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G = C37⋊C6order 222 = 2·3·37

The semidirect product of C37 and C6 acting faithfully

metacyclic, supersoluble, monomial, Z-group

Aliases: C37⋊C6, D37⋊C3, C37⋊C3⋊C2, SmallGroup(222,1)

Series: Derived Chief Lower central Upper central

C1C37 — C37⋊C6
C1C37C37⋊C3 — C37⋊C6
C37 — C37⋊C6
C1

Generators and relations for C37⋊C6
 G = < a,b | a37=b6=1, bab-1=a11 >

37C2
37C3
37C6

Character table of C37⋊C6

 class 123A3B6A6B37A37B37C37D37E37F
 size 13737373737666666
ρ1111111111111    trivial
ρ21-111-1-1111111    linear of order 2
ρ31-1ζ32ζ3ζ6ζ65111111    linear of order 6
ρ41-1ζ3ζ32ζ65ζ6111111    linear of order 6
ρ511ζ32ζ3ζ32ζ3111111    linear of order 3
ρ611ζ3ζ32ζ3ζ32111111    linear of order 3
ρ7600000ζ373437333730377374373ζ37283725372137163712379ζ37323724371937183713375ζ3736372737263711371037ζ3731372937233714378376ζ37353722372037173715372    orthogonal faithful
ρ8600000ζ37323724371937183713375ζ37353722372037173715372ζ373437333730377374373ζ3731372937233714378376ζ3736372737263711371037ζ37283725372137163712379    orthogonal faithful
ρ9600000ζ37283725372137163712379ζ3736372737263711371037ζ37353722372037173715372ζ373437333730377374373ζ37323724371937183713375ζ3731372937233714378376    orthogonal faithful
ρ10600000ζ3731372937233714378376ζ37323724371937183713375ζ3736372737263711371037ζ37353722372037173715372ζ37283725372137163712379ζ373437333730377374373    orthogonal faithful
ρ11600000ζ37353722372037173715372ζ3731372937233714378376ζ37283725372137163712379ζ37323724371937183713375ζ373437333730377374373ζ3736372737263711371037    orthogonal faithful
ρ12600000ζ3736372737263711371037ζ373437333730377374373ζ3731372937233714378376ζ37283725372137163712379ζ37353722372037173715372ζ37323724371937183713375    orthogonal faithful

Smallest permutation representation of C37⋊C6
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 28 27 37 11 12)(3 18 16 36 21 23)(4 8 5 35 31 34)(6 25 20 33 14 19)(7 15 9 32 24 30)(10 22 13 29 17 26)

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

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

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

C37⋊C6 is a maximal subgroup of   C37⋊C12
C37⋊C6 is a maximal quotient of   C74.C6

Matrix representation of C37⋊C6 in GL6(𝔽223)

010000
001000
000100
000010
000001
222125178177178125
,
100000
1495457719155
20416454198217162
531391031616855
9814753127193
1631593413618049

G:=sub<GL(6,GF(223))| [0,0,0,0,0,222,1,0,0,0,0,125,0,1,0,0,0,178,0,0,1,0,0,177,0,0,0,1,0,178,0,0,0,0,1,125],[1,149,204,53,98,163,0,54,164,139,14,159,0,5,54,103,75,34,0,77,198,161,3,136,0,191,217,68,127,180,0,55,162,55,193,49] >;

C37⋊C6 in GAP, Magma, Sage, TeX

C_{37}\rtimes C_6
% in TeX

G:=Group("C37:C6");
// GroupNames label

G:=SmallGroup(222,1);
// by ID

G=gap.SmallGroup(222,1);
# by ID

G:=PCGroup([3,-2,-3,-37,1946,707]);
// Polycyclic

G:=Group<a,b|a^37=b^6=1,b*a*b^-1=a^11>;
// generators/relations

Export

Subgroup lattice of C37⋊C6 in TeX
Character table of C37⋊C6 in TeX

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