000000ζ1311138137ζ136135132ζ13913313ζ1312131013413111381371361351321312131013413913313ζ4ζ13114ζ1384ζ137ζ4ζ1394ζ1334ζ13ζ4ζ1364ζ1354ζ132ζ4ζ13124ζ13104ζ134ζ43ζ13943ζ13343ζ13ζ43ζ13643ζ13543ζ132ζ43ζ131243ζ131043ζ134ζ43ζ131143ζ13843ζ137    complex faithful

Smallest permutation representation of C4×C13⋊C3
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)
(2 4 10)(3 7 6)(5 13 11)(8 9 12)(15 17 23)(16 20 19)(18 26 24)(21 22 25)(28 30 36)(29 33 32)(31 39 37)(34 35 38)(41 43 49)(42 46 45)(44 52 50)(47 48 51)

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

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

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

C4×C13⋊C3 is a maximal subgroup of   C132C24  Dic26⋊C3  D52⋊C3

Matrix representation of C4×C13⋊C3 in GL4(𝔽157) generated by

28000
0100
0010
0001
,
1000
0119521
0100
0010
,
144000
0100
010411852
0735338
G:=sub<GL(4,GF(157))| [28,0,0,0,0,1,0,0,0,0,1,0,0,0,0,1],[1,0,0,0,0,119,1,0,0,52,0,1,0,1,0,0],[144,0,0,0,0,1,104,73,0,0,118,53,0,0,52,38] >;

C4×C13⋊C3 in GAP, Magma, Sage, TeX

C_4\times C_{13}\rtimes C_3
% in TeX

G:=Group("C4xC13:C3");
// GroupNames label

G:=SmallGroup(156,2);
// by ID

G=gap.SmallGroup(156,2);
# by ID

G:=PCGroup([4,-2,-3,-2,-13,24,295]);
// Polycyclic

G:=Group<a,b,c|a^4=b^13=c^3=1,a*b=b*a,a*c=c*a,c*b*c^-1=b^9>;
// generators/relations

Export

Subgroup lattice of C4×C13⋊C3 in TeX
Character table of C4×C13⋊C3 in TeX

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