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G = C37⋊C4order 148 = 22·37

The semidirect product of C37 and C4 acting faithfully

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

Aliases: C37⋊C4, D37.C2, SmallGroup(148,3)

Series: Derived Chief Lower central Upper central

C1C37 — C37⋊C4
C1C37D37 — C37⋊C4
C37 — C37⋊C4
C1

Generators and relations for C37⋊C4
 G = < a,b | a37=b4=1, bab-1=a6 >

37C2
37C4

Character table of C37⋊C4

 class 124A4B37A37B37C37D37E37F37G37H37I
 size 1373737444444444
ρ11111111111111    trivial
ρ211-1-1111111111    linear of order 2
ρ31-1i-i111111111    linear of order 4
ρ41-1-ii111111111    linear of order 4
ρ54000ζ373437193718373ζ373337243713374ζ3736373137637ζ3722372137163715ζ372937263711378ζ373537253712372ζ3727372337143710ζ372837203717379ζ37323730377375    orthogonal faithful
ρ64000ζ373337243713374ζ37323730377375ζ372937263711378ζ372837203717379ζ3727372337143710ζ3722372137163715ζ3736373137637ζ373537253712372ζ373437193718373    orthogonal faithful
ρ74000ζ3727372337143710ζ3736373137637ζ372837203717379ζ373337243713374ζ373537253712372ζ373437193718373ζ3722372137163715ζ37323730377375ζ372937263711378    orthogonal faithful
ρ84000ζ3722372137163715ζ372837203717379ζ37323730377375ζ3736373137637ζ373437193718373ζ3727372337143710ζ373337243713374ζ372937263711378ζ373537253712372    orthogonal faithful
ρ94000ζ372837203717379ζ373537253712372ζ373437193718373ζ372937263711378ζ373337243713374ζ3736373137637ζ37323730377375ζ3727372337143710ζ3722372137163715    orthogonal faithful
ρ104000ζ373537253712372ζ3722372137163715ζ373337243713374ζ3727372337143710ζ37323730377375ζ372937263711378ζ373437193718373ζ3736373137637ζ372837203717379    orthogonal faithful
ρ114000ζ3736373137637ζ372937263711378ζ373537253712372ζ37323730377375ζ3722372137163715ζ373337243713374ζ372837203717379ζ373437193718373ζ3727372337143710    orthogonal faithful
ρ124000ζ37323730377375ζ373437193718373ζ3727372337143710ζ373537253712372ζ3736373137637ζ372837203717379ζ372937263711378ζ3722372137163715ζ373337243713374    orthogonal faithful
ρ134000ζ372937263711378ζ3727372337143710ζ3722372137163715ζ373437193718373ζ372837203717379ζ37323730377375ζ373537253712372ζ373337243713374ζ3736373137637    orthogonal faithful

Smallest permutation representation of C37⋊C4
On 37 points: primitive
Generators in S37
(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)
(2 32 37 7)(3 26 36 13)(4 20 35 19)(5 14 34 25)(6 8 33 31)(9 27 30 12)(10 21 29 18)(11 15 28 24)(16 22 23 17)

G:=sub<Sym(37)| (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), (2,32,37,7)(3,26,36,13)(4,20,35,19)(5,14,34,25)(6,8,33,31)(9,27,30,12)(10,21,29,18)(11,15,28,24)(16,22,23,17)>;

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), (2,32,37,7)(3,26,36,13)(4,20,35,19)(5,14,34,25)(6,8,33,31)(9,27,30,12)(10,21,29,18)(11,15,28,24)(16,22,23,17) );

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)], [(2,32,37,7),(3,26,36,13),(4,20,35,19),(5,14,34,25),(6,8,33,31),(9,27,30,12),(10,21,29,18),(11,15,28,24),(16,22,23,17)])

C37⋊C4 is a maximal subgroup of   C37⋊C12  C37⋊Dic3
C37⋊C4 is a maximal quotient of   C37⋊C8  C37⋊Dic3

Matrix representation of C37⋊C4 in GL4(𝔽149) generated by

0100
0010
0001
1481343134
,
1000
70221496
737113656
647087139
G:=sub<GL(4,GF(149))| [0,0,0,148,1,0,0,134,0,1,0,3,0,0,1,134],[1,70,73,64,0,22,71,70,0,14,136,87,0,96,56,139] >;

C37⋊C4 in GAP, Magma, Sage, TeX

C_{37}\rtimes C_4
% in TeX

G:=Group("C37:C4");
// GroupNames label

G:=SmallGroup(148,3);
// by ID

G=gap.SmallGroup(148,3);
# by ID

G:=PCGroup([3,-2,-2,-37,6,1118,653]);
// Polycyclic

G:=Group<a,b|a^37=b^4=1,b*a*b^-1=a^6>;
// generators/relations

Export

Subgroup lattice of C37⋊C4 in TeX
Character table of C37⋊C4 in TeX

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