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G = C73⋊C6order 438 = 2·3·73

The semidirect product of C73 and C6 acting faithfully

metacyclic, supersoluble, monomial, Z-group

Aliases: C73⋊C6, D73⋊C3, C73⋊C3⋊C2, SmallGroup(438,1)

Series: Derived Chief Lower central Upper central

C1C73 — C73⋊C6
C1C73C73⋊C3 — C73⋊C6
C73 — C73⋊C6
C1

Generators and relations for C73⋊C6
 G = < a,b | a73=b6=1, bab-1=a65 >

73C2
73C3
73C6

Character table of C73⋊C6

 class 123A3B6A6B73A73B73C73D73E73F73G73H73I73J73K73L
 size 17373737373666666666666
ρ1111111111111111111    trivial
ρ21-111-1-1111111111111    linear of order 2
ρ31-1ζ32ζ3ζ6ζ65111111111111    linear of order 6
ρ41-1ζ3ζ32ζ65ζ6111111111111    linear of order 6
ρ511ζ32ζ3ζ32ζ3111111111111    linear of order 3
ρ611ζ3ζ32ζ3ζ32111111111111    linear of order 3
ρ7600000ζ736073447342733173297313ζ73707349734673277324733ζ73697341733773367332734ζ736273587347732673157311ζ73717357735573187316732ζ73677354734873257319736ζ73727365736473973873ζ73667363735673177310737ζ735973537339733473207314ζ735273517343733073227321ζ73687345734073337328735ζ736173507338733573237312    orthogonal faithful
ρ8600000ζ735973537339733473207314ζ73727365736473973873ζ736173507338733573237312ζ73687345734073337328735ζ73677354734873257319736ζ73717357735573187316732ζ73707349734673277324733ζ735273517343733073227321ζ736073447342733173297313ζ73667363735673177310737ζ736273587347732673157311ζ73697341733773367332734    orthogonal faithful
ρ9600000ζ73667363735673177310737ζ73697341733773367332734ζ73677354734873257319736ζ735973537339733473207314ζ73707349734673277324733ζ73727365736473973873ζ736173507338733573237312ζ736273587347732673157311ζ735273517343733073227321ζ73687345734073337328735ζ736073447342733173297313ζ73717357735573187316732    orthogonal faithful
ρ10600000ζ736173507338733573237312ζ735973537339733473207314ζ735273517343733073227321ζ73707349734673277324733ζ736273587347732673157311ζ73687345734073337328735ζ736073447342733173297313ζ73717357735573187316732ζ73697341733773367332734ζ73677354734873257319736ζ73727365736473973873ζ73667363735673177310737    orthogonal faithful
ρ11600000ζ73727365736473973873ζ736273587347732673157311ζ735973537339733473207314ζ73717357735573187316732ζ73667363735673177310737ζ735273517343733073227321ζ73687345734073337328735ζ736173507338733573237312ζ73707349734673277324733ζ73697341733773367332734ζ73677354734873257319736ζ736073447342733173297313    orthogonal faithful
ρ12600000ζ73687345734073337328735ζ73717357735573187316732ζ73707349734673277324733ζ73667363735673177310737ζ736173507338733573237312ζ73697341733773367332734ζ73677354734873257319736ζ736073447342733173297313ζ736273587347732673157311ζ735973537339733473207314ζ735273517343733073227321ζ73727365736473973873    orthogonal faithful
ρ13600000ζ735273517343733073227321ζ736173507338733573237312ζ73717357735573187316732ζ736073447342733173297313ζ73727365736473973873ζ73707349734673277324733ζ73697341733773367332734ζ73687345734073337328735ζ73667363735673177310737ζ736273587347732673157311ζ735973537339733473207314ζ73677354734873257319736    orthogonal faithful
ρ14600000ζ73707349734673277324733ζ73687345734073337328735ζ736073447342733173297313ζ73677354734873257319736ζ735273517343733073227321ζ73667363735673177310737ζ736273587347732673157311ζ73697341733773367332734ζ73727365736473973873ζ736173507338733573237312ζ73717357735573187316732ζ735973537339733473207314    orthogonal faithful
ρ15600000ζ73717357735573187316732ζ735273517343733073227321ζ73687345734073337328735ζ73697341733773367332734ζ735973537339733473207314ζ736073447342733173297313ζ73667363735673177310737ζ73707349734673277324733ζ73677354734873257319736ζ73727365736473973873ζ736173507338733573237312ζ736273587347732673157311    orthogonal faithful
ρ16600000ζ73697341733773367332734ζ736073447342733173297313ζ73667363735673177310737ζ73727365736473973873ζ73687345734073337328735ζ736273587347732673157311ζ735973537339733473207314ζ73677354734873257319736ζ736173507338733573237312ζ73717357735573187316732ζ73707349734673277324733ζ735273517343733073227321    orthogonal faithful
ρ17600000ζ73677354734873257319736ζ73667363735673177310737ζ736273587347732673157311ζ736173507338733573237312ζ736073447342733173297313ζ735973537339733473207314ζ735273517343733073227321ζ73727365736473973873ζ73717357735573187316732ζ73707349734673277324733ζ73697341733773367332734ζ73687345734073337328735    orthogonal faithful
ρ18600000ζ736273587347732673157311ζ73677354734873257319736ζ73727365736473973873ζ735273517343733073227321ζ73697341733773367332734ζ736173507338733573237312ζ73717357735573187316732ζ735973537339733473207314ζ73687345734073337328735ζ736073447342733173297313ζ73667363735673177310737ζ73707349734673277324733    orthogonal faithful

Smallest permutation representation of C73⋊C6
On 73 points: primitive
Generators in S73
(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)
(2 10 9 73 65 66)(3 19 17 72 56 58)(4 28 25 71 47 50)(5 37 33 70 38 42)(6 46 41 69 29 34)(7 55 49 68 20 26)(8 64 57 67 11 18)(12 27 16 63 48 59)(13 36 24 62 39 51)(14 45 32 61 30 43)(15 54 40 60 21 35)(22 44 23 53 31 52)

G:=sub<Sym(73)| (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), (2,10,9,73,65,66)(3,19,17,72,56,58)(4,28,25,71,47,50)(5,37,33,70,38,42)(6,46,41,69,29,34)(7,55,49,68,20,26)(8,64,57,67,11,18)(12,27,16,63,48,59)(13,36,24,62,39,51)(14,45,32,61,30,43)(15,54,40,60,21,35)(22,44,23,53,31,52)>;

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), (2,10,9,73,65,66)(3,19,17,72,56,58)(4,28,25,71,47,50)(5,37,33,70,38,42)(6,46,41,69,29,34)(7,55,49,68,20,26)(8,64,57,67,11,18)(12,27,16,63,48,59)(13,36,24,62,39,51)(14,45,32,61,30,43)(15,54,40,60,21,35)(22,44,23,53,31,52) );

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)], [(2,10,9,73,65,66),(3,19,17,72,56,58),(4,28,25,71,47,50),(5,37,33,70,38,42),(6,46,41,69,29,34),(7,55,49,68,20,26),(8,64,57,67,11,18),(12,27,16,63,48,59),(13,36,24,62,39,51),(14,45,32,61,30,43),(15,54,40,60,21,35),(22,44,23,53,31,52)])

Matrix representation of C73⋊C6 in GL6(𝔽439)

8910000
26401000
39900100
24300010
7000001
66400344242156131
,
38144257194214
37841215737019972
5610518740775285
37326526377185411
4273234038025242
3853926334345147

G:=sub<GL(6,GF(439))| [89,264,399,243,70,66,1,0,0,0,0,400,0,1,0,0,0,344,0,0,1,0,0,242,0,0,0,1,0,156,0,0,0,0,1,131],[381,378,56,373,427,385,4,412,105,265,32,39,425,157,187,26,340,26,7,370,407,377,380,334,194,199,75,185,252,345,214,72,285,411,42,147] >;

C73⋊C6 in GAP, Magma, Sage, TeX

C_{73}\rtimes C_6
% in TeX

G:=Group("C73:C6");
// GroupNames label

G:=SmallGroup(438,1);
// by ID

G=gap.SmallGroup(438,1);
# by ID

G:=PCGroup([3,-2,-3,-73,3890,221]);
// Polycyclic

G:=Group<a,b|a^73=b^6=1,b*a*b^-1=a^65>;
// generators/relations

Export

Subgroup lattice of C73⋊C6 in TeX
Character table of C73⋊C6 in TeX

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