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G = C37⋊C12order 444 = 22·3·37

The semidirect product of C37 and C12 acting faithfully

metacyclic, supersoluble, monomial, Z-group

Aliases: C37⋊C12, D37.C6, C37⋊C4⋊C3, C37⋊C3⋊C4, C37⋊C6.C2, SmallGroup(444,7)

Series: Derived Chief Lower central Upper central

C1C37 — C37⋊C12
C1C37D37C37⋊C6 — C37⋊C12
C37 — C37⋊C12
C1

Generators and relations for C37⋊C12
 G = < a,b | a37=b12=1, bab-1=a23 >

37C2
37C3
37C4
37C6
37C12

Character table of C37⋊C12

 class 123A3B4A4B6A6B12A12B12C12D37A37B37C
 size 13737373737373737373737121212
ρ1111111111111111    trivial
ρ21111-1-111-1-1-1-1111    linear of order 2
ρ311ζ3ζ32-1-1ζ3ζ32ζ65ζ6ζ6ζ65111    linear of order 6
ρ411ζ32ζ3-1-1ζ32ζ3ζ6ζ65ζ65ζ6111    linear of order 6
ρ511ζ3ζ3211ζ3ζ32ζ3ζ32ζ32ζ3111    linear of order 3
ρ611ζ32ζ311ζ32ζ3ζ32ζ3ζ3ζ32111    linear of order 3
ρ71-111-ii-1-1-ii-ii111    linear of order 4
ρ81-111i-i-1-1i-ii-i111    linear of order 4
ρ91-1ζ32ζ3-iiζ6ζ65ζ43ζ32ζ4ζ3ζ43ζ3ζ4ζ32111    linear of order 12
ρ101-1ζ3ζ32i-iζ65ζ6ζ4ζ3ζ43ζ32ζ4ζ32ζ43ζ3111    linear of order 12
ρ111-1ζ32ζ3i-iζ6ζ65ζ4ζ32ζ43ζ3ζ4ζ3ζ43ζ32111    linear of order 12
ρ121-1ζ3ζ32-iiζ65ζ6ζ43ζ3ζ4ζ32ζ43ζ32ζ4ζ3111    linear of order 12
ρ131200000000000ζ37343733373237303724371937183713377375374373ζ37363731372937273726372337143711371037837637ζ3735372837253722372137203717371637153712379372    orthogonal faithful
ρ141200000000000ζ3735372837253722372137203717371637153712379372ζ37343733373237303724371937183713377375374373ζ37363731372937273726372337143711371037837637    orthogonal faithful
ρ151200000000000ζ37363731372937273726372337143711371037837637ζ3735372837253722372137203717371637153712379372ζ37343733373237303724371937183713377375374373    orthogonal faithful

Smallest permutation representation of C37⋊C12
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 30 28 7 27 15 37 9 11 32 12 24)(3 22 18 13 16 29 36 17 21 26 23 10)(4 14 8 19 5 6 35 25 31 20 34 33)

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

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

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

Matrix representation of C37⋊C12 in GL12(𝔽1777)

13310000000000
12901000000000
173100100000000
30900010000000
146900001000000
4500000100000
146900000010000
30900000001000
173100000000100
12900000000010
13300000000001
177600000000000
,
394161366100111616413830133182130
38312319020092145082101291511399
12641559165301150913992130173116001772
568923242001495150542503098841338
15551674895005773561776014691685405
5991651648001567107312190453061339
1389174225300156614701349014691339306
17353178000014156308313094051685
866127668300789754688017311338884
7791069169200451359214012917721600
92012311536101626172755701333991511
172993881001742743125201776130182

G:=sub<GL(12,GF(1777))| [133,129,1731,309,1469,45,1469,309,1731,129,133,1776,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0],[394,383,1264,568,1555,599,1389,1735,866,779,920,1729,1613,1231,1559,923,1674,1651,1742,317,1276,1069,1231,938,661,902,1653,242,895,648,253,800,683,1692,1536,81,0,0,0,0,0,0,0,0,0,0,1,0,0,0,1,0,0,0,0,0,0,0,0,0,1116,921,1509,1495,577,1567,1566,1415,789,45,1626,1742,164,450,1399,1505,356,1073,1470,630,754,1359,1727,743,1383,821,213,425,1776,1219,1349,83,688,214,557,1252,0,0,0,0,0,0,0,1,0,0,0,0,133,129,1731,309,1469,45,1469,309,1731,129,133,1776,182,1511,1600,884,1685,306,1339,405,1338,1772,399,130,130,399,1772,1338,405,1339,306,1685,884,1600,1511,182] >;

C37⋊C12 in GAP, Magma, Sage, TeX

C_{37}\rtimes C_{12}
% in TeX

G:=Group("C37:C12");
// GroupNames label

G:=SmallGroup(444,7);
// by ID

G=gap.SmallGroup(444,7);
# by ID

G:=PCGroup([4,-2,-3,-2,-37,24,5955,2503,1163]);
// Polycyclic

G:=Group<a,b|a^37=b^12=1,b*a*b^-1=a^23>;
// generators/relations

Export

Subgroup lattice of C37⋊C12 in TeX
Character table of C37⋊C12 in TeX

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