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G = C61⋊C3order 183 = 3·61

The semidirect product of C61 and C3 acting faithfully

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

Aliases: C61⋊C3, SmallGroup(183,1)

Series: Derived Chief Lower central Upper central

C1C61 — C61⋊C3
C1C61 — C61⋊C3
C61 — C61⋊C3
C1

Generators and relations for C61⋊C3
 G = < a,b | a61=b3=1, bab-1=a13 >

61C3

Character table of C61⋊C3

 class 13A3B61A61B61C61D61E61F61G61H61I61J61K61L61M61N61O61P61Q61R61S61T
 size 1616133333333333333333333
ρ111111111111111111111111    trivial
ρ21ζ3ζ3211111111111111111111    linear of order 3
ρ31ζ32ζ311111111111111111111    linear of order 3
ρ4300ζ614961466127ζ614561416136ζ615461376131ζ61336126612ζ612961216111ζ612561206116ζ61386117616ζ6147611361ζ6152615614ζ615861426122ζ615061406132ζ613461156112ζ61306124617ζ616061486114ζ61436110618ζ615961356128ζ615561446123ζ61396119613ζ61576156619ζ615361516118    complex faithful
ρ5300ζ6147611361ζ615861426122ζ61336126612ζ61436110618ζ615561446123ζ61396119613ζ61306124617ζ6152615614ζ612561206116ζ614961466127ζ61386117616ζ616061486114ζ615961356128ζ61576156619ζ615061406132ζ615361516118ζ615461376131ζ613461156112ζ614561416136ζ612961216111    complex faithful
ρ6300ζ6152615614ζ614961466127ζ61436110618ζ615061406132ζ615461376131ζ613461156112ζ615961356128ζ612561206116ζ61396119613ζ6147611361ζ61306124617ζ61576156619ζ615361516118ζ614561416136ζ61386117616ζ612961216111ζ61336126612ζ616061486114ζ615861426122ζ615561446123    complex faithful
ρ7300ζ61436110618ζ615461376131ζ612561206116ζ61396119613ζ6147611361ζ61306124617ζ61576156619ζ615061406132ζ61386117616ζ61336126612ζ616061486114ζ615361516118ζ614561416136ζ612961216111ζ613461156112ζ615861426122ζ6152615614ζ615961356128ζ615561446123ζ614961466127    complex faithful
ρ8300ζ61576156619ζ613461156112ζ615361516118ζ612961216111ζ61306124617ζ614961466127ζ61336126612ζ614561416136ζ615861426122ζ616061486114ζ615461376131ζ6152615614ζ61436110618ζ612561206116ζ615561446123ζ615061406132ζ615961356128ζ6147611361ζ61396119613ζ61386117616    complex faithful
ρ9300ζ612961216111ζ615961356128ζ615861426122ζ614961466127ζ61576156619ζ61336126612ζ612561206116ζ615561446123ζ615461376131ζ615361516118ζ6152615614ζ615061406132ζ61396119613ζ61386117616ζ6147611361ζ613461156112ζ614561416136ζ61436110618ζ61306124617ζ616061486114    complex faithful
ρ10300ζ615361516118ζ61306124617ζ614561416136ζ615861426122ζ616061486114ζ615461376131ζ6152615614ζ612961216111ζ615561446123ζ615961356128ζ6147611361ζ61436110618ζ612561206116ζ615061406132ζ614961466127ζ61396119613ζ61576156619ζ61336126612ζ61386117616ζ613461156112    complex faithful
ρ11300ζ61306124617ζ615061406132ζ616061486114ζ61576156619ζ61396119613ζ612961216111ζ614961466127ζ615961356128ζ615361516118ζ61386117616ζ615861426122ζ615461376131ζ6147611361ζ61336126612ζ614561416136ζ6152615614ζ613461156112ζ615561446123ζ61436110618ζ612561206116    complex faithful
ρ12300ζ61396119613ζ6152615614ζ61386117616ζ61306124617ζ61436110618ζ61576156619ζ612961216111ζ613461156112ζ616061486114ζ612561206116ζ615361516118ζ615861426122ζ615561446123ζ614961466127ζ615961356128ζ615461376131ζ615061406132ζ614561416136ζ6147611361ζ61336126612    complex faithful
ρ13300ζ615561446123ζ615361516118ζ614961466127ζ6147611361ζ614561416136ζ61436110618ζ61396119613ζ615461376131ζ61336126612ζ612961216111ζ612561206116ζ61386117616ζ613461156112ζ61306124617ζ6152615614ζ616061486114ζ615861426122ζ615061406132ζ615961356128ζ61576156619    complex faithful
ρ14300ζ61336126612ζ615561446123ζ6152615614ζ612561206116ζ614961466127ζ61386117616ζ616061486114ζ61436110618ζ615061406132ζ615461376131ζ613461156112ζ615961356128ζ61576156619ζ615361516118ζ61396119613ζ614561416136ζ6147611361ζ61306124617ζ612961216111ζ615861426122    complex faithful
ρ15300ζ61386117616ζ61436110618ζ613461156112ζ616061486114ζ612561206116ζ615361516118ζ615861426122ζ61306124617ζ615961356128ζ615061406132ζ614561416136ζ615561446123ζ614961466127ζ615461376131ζ61576156619ζ6147611361ζ61396119613ζ612961216111ζ61336126612ζ6152615614    complex faithful
ρ16300ζ615461376131ζ612961216111ζ6147611361ζ6152615614ζ615861426122ζ615061406132ζ613461156112ζ61336126612ζ61436110618ζ615561446123ζ61396119613ζ61306124617ζ616061486114ζ615961356128ζ612561206116ζ61576156619ζ614961466127ζ61386117616ζ615361516118ζ614561416136    complex faithful
ρ17300ζ615961356128ζ61386117616ζ61576156619ζ614561416136ζ613461156112ζ615561446123ζ6147611361ζ615361516118ζ612961216111ζ61306124617ζ614961466127ζ61336126612ζ6152615614ζ61436110618ζ615861426122ζ612561206116ζ616061486114ζ615461376131ζ615061406132ζ61396119613    complex faithful
ρ18300ζ612561206116ζ6147611361ζ615061406132ζ61386117616ζ61336126612ζ616061486114ζ615361516118ζ61396119613ζ613461156112ζ6152615614ζ615961356128ζ614561416136ζ612961216111ζ615861426122ζ61306124617ζ615561446123ζ61436110618ζ61576156619ζ614961466127ζ615461376131    complex faithful
ρ19300ζ615861426122ζ61576156619ζ615561446123ζ615461376131ζ615361516118ζ6152615614ζ615061406132ζ614961466127ζ6147611361ζ614561416136ζ61436110618ζ61396119613ζ61386117616ζ613461156112ζ61336126612ζ61306124617ζ612961216111ζ612561206116ζ616061486114ζ615961356128    complex faithful
ρ20300ζ615061406132ζ61336126612ζ61396119613ζ613461156112ζ6152615614ζ615961356128ζ614561416136ζ61386117616ζ61306124617ζ61436110618ζ61576156619ζ612961216111ζ615861426122ζ615561446123ζ616061486114ζ614961466127ζ612561206116ζ615361516118ζ615461376131ζ6147611361    complex faithful
ρ21300ζ613461156112ζ612561206116ζ61306124617ζ615961356128ζ615061406132ζ614561416136ζ615561446123ζ616061486114ζ61576156619ζ61396119613ζ612961216111ζ614961466127ζ615461376131ζ6147611361ζ615361516118ζ61336126612ζ61386117616ζ615861426122ζ6152615614ζ61436110618    complex faithful
ρ22300ζ614561416136ζ616061486114ζ612961216111ζ615561446123ζ615961356128ζ6147611361ζ61436110618ζ615861426122ζ614961466127ζ61576156619ζ61336126612ζ612561206116ζ615061406132ζ61396119613ζ615461376131ζ61386117616ζ615361516118ζ6152615614ζ613461156112ζ61306124617    complex faithful
ρ23300ζ616061486114ζ61396119613ζ615961356128ζ615361516118ζ61386117616ζ615861426122ζ615461376131ζ61576156619ζ614561416136ζ613461156112ζ615561446123ζ6147611361ζ61336126612ζ6152615614ζ612961216111ζ61436110618ζ61306124617ζ614961466127ζ612561206116ζ615061406132    complex faithful

Smallest permutation representation of C61⋊C3
On 61 points: primitive
Generators in S61
(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)
(2 48 14)(3 34 27)(4 20 40)(5 6 53)(7 39 18)(8 25 31)(9 11 44)(10 58 57)(12 30 22)(13 16 35)(15 49 61)(17 21 26)(19 54 52)(23 59 43)(24 45 56)(28 50 47)(29 36 60)(32 55 38)(33 41 51)(37 46 42)

G:=sub<Sym(61)| (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), (2,48,14)(3,34,27)(4,20,40)(5,6,53)(7,39,18)(8,25,31)(9,11,44)(10,58,57)(12,30,22)(13,16,35)(15,49,61)(17,21,26)(19,54,52)(23,59,43)(24,45,56)(28,50,47)(29,36,60)(32,55,38)(33,41,51)(37,46,42)>;

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), (2,48,14)(3,34,27)(4,20,40)(5,6,53)(7,39,18)(8,25,31)(9,11,44)(10,58,57)(12,30,22)(13,16,35)(15,49,61)(17,21,26)(19,54,52)(23,59,43)(24,45,56)(28,50,47)(29,36,60)(32,55,38)(33,41,51)(37,46,42) );

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)], [(2,48,14),(3,34,27),(4,20,40),(5,6,53),(7,39,18),(8,25,31),(9,11,44),(10,58,57),(12,30,22),(13,16,35),(15,49,61),(17,21,26),(19,54,52),(23,59,43),(24,45,56),(28,50,47),(29,36,60),(32,55,38),(33,41,51),(37,46,42)])

C61⋊C3 is a maximal subgroup of   C61⋊C6

Matrix representation of C61⋊C3 in GL3(𝔽13) generated by

1190
991
420
,
104
0012
0112
G:=sub<GL(3,GF(13))| [11,9,4,9,9,2,0,1,0],[1,0,0,0,0,1,4,12,12] >;

C61⋊C3 in GAP, Magma, Sage, TeX

C_{61}\rtimes C_3
% in TeX

G:=Group("C61:C3");
// GroupNames label

G:=SmallGroup(183,1);
// by ID

G=gap.SmallGroup(183,1);
# by ID

G:=PCGroup([2,-3,-61,565]);
// Polycyclic

G:=Group<a,b|a^61=b^3=1,b*a*b^-1=a^13>;
// generators/relations

Export

Subgroup lattice of C61⋊C3 in TeX
Character table of C61⋊C3 in TeX

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