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G = C101⋊C4order 404 = 22·101

The semidirect product of C101 and C4 acting faithfully

metacyclic, supersoluble, monomial, Z-group, 2-hyperelementary

Aliases: C101⋊C4, D101.C2, SmallGroup(404,3)

Series: Derived Chief Lower central Upper central

C1C101 — C101⋊C4
C1C101D101 — C101⋊C4
C101 — C101⋊C4
C1

Generators and relations for C101⋊C4
 G = < a,b | a101=b4=1, bab-1=a91 >

101C2
101C4

Character table of C101⋊C4

 class 124A4B101A101B101C101D101E101F101G101H101I101J101K101L101M101N101O101P101Q101R101S101T101U101V101W101X101Y
 size 11011011014444444444444444444444444
ρ111111111111111111111111111111    trivial
ρ211-1-11111111111111111111111111    linear of order 2
ρ31-1-ii1111111111111111111111111    linear of order 4
ρ41-1i-i1111111111111111111111111    linear of order 4
ρ54000ζ10167101641013710134ζ1019310180101211018ζ1019610151101501015ζ1019210190101111019ζ10174101681013310127ζ10175101581014310126ζ1011001019110110101ζ1019510160101411016ζ10183101791012210118ζ10177101631013810124ζ10166101541014710135ζ1019410170101311017ζ10186101521014910115ζ10165101571014410136ζ1019910181101201012ζ1019710161101401014ζ10189101821011910112ζ10178101731012810123ζ10187101621013910114ζ10156101551014610145ζ1019810171101301013ζ10188101721012910113ζ10185101591014210116ζ10184101691013210117ζ10176101531014810125    orthogonal faithful
ρ64000ζ1019810171101301013ζ10188101721012910113ζ10156101551014610145ζ1019910181101201012ζ1019510160101411016ζ10184101691013210117ζ1019210190101111019ζ10166101541014710135ζ1019710161101401014ζ10187101621013910114ζ10189101821011910112ζ10177101631013810124ζ10167101641013710134ζ1019310180101211018ζ10183101791012210118ζ10165101571014410136ζ1019410170101311017ζ1019610151101501015ζ10176101531014810125ζ1011001019110110101ζ10174101681013310127ζ10185101591014210116ζ10175101581014310126ζ10186101521014910115ζ10178101731012810123    orthogonal faithful
ρ74000ζ10176101531014810125ζ1019510160101411016ζ10188101721012910113ζ10184101691013210117ζ1019610151101501015ζ1019410170101311017ζ10175101581014310126ζ10156101551014610145ζ10167101641013710134ζ10183101791012210118ζ1011001019110110101ζ1019910181101201012ζ10187101621013910114ζ10174101681013310127ζ10186101521014910115ζ1019810171101301013ζ1019210190101111019ζ1019310180101211018ζ1019710161101401014ζ10185101591014210116ζ10178101731012810123ζ10166101541014710135ζ10189101821011910112ζ10177101631013810124ζ10165101571014410136    orthogonal faithful
ρ84000ζ10186101521014910115ζ10165101571014410136ζ10178101731012810123ζ1011001019110110101ζ1019810171101301013ζ10185101591014210116ζ10156101551014610145ζ10174101681013310127ζ1019910181101201012ζ1019410170101311017ζ1019510160101411016ζ10189101821011910112ζ10184101691013210117ζ1019710161101401014ζ1019210190101111019ζ10183101791012210118ζ10166101541014710135ζ10176101531014810125ζ10177101631013810124ζ1019610151101501015ζ10167101641013710134ζ1019310180101211018ζ10188101721012910113ζ10175101581014310126ζ10187101621013910114    orthogonal faithful
ρ94000ζ10189101821011910112ζ10186101521014910115ζ10183101791012210118ζ1019310180101211018ζ10177101631013810124ζ10174101681013310127ζ10165101571014410136ζ10187101621013910114ζ10185101591014210116ζ10156101551014610145ζ10176101531014810125ζ1019610151101501015ζ10166101541014710135ζ10184101691013210117ζ10188101721012910113ζ10175101581014310126ζ10178101731012810123ζ1019910181101201012ζ1011001019110110101ζ1019710161101401014ζ1019410170101311017ζ10167101641013710134ζ1019810171101301013ζ1019510160101411016ζ1019210190101111019    orthogonal faithful
ρ104000ζ1019310180101211018ζ1011001019110110101ζ10189101821011910112ζ10187101621013910114ζ10185101591014210116ζ10183101791012210118ζ10177101631013810124ζ10175101581014310126ζ10178101731012810123ζ1019810171101301013ζ10184101691013210117ζ10167101641013710134ζ10165101571014410136ζ10156101551014610145ζ10176101531014810125ζ1019610151101501015ζ10186101521014910115ζ10166101541014710135ζ10174101681013310127ζ1019410170101311017ζ10188101721012910113ζ1019210190101111019ζ1019910181101201012ζ1019710161101401014ζ1019510160101411016    orthogonal faithful
ρ114000ζ10165101571014410136ζ10156101551014610145ζ10166101541014710135ζ10177101631013810124ζ10188101721012910113ζ1019910181101201012ζ1019410170101311017ζ10185101591014210116ζ10176101531014810125ζ10167101641013710134ζ10175101581014310126ζ10186101521014910115ζ1019710161101401014ζ1019610151101501015ζ10187101621013910114ζ10178101731012810123ζ10184101691013210117ζ1019510160101411016ζ1019810171101301013ζ10189101821011910112ζ1019310180101211018ζ1011001019110110101ζ1019210190101111019ζ10183101791012210118ζ10174101681013310127    orthogonal faithful
ρ124000ζ10156101551014610145ζ1019410170101311017ζ10184101691013210117ζ1019810171101301013ζ1019210190101111019ζ10176101531014810125ζ10167101641013710134ζ1019910181101201012ζ1019510160101411016ζ1019310180101211018ζ10183101791012210118ζ10165101571014410136ζ1019610151101501015ζ10189101821011910112ζ10174101681013310127ζ10166101541014710135ζ1019710161101401014ζ10175101581014310126ζ10188101721012910113ζ10186101521014910115ζ1011001019110110101ζ10177101631013810124ζ10187101621013910114ζ10178101731012810123ζ10185101591014210116    orthogonal faithful
ρ134000ζ10183101791012210118ζ10178101731012810123ζ10174101681013310127ζ10189101821011910112ζ10165101571014410136ζ1011001019110110101ζ10166101541014710135ζ1019310180101211018ζ10177101631013810124ζ10184101691013210117ζ10188101721012910113ζ10175101581014310126ζ1019910181101201012ζ10176101531014810125ζ1019410170101311017ζ10187101621013910114ζ10185101591014210116ζ1019810171101301013ζ10186101521014910115ζ1019510160101411016ζ1019710161101401014ζ1019610151101501015ζ10156101551014610145ζ1019210190101111019ζ10167101641013710134    orthogonal faithful
ρ144000ζ10184101691013210117ζ1019710161101401014ζ10176101531014810125ζ10156101551014610145ζ10167101641013710134ζ10188101721012910113ζ1019610151101501015ζ1019810171101301013ζ1019210190101111019ζ10189101821011910112ζ10174101681013310127ζ10166101541014710135ζ10175101581014310126ζ10183101791012210118ζ1011001019110110101ζ1019910181101201012ζ1019510160101411016ζ10187101621013910114ζ1019410170101311017ζ10178101731012810123ζ10186101521014910115ζ10165101571014410136ζ1019310180101211018ζ10185101591014210116ζ10177101631013810124    orthogonal faithful
ρ154000ζ1019410170101311017ζ10167101641013710134ζ1019710161101401014ζ10188101721012910113ζ10187101621013910114ζ1019510160101411016ζ1019310180101211018ζ10176101531014810125ζ10175101581014310126ζ1011001019110110101ζ10178101731012810123ζ10156101551014610145ζ10189101821011910112ζ10186101521014910115ζ10185101591014210116ζ10184101691013210117ζ1019610151101501015ζ10183101791012210118ζ1019210190101111019ζ10165101571014410136ζ10177101631013810124ζ1019810171101301013ζ10174101681013310127ζ10166101541014710135ζ1019910181101201012    orthogonal faithful
ρ164000ζ1019510160101411016ζ10175101581014310126ζ1019210190101111019ζ1019710161101401014ζ10189101821011910112ζ10167101641013710134ζ10183101791012210118ζ1019410170101311017ζ1019310180101211018ζ10178101731012810123ζ10177101631013810124ζ10176101531014810125ζ10174101681013310127ζ10185101591014210116ζ10165101571014410136ζ10188101721012910113ζ10187101621013910114ζ1011001019110110101ζ1019610151101501015ζ1019910181101201012ζ10166101541014710135ζ10184101691013210117ζ10186101521014910115ζ1019810171101301013ζ10156101551014610145    orthogonal faithful
ρ174000ζ10188101721012910113ζ1019210190101111019ζ1019410170101311017ζ10176101531014810125ζ10175101581014310126ζ1019710161101401014ζ10187101621013910114ζ10184101691013210117ζ1019610151101501015ζ10174101681013310127ζ10186101521014910115ζ1019810171101301013ζ1019310180101211018ζ1011001019110110101ζ10178101731012810123ζ10156101551014610145ζ10167101641013710134ζ10189101821011910112ζ1019510160101411016ζ10177101631013810124ζ10185101591014210116ζ1019910181101201012ζ10183101791012210118ζ10165101571014410136ζ10166101541014710135    orthogonal faithful
ρ184000ζ1019910181101201012ζ10176101531014810125ζ1019810171101301013ζ10166101541014710135ζ1019710161101401014ζ10156101551014610145ζ1019510160101411016ζ10165101571014410136ζ1019410170101311017ζ10175101581014310126ζ1019310180101211018ζ10185101591014210116ζ1019210190101111019ζ10187101621013910114ζ10189101821011910112ζ10177101631013810124ζ10188101721012910113ζ10167101641013710134ζ10184101691013210117ζ10174101681013310127ζ10183101791012210118ζ10178101731012810123ζ1019610151101501015ζ1011001019110110101ζ10186101521014910115    orthogonal faithful
ρ194000ζ10187101621013910114ζ10174101681013310127ζ1019310180101211018ζ10175101581014310126ζ10178101731012810123ζ10189101821011910112ζ10185101591014210116ζ1019610151101501015ζ10186101521014910115ζ1019910181101201012ζ10156101551014610145ζ1019210190101111019ζ10177101631013810124ζ1019810171101301013ζ10184101691013210117ζ10167101641013710134ζ1011001019110110101ζ10165101571014410136ζ10183101791012210118ζ10188101721012910113ζ10176101531014810125ζ1019510160101411016ζ10166101541014710135ζ1019410170101311017ζ1019710161101401014    orthogonal faithful
ρ204000ζ10177101631013810124ζ1019810171101301013ζ10165101571014410136ζ10185101591014210116ζ10176101531014810125ζ10166101541014710135ζ10188101721012910113ζ10178101731012810123ζ10184101691013210117ζ1019210190101111019ζ1019610151101501015ζ1011001019110110101ζ1019410170101311017ζ10167101641013710134ζ10175101581014310126ζ10186101521014910115ζ10156101551014610145ζ1019710161101401014ζ1019910181101201012ζ1019310180101211018ζ10187101621013910114ζ10174101681013310127ζ1019510160101411016ζ10189101821011910112ζ10183101791012210118    orthogonal faithful
ρ214000ζ10174101681013310127ζ10185101591014210116ζ1011001019110110101ζ10183101791012210118ζ10166101541014710135ζ10186101521014910115ζ1019910181101201012ζ10189101821011910112ζ10165101571014410136ζ10176101531014810125ζ1019410170101311017ζ10187101621013910114ζ1019810171101301013ζ10188101721012910113ζ1019710161101401014ζ1019310180101211018ζ10177101631013810124ζ10156101551014610145ζ10178101731012810123ζ1019210190101111019ζ1019510160101411016ζ10175101581014310126ζ10184101691013210117ζ10167101641013710134ζ1019610151101501015    orthogonal faithful
ρ224000ζ10185101591014210116ζ1019910181101201012ζ10177101631013810124ζ10178101731012810123ζ10184101691013210117ζ10165101571014410136ζ10176101531014810125ζ10186101521014910115ζ10156101551014610145ζ1019510160101411016ζ10167101641013710134ζ10174101681013310127ζ10188101721012910113ζ1019210190101111019ζ1019610151101501015ζ1011001019110110101ζ1019810171101301013ζ1019410170101311017ζ10166101541014710135ζ10187101621013910114ζ10175101581014310126ζ10183101791012210118ζ1019710161101401014ζ1019310180101211018ζ10189101821011910112    orthogonal faithful
ρ234000ζ1011001019110110101ζ10177101631013810124ζ10186101521014910115ζ10174101681013310127ζ1019910181101201012ζ10178101731012810123ζ1019810171101301013ζ10183101791012210118ζ10166101541014710135ζ10188101721012910113ζ1019710161101401014ζ1019310180101211018ζ10156101551014610145ζ1019410170101311017ζ1019510160101411016ζ10189101821011910112ζ10165101571014410136ζ10184101691013210117ζ10185101591014210116ζ10167101641013710134ζ1019210190101111019ζ10187101621013910114ζ10176101531014810125ζ1019610151101501015ζ10175101581014310126    orthogonal faithful
ρ244000ζ10166101541014710135ζ10184101691013210117ζ1019910181101201012ζ10165101571014410136ζ1019410170101311017ζ1019810171101301013ζ1019710161101401014ζ10177101631013810124ζ10188101721012910113ζ1019610151101501015ζ10187101621013910114ζ10178101731012810123ζ1019510160101411016ζ10175101581014310126ζ1019310180101211018ζ10185101591014210116ζ10176101531014810125ζ1019210190101111019ζ10156101551014610145ζ10183101791012210118ζ10189101821011910112ζ10186101521014910115ζ10167101641013710134ζ10174101681013310127ζ1011001019110110101    orthogonal faithful
ρ254000ζ1019210190101111019ζ10187101621013910114ζ10167101641013710134ζ1019510160101411016ζ10183101791012210118ζ1019610151101501015ζ10174101681013310127ζ1019710161101401014ζ10189101821011910112ζ10185101591014210116ζ10165101571014410136ζ10188101721012910113ζ1011001019110110101ζ10177101631013810124ζ10166101541014710135ζ1019410170101311017ζ1019310180101211018ζ10186101521014910115ζ10175101581014310126ζ1019810171101301013ζ1019910181101201012ζ10176101531014810125ζ10178101731012810123ζ10156101551014610145ζ10184101691013210117    orthogonal faithful
ρ264000ζ1019710161101401014ζ1019610151101501015ζ1019510160101411016ζ1019410170101311017ζ1019310180101211018ζ1019210190101111019ζ10189101821011910112ζ10188101721012910113ζ10187101621013910114ζ10186101521014910115ζ10185101591014210116ζ10184101691013210117ζ10183101791012210118ζ10178101731012810123ζ10177101631013810124ζ10176101531014810125ζ10175101581014310126ζ10174101681013310127ζ10167101641013710134ζ10166101541014710135ζ10165101571014410136ζ10156101551014610145ζ1011001019110110101ζ1019910181101201012ζ1019810171101301013    orthogonal faithful
ρ274000ζ1019610151101501015ζ10189101821011910112ζ10175101581014310126ζ10167101641013710134ζ1011001019110110101ζ10187101621013910114ζ10186101521014910115ζ1019210190101111019ζ10174101681013310127ζ10165101571014410136ζ1019910181101201012ζ1019710161101401014ζ10178101731012810123ζ10166101541014710135ζ1019810171101301013ζ1019510160101411016ζ10183101791012210118ζ10185101591014210116ζ1019310180101211018ζ10184101691013210117ζ10156101551014610145ζ1019410170101311017ζ10177101631013810124ζ10176101531014810125ζ10188101721012910113    orthogonal faithful
ρ284000ζ10178101731012810123ζ10166101541014710135ζ10185101591014210116ζ10186101521014910115ζ10156101551014610145ζ10177101631013810124ζ10184101691013210117ζ1011001019110110101ζ1019810171101301013ζ1019710161101401014ζ1019210190101111019ζ10183101791012210118ζ10176101531014810125ζ1019510160101411016ζ10167101641013710134ζ10174101681013310127ζ1019910181101201012ζ10188101721012910113ζ10165101571014410136ζ10175101581014310126ζ1019610151101501015ζ10189101821011910112ζ1019410170101311017ζ10187101621013910114ζ1019310180101211018    orthogonal faithful
ρ294000ζ10175101581014310126ζ10183101791012210118ζ10187101621013910114ζ1019610151101501015ζ10186101521014910115ζ1019310180101211018ζ10178101731012810123ζ10167101641013710134ζ1011001019110110101ζ10166101541014710135ζ1019810171101301013ζ1019510160101411016ζ10185101591014210116ζ1019910181101201012ζ10156101551014610145ζ1019210190101111019ζ10174101681013310127ζ10177101631013810124ζ10189101821011910112ζ10176101531014810125ζ10184101691013210117ζ1019710161101401014ζ10165101571014410136ζ10188101721012910113ζ1019410170101311017    orthogonal faithful

Smallest permutation representation of C101⋊C4
On 101 points: primitive
Generators in S101
(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 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101)
(2 11 101 92)(3 21 100 82)(4 31 99 72)(5 41 98 62)(6 51 97 52)(7 61 96 42)(8 71 95 32)(9 81 94 22)(10 91 93 12)(13 20 90 83)(14 30 89 73)(15 40 88 63)(16 50 87 53)(17 60 86 43)(18 70 85 33)(19 80 84 23)(24 29 79 74)(25 39 78 64)(26 49 77 54)(27 59 76 44)(28 69 75 34)(35 38 68 65)(36 48 67 55)(37 58 66 45)(46 47 57 56)

G:=sub<Sym(101)| (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101), (2,11,101,92)(3,21,100,82)(4,31,99,72)(5,41,98,62)(6,51,97,52)(7,61,96,42)(8,71,95,32)(9,81,94,22)(10,91,93,12)(13,20,90,83)(14,30,89,73)(15,40,88,63)(16,50,87,53)(17,60,86,43)(18,70,85,33)(19,80,84,23)(24,29,79,74)(25,39,78,64)(26,49,77,54)(27,59,76,44)(28,69,75,34)(35,38,68,65)(36,48,67,55)(37,58,66,45)(46,47,57,56)>;

G:=Group( (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101), (2,11,101,92)(3,21,100,82)(4,31,99,72)(5,41,98,62)(6,51,97,52)(7,61,96,42)(8,71,95,32)(9,81,94,22)(10,91,93,12)(13,20,90,83)(14,30,89,73)(15,40,88,63)(16,50,87,53)(17,60,86,43)(18,70,85,33)(19,80,84,23)(24,29,79,74)(25,39,78,64)(26,49,77,54)(27,59,76,44)(28,69,75,34)(35,38,68,65)(36,48,67,55)(37,58,66,45)(46,47,57,56) );

G=PermutationGroup([(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101)], [(2,11,101,92),(3,21,100,82),(4,31,99,72),(5,41,98,62),(6,51,97,52),(7,61,96,42),(8,71,95,32),(9,81,94,22),(10,91,93,12),(13,20,90,83),(14,30,89,73),(15,40,88,63),(16,50,87,53),(17,60,86,43),(18,70,85,33),(19,80,84,23),(24,29,79,74),(25,39,78,64),(26,49,77,54),(27,59,76,44),(28,69,75,34),(35,38,68,65),(36,48,67,55),(37,58,66,45),(46,47,57,56)])

Matrix representation of C101⋊C4 in GL4(𝔽809) generated by

0100
0010
0001
808309772309
,
1000
681155724329
173768706766
126322666756
G:=sub<GL(4,GF(809))| [0,0,0,808,1,0,0,309,0,1,0,772,0,0,1,309],[1,681,173,126,0,155,768,322,0,724,706,666,0,329,766,756] >;

C101⋊C4 in GAP, Magma, Sage, TeX

C_{101}\rtimes C_4
% in TeX

G:=Group("C101:C4");
// GroupNames label

G:=SmallGroup(404,3);
// by ID

G=gap.SmallGroup(404,3);
# by ID

G:=PCGroup([3,-2,-2,-101,6,362,1805]);
// Polycyclic

G:=Group<a,b|a^101=b^4=1,b*a*b^-1=a^91>;
// generators/relations

Export

Subgroup lattice of C101⋊C4 in TeX
Character table of C101⋊C4 in TeX

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