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G = C61⋊C5order 305 = 5·61

The semidirect product of C61 and C5 acting faithfully

metacyclic, supersoluble, monomial, Z-group, 5-hyperelementary

Aliases: C61⋊C5, SmallGroup(305,1)

Series: Derived Chief Lower central Upper central

C1C61 — C61⋊C5
C1C61 — C61⋊C5
C61 — C61⋊C5
C1

Generators and relations for C61⋊C5
 G = < a,b | a61=b5=1, bab-1=a34 >

61C5

Character table of C61⋊C5

 class 15A5B5C5D61A61B61C61D61E61F61G61H61I61J61K61L
 size 161616161555555555555
ρ111111111111111111    trivial
ρ21ζ5ζ53ζ52ζ54111111111111    linear of order 5
ρ31ζ54ζ52ζ53ζ5111111111111    linear of order 5
ρ41ζ52ζ5ζ54ζ53111111111111    linear of order 5
ρ51ζ53ζ54ζ5ζ52111111111111    linear of order 5
ρ650000ζ61516144613261306126ζ6149613661196114614ζ6148614661456139615ζ6160615261416127613ζ6138613761286111618ζ61356131612961176110ζ61566122611661156113ζ61586134612061961ζ6159615461436121616ζ61576147614261256112ζ615561406118617612ζ61536150613361246123    complex faithful
ρ750000ζ6149613661196114614ζ61356131612961176110ζ6159615461436121616ζ6138613761286111618ζ61586134612061961ζ61576147614261256112ζ615561406118617612ζ61536150613361246123ζ61566122611661156113ζ61516144613261306126ζ6148614661456139615ζ6160615261416127613    complex faithful
ρ850000ζ61586134612061961ζ61536150613361246123ζ61516144613261306126ζ615561406118617612ζ6148614661456139615ζ6160615261416127613ζ61356131612961176110ζ6159615461436121616ζ6149613661196114614ζ6138613761286111618ζ61576147614261256112ζ61566122611661156113    complex faithful
ρ950000ζ6160615261416127613ζ6138613761286111618ζ61356131612961176110ζ6159615461436121616ζ61566122611661156113ζ61586134612061961ζ61516144613261306126ζ615561406118617612ζ61576147614261256112ζ61536150613361246123ζ6149613661196114614ζ6148614661456139615    complex faithful
ρ1050000ζ61356131612961176110ζ61576147614261256112ζ61566122611661156113ζ61586134612061961ζ61536150613361246123ζ61516144613261306126ζ6148614661456139615ζ6160615261416127613ζ615561406118617612ζ6149613661196114614ζ6159615461436121616ζ6138613761286111618    complex faithful
ρ1150000ζ6138613761286111618ζ61586134612061961ζ61576147614261256112ζ61566122611661156113ζ615561406118617612ζ61536150613361246123ζ6149613661196114614ζ6148614661456139615ζ61516144613261306126ζ6160615261416127613ζ61356131612961176110ζ6159615461436121616    complex faithful
ρ1250000ζ61576147614261256112ζ61516144613261306126ζ615561406118617612ζ61536150613361246123ζ6160615261416127613ζ6149613661196114614ζ6159615461436121616ζ6138613761286111618ζ6148614661456139615ζ61356131612961176110ζ61566122611661156113ζ61586134612061961    complex faithful
ρ1350000ζ6148614661456139615ζ6159615461436121616ζ6138613761286111618ζ61356131612961176110ζ61576147614261256112ζ61566122611661156113ζ61536150613361246123ζ61516144613261306126ζ61586134612061961ζ615561406118617612ζ6160615261416127613ζ6149613661196114614    complex faithful
ρ1450000ζ61566122611661156113ζ615561406118617612ζ61536150613361246123ζ61516144613261306126ζ6149613661196114614ζ6148614661456139615ζ6138613761286111618ζ61356131612961176110ζ6160615261416127613ζ6159615461436121616ζ61586134612061961ζ61576147614261256112    complex faithful
ρ1550000ζ6159615461436121616ζ61566122611661156113ζ61586134612061961ζ61576147614261256112ζ61516144613261306126ζ615561406118617612ζ6160615261416127613ζ6149613661196114614ζ61536150613361246123ζ6148614661456139615ζ6138613761286111618ζ61356131612961176110    complex faithful
ρ1650000ζ61536150613361246123ζ6160615261416127613ζ6149613661196114614ζ6148614661456139615ζ6159615461436121616ζ6138613761286111618ζ61576147614261256112ζ61566122611661156113ζ61356131612961176110ζ61586134612061961ζ61516144613261306126ζ615561406118617612    complex faithful
ρ1750000ζ615561406118617612ζ6148614661456139615ζ6160615261416127613ζ6149613661196114614ζ61356131612961176110ζ6159615461436121616ζ61586134612061961ζ61576147614261256112ζ6138613761286111618ζ61566122611661156113ζ61536150613361246123ζ61516144613261306126    complex faithful

Smallest permutation representation of C61⋊C5
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 10 21 59 35)(3 19 41 56 8)(4 28 61 53 42)(5 37 20 50 15)(6 46 40 47 49)(7 55 60 44 22)(9 12 39 38 29)(11 30 18 32 36)(13 48 58 26 43)(14 57 17 23 16)(24 25 34 54 51)(27 52 33 45 31)

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

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

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

Matrix representation of C61⋊C5 in GL5(𝔽1831)

01000
00100
00010
00001
1111012871215848
,
10000
134016021813393151
736118275291511
9756071232675601
11411641331323632

G:=sub<GL(5,GF(1831))| [0,0,0,0,1,1,0,0,0,1110,0,1,0,0,1287,0,0,1,0,1215,0,0,0,1,848],[1,1340,736,975,114,0,1602,1182,607,1164,0,1813,752,1232,133,0,393,91,675,1323,0,151,511,601,632] >;

C61⋊C5 in GAP, Magma, Sage, TeX

C_{61}\rtimes C_5
% in TeX

G:=Group("C61:C5");
// GroupNames label

G:=SmallGroup(305,1);
// by ID

G=gap.SmallGroup(305,1);
# by ID

G:=PCGroup([2,-5,-61,181]);
// Polycyclic

G:=Group<a,b|a^61=b^5=1,b*a*b^-1=a^34>;
// generators/relations

Export

Subgroup lattice of C61⋊C5 in TeX
Character table of C61⋊C5 in TeX

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