Copied to
clipboard

G = C71⋊C7order 497 = 7·71

The semidirect product of C71 and C7 acting faithfully

metacyclic, supersoluble, monomial, Z-group, 7-hyperelementary

Aliases: C71⋊C7, SmallGroup(497,1)

Series: Derived Chief Lower central Upper central

C1C71 — C71⋊C7
C1C71 — C71⋊C7
C71 — C71⋊C7
C1

Generators and relations for C71⋊C7
 G = < a,b | a71=b7=1, bab-1=a30 >

71C7

Character table of C71⋊C7

 class 17A7B7C7D7E7F71A71B71C71D71E71F71G71H71I71J
 size 17171717171717777777777
ρ111111111111111111    trivial
ρ21ζ74ζ76ζ72ζ75ζ7ζ731111111111    linear of order 7
ρ31ζ72ζ73ζ7ζ76ζ74ζ751111111111    linear of order 7
ρ41ζ73ζ7ζ75ζ72ζ76ζ741111111111    linear of order 7
ρ51ζ76ζ72ζ73ζ74ζ75ζ71111111111    linear of order 7
ρ61ζ75ζ74ζ76ζ7ζ73ζ721111111111    linear of order 7
ρ71ζ7ζ75ζ74ζ73ζ72ζ761111111111    linear of order 7
ρ87000000ζ71437129712771187112718715ζ71647160714071257119713712ζ7158715471367124711671157110ζ7157715071497138719716714ζ7161715671557147713571177113ζ7170715171417139713471267123ζ716971687152714671317111717ζ7167716571627133712271217114ζ71487145713771327130712071ζ7166716371597153714471427128    complex faithful
ρ97000000ζ71647160714071257119713712ζ7158715471367124711671157110ζ7157715071497138719716714ζ71487145713771327130712071ζ7167716571627133712271217114ζ7166716371597153714471427128ζ7161715671557147713571177113ζ7170715171417139713471267123ζ71437129712771187112718715ζ716971687152714671317111717    complex faithful
ρ107000000ζ7167716571627133712271217114ζ7170715171417139713471267123ζ7166716371597153714471427128ζ716971687152714671317111717ζ71437129712771187112718715ζ7158715471367124711671157110ζ71487145713771327130712071ζ71647160714071257119713712ζ7161715671557147713571177113ζ7157715071497138719716714    complex faithful
ρ117000000ζ7157715071497138719716714ζ71487145713771327130712071ζ71437129712771187112718715ζ71647160714071257119713712ζ7166716371597153714471427128ζ7161715671557147713571177113ζ7170715171417139713471267123ζ716971687152714671317111717ζ7158715471367124711671157110ζ7167716571627133712271217114    complex faithful
ρ127000000ζ71487145713771327130712071ζ71437129712771187112718715ζ71647160714071257119713712ζ7158715471367124711671157110ζ716971687152714671317111717ζ7167716571627133712271217114ζ7166716371597153714471427128ζ7161715671557147713571177113ζ7157715071497138719716714ζ7170715171417139713471267123    complex faithful
ρ137000000ζ716971687152714671317111717ζ7161715671557147713571177113ζ7167716571627133712271217114ζ7170715171417139713471267123ζ7157715071497138719716714ζ71437129712771187112718715ζ7158715471367124711671157110ζ71487145713771327130712071ζ7166716371597153714471427128ζ71647160714071257119713712    complex faithful
ρ147000000ζ7161715671557147713571177113ζ7167716571627133712271217114ζ7170715171417139713471267123ζ7166716371597153714471427128ζ71487145713771327130712071ζ71647160714071257119713712ζ7157715071497138719716714ζ71437129712771187112718715ζ716971687152714671317111717ζ7158715471367124711671157110    complex faithful
ρ157000000ζ7158715471367124711671157110ζ7157715071497138719716714ζ71487145713771327130712071ζ71437129712771187112718715ζ7170715171417139713471267123ζ716971687152714671317111717ζ7167716571627133712271217114ζ7166716371597153714471427128ζ71647160714071257119713712ζ7161715671557147713571177113    complex faithful
ρ167000000ζ7166716371597153714471427128ζ716971687152714671317111717ζ7161715671557147713571177113ζ7167716571627133712271217114ζ7158715471367124711671157110ζ71487145713771327130712071ζ71647160714071257119713712ζ7157715071497138719716714ζ7170715171417139713471267123ζ71437129712771187112718715    complex faithful
ρ177000000ζ7170715171417139713471267123ζ7166716371597153714471427128ζ716971687152714671317111717ζ7161715671557147713571177113ζ71647160714071257119713712ζ7157715071497138719716714ζ71437129712771187112718715ζ7158715471367124711671157110ζ7167716571627133712271217114ζ71487145713771327130712071    complex faithful

Smallest permutation representation of C71⋊C7
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 46 38 33 21 49 31)(3 20 4 65 41 26 61)(5 39 7 58 10 51 50)(6 13 44 19 30 28 9)(8 32 47 12 70 53 69)(11 25 16 37 59 55 17)(14 18 56 62 48 57 36)(15 63 22 23 68 34 66)(24 42 71 27 35 40 52)(29 54 43 45 64 67 60)

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

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

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

Matrix representation of C71⋊C7 in GL7(𝔽6959)

0100000
0010000
0001000
0000100
0000010
0000001
191814221623101147881992
,
1000000
1801439629872575486624662647
367789729421850315063553225
3172604520766690461410631445
337456736864677304420362980
1992539912295462438448776222
42451543658533506598205886

G:=sub<GL(7,GF(6959))| [0,0,0,0,0,0,1,1,0,0,0,0,0,918,0,1,0,0,0,0,1422,0,0,1,0,0,0,1623,0,0,0,1,0,0,1011,0,0,0,0,1,0,4788,0,0,0,0,0,1,1992],[1,1801,3677,3172,3374,1992,4245,0,4396,897,6045,5673,5399,1543,0,2987,2942,2076,686,1229,6585,0,2575,1850,6690,4677,5462,3350,0,4866,3150,4614,3044,4384,659,0,2466,6355,1063,2036,4877,820,0,2647,3225,1445,2980,6222,5886] >;

C71⋊C7 in GAP, Magma, Sage, TeX

C_{71}\rtimes C_7
% in TeX

G:=Group("C71:C7");
// GroupNames label

G:=SmallGroup(497,1);
// by ID

G=gap.SmallGroup(497,1);
# by ID

G:=PCGroup([2,-7,-71,1261]);
// Polycyclic

G:=Group<a,b|a^71=b^7=1,b*a*b^-1=a^30>;
// generators/relations

Export

Subgroup lattice of C71⋊C7 in TeX
Character table of C71⋊C7 in TeX

׿
×
𝔽