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G = C71⋊C5order 355 = 5·71

The semidirect product of C71 and C5 acting faithfully

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

Aliases: C71⋊C5, SmallGroup(355,1)

Series: Derived Chief Lower central Upper central

C1C71 — C71⋊C5
C1C71 — C71⋊C5
C71 — C71⋊C5
C1

Generators and relations for C71⋊C5
 G = < a,b | a71=b5=1, bab-1=a25 >

71C5

Character table of C71⋊C5

 class 15A5B5C5D71A71B71C71D71E71F71G71H71I71J71K71L71M71N
 size 17171717155555555555555
ρ11111111111111111111    trivial
ρ21ζ52ζ5ζ54ζ5311111111111111    linear of order 5
ρ31ζ54ζ52ζ53ζ511111111111111    linear of order 5
ρ41ζ53ζ54ζ5ζ5211111111111111    linear of order 5
ρ51ζ5ζ53ζ52ζ5411111111111111    linear of order 5
ρ650000ζ71687167715671517142ζ71577154712571571ζ71707166714671177114ζ7144713571337123717ζ71657163714171317113ζ7150714371377110712ζ71627159715571267111ζ71697161713471287121ζ712971207115714713ζ71537152714771397122ζ715871407130718716ζ7160714571167112719ζ71497132712471197118ζ71647148713871367127    complex faithful
ρ750000ζ715871407130718716ζ71697161713471287121ζ7150714371377110712ζ71577154712571571ζ7160714571167112719ζ71687167715671517142ζ71497132712471197118ζ712971207115714713ζ71657163714171317113ζ71647148713871367127ζ71627159715571267111ζ71537152714771397122ζ7144713571337123717ζ71707166714671177114    complex faithful
ρ850000ζ71537152714771397122ζ715871407130718716ζ71657163714171317113ζ71687167715671517142ζ7144713571337123717ζ7160714571167112719ζ71707166714671177114ζ71627159715571267111ζ71497132712471197118ζ71697161713471287121ζ71647148713871367127ζ71577154712571571ζ7150714371377110712ζ712971207115714713    complex faithful
ρ950000ζ71707166714671177114ζ71497132712471197118ζ71537152714771397122ζ71627159715571267111ζ71697161713471287121ζ71647148713871367127ζ71687167715671517142ζ7144713571337123717ζ71577154712571571ζ71657163714171317113ζ7150714371377110712ζ712971207115714713ζ715871407130718716ζ7160714571167112719    complex faithful
ρ1050000ζ71657163714171317113ζ7150714371377110712ζ71697161713471287121ζ71707166714671177114ζ71627159715571267111ζ712971207115714713ζ71537152714771397122ζ71687167715671517142ζ715871407130718716ζ7144713571337123717ζ7160714571167112719ζ71497132712471197118ζ71647148713871367127ζ71577154712571571    complex faithful
ρ1150000ζ712971207115714713ζ71707166714671177114ζ71577154712571571ζ71647148713871367127ζ715871407130718716ζ71697161713471287121ζ7160714571167112719ζ7150714371377110712ζ71687167715671517142ζ71497132712471197118ζ71657163714171317113ζ71627159715571267111ζ71537152714771397122ζ7144713571337123717    complex faithful
ρ1250000ζ71497132712471197118ζ71657163714171317113ζ715871407130718716ζ712971207115714713ζ71647148713871367127ζ71627159715571267111ζ71577154712571571ζ7160714571167112719ζ71537152714771397122ζ7150714371377110712ζ7144713571337123717ζ71707166714671177114ζ71697161713471287121ζ71687167715671517142    complex faithful
ρ1350000ζ71577154712571571ζ71537152714771397122ζ71497132712471197118ζ7160714571167112719ζ7150714371377110712ζ7144713571337123717ζ712971207115714713ζ71647148713871367127ζ71707166714671177114ζ715871407130718716ζ71697161713471287121ζ71687167715671517142ζ71657163714171317113ζ71627159715571267111    complex faithful
ρ1450000ζ71647148713871367127ζ71627159715571267111ζ7160714571167112719ζ715871407130718716ζ71577154712571571ζ71537152714771397122ζ7150714371377110712ζ71497132712471197118ζ7144713571337123717ζ712971207115714713ζ71707166714671177114ζ71697161713471287121ζ71687167715671517142ζ71657163714171317113    complex faithful
ρ1550000ζ7160714571167112719ζ71687167715671517142ζ712971207115714713ζ7150714371377110712ζ71497132712471197118ζ71657163714171317113ζ71647148713871367127ζ715871407130718716ζ71627159715571267111ζ71577154712571571ζ71537152714771397122ζ7144713571337123717ζ71707166714671177114ζ71697161713471287121    complex faithful
ρ1650000ζ7144713571337123717ζ7160714571167112719ζ71627159715571267111ζ71657163714171317113ζ71707166714671177114ζ71497132712471197118ζ71697161713471287121ζ71537152714771397122ζ71647148713871367127ζ71687167715671517142ζ71577154712571571ζ7150714371377110712ζ712971207115714713ζ715871407130718716    complex faithful
ρ1750000ζ7150714371377110712ζ7144713571337123717ζ71647148713871367127ζ71497132712471197118ζ712971207115714713ζ71707166714671177114ζ715871407130718716ζ71577154712571571ζ71697161713471287121ζ7160714571167112719ζ71687167715671517142ζ71657163714171317113ζ71627159715571267111ζ71537152714771397122    complex faithful
ρ1850000ζ71697161713471287121ζ71647148713871367127ζ7144713571337123717ζ71537152714771397122ζ71687167715671517142ζ71577154712571571ζ71657163714171317113ζ71707166714671177114ζ7150714371377110712ζ71627159715571267111ζ712971207115714713ζ715871407130718716ζ7160714571167112719ζ71497132712471197118    complex faithful
ρ1950000ζ71627159715571267111ζ712971207115714713ζ71687167715671517142ζ71697161713471287121ζ71537152714771397122ζ715871407130718716ζ7144713571337123717ζ71657163714171317113ζ7160714571167112719ζ71707166714671177114ζ71497132712471197118ζ71647148713871367127ζ71577154712571571ζ7150714371377110712    complex faithful

Smallest permutation representation of C71⋊C5
On 71 points: primitive
Generators in S71
(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)
(2 55 6 58 26)(3 38 11 44 51)(4 21 16 30 5)(7 41 31 59 9)(8 24 36 45 34)(10 61 46 17 13)(12 27 56 60 63)(14 64 66 32 42)(15 47 71 18 67)(19 50 20 33 25)(22 70 35 62 29)(23 53 40 48 54)(28 39 65 49 37)(43 68 69 52 57)

G:=sub<Sym(71)| (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), (2,55,6,58,26)(3,38,11,44,51)(4,21,16,30,5)(7,41,31,59,9)(8,24,36,45,34)(10,61,46,17,13)(12,27,56,60,63)(14,64,66,32,42)(15,47,71,18,67)(19,50,20,33,25)(22,70,35,62,29)(23,53,40,48,54)(28,39,65,49,37)(43,68,69,52,57)>;

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), (2,55,6,58,26)(3,38,11,44,51)(4,21,16,30,5)(7,41,31,59,9)(8,24,36,45,34)(10,61,46,17,13)(12,27,56,60,63)(14,64,66,32,42)(15,47,71,18,67)(19,50,20,33,25)(22,70,35,62,29)(23,53,40,48,54)(28,39,65,49,37)(43,68,69,52,57) );

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)], [(2,55,6,58,26),(3,38,11,44,51),(4,21,16,30,5),(7,41,31,59,9),(8,24,36,45,34),(10,61,46,17,13),(12,27,56,60,63),(14,64,66,32,42),(15,47,71,18,67),(19,50,20,33,25),(22,70,35,62,29),(23,53,40,48,54),(28,39,65,49,37),(43,68,69,52,57)])

Matrix representation of C71⋊C5 in GL5(𝔽2131)

01000
00100
00010
00001
119307031191108
,
10000
749169711051411882
13054368587441817
00001
29236049891706

G:=sub<GL(5,GF(2131))| [0,0,0,0,1,1,0,0,0,1930,0,1,0,0,703,0,0,1,0,119,0,0,0,1,1108],[1,749,1305,0,2,0,1697,436,0,923,0,1105,858,0,604,0,141,744,0,989,0,1882,1817,1,1706] >;

C71⋊C5 in GAP, Magma, Sage, TeX

C_{71}\rtimes C_5
% in TeX

G:=Group("C71:C5");
// GroupNames label

G:=SmallGroup(355,1);
// by ID

G=gap.SmallGroup(355,1);
# by ID

G:=PCGroup([2,-5,-71,1081]);
// Polycyclic

G:=Group<a,b|a^71=b^5=1,b*a*b^-1=a^25>;
// generators/relations

Export

Subgroup lattice of C71⋊C5 in TeX
Character table of C71⋊C5 in TeX

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