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G = C4.19S3≀C2order 288 = 25·32

4th central extension by C4 of S3≀C2

non-abelian, soluble, monomial

Aliases: C4.19S3≀C2, (C3×C12).19D4, C322C8.4C4, C321(C8.C4), C12.31D6.2C2, (C2×C3⋊S3).1Q8, (C3×C6).2(C4⋊C4), C3⋊S33C8.3C2, C3⋊Dic3.8(C2×C4), C2.3(C3⋊S3.Q8), (C4×C3⋊S3).53C22, SmallGroup(288,381)

Series: Derived Chief Lower central Upper central

C1C32C3⋊Dic3 — C4.19S3≀C2
C1C32C3×C6C3⋊Dic3C4×C3⋊S3C12.31D6 — C4.19S3≀C2
C32C3×C6C3⋊Dic3 — C4.19S3≀C2
C1C4

Generators and relations for C4.19S3≀C2
 G = < a,b,c,d,e | a4=b3=c3=1, d4=a2, e2=a, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, dbd-1=c, ebe-1=dcd-1=b-1, ce=ec, ede-1=d3 >

18C2
2C3
2C3
9C4
9C22
2C6
2C6
12S3
12S3
6C8
6C8
9C2×C4
9C8
9C8
2C12
2C12
6Dic3
6Dic3
6D6
6D6
2C3⋊S3
9M4(2)
9C2×C8
9M4(2)
2C3⋊C8
2C3⋊C8
6C24
6C24
6C4×S3
6C4×S3
9C8.C4
6C8⋊S3
6C8⋊S3
2C3×C3⋊C8
2C3×C3⋊C8

Character table of C4.19S3≀C2

 class 12A2B3A3B4A4B4C6A6B8A8B8C8D8E8F8G8H12A12B12C12D24A24B24C24D24E24F24G24H
 size 111844111844121212121818181844441212121212121212
ρ1111111111111111111111111111111    trivial
ρ211111111111-11-1-1-1-1-111111-1-111-1-11    linear of order 2
ρ31111111111-1-1-1-111111111-1-1-1-1-1-1-1-1    linear of order 2
ρ41111111111-11-11-1-1-1-11111-111-1-111-1    linear of order 2
ρ511-111-1-1111ii-i-i1-11-1-1-1-1-1iiii-i-i-i-i    linear of order 4
ρ611-111-1-1111i-i-ii-11-11-1-1-1-1i-i-ii-iii-i    linear of order 4
ρ711-111-1-1111-i-iii1-11-1-1-1-1-1-i-i-i-iiiii    linear of order 4
ρ811-111-1-1111-iii-i-11-11-1-1-1-1-iii-ii-i-ii    linear of order 4
ρ922-22222-22200000000222200000000    orthogonal lifted from D4
ρ1022222-2-2-22200000000-2-2-2-200000000    symplectic lifted from Q8, Schur index 2
ρ112-20222i-2i0-2-200002-2-2--22i-2i-2i2i00000000    complex lifted from C8.C4
ρ122-2022-2i2i0-2-200002--2-2-2-2i2i2i-2i00000000    complex lifted from C8.C4
ρ132-2022-2i2i0-2-20000-2-22--2-2i2i2i-2i00000000    complex lifted from C8.C4
ρ142-20222i-2i0-2-20000-2--22-22i-2i-2i2i00000000    complex lifted from C8.C4
ρ15440-214401-20-20-2000011-2-201100110    orthogonal lifted from S3≀C2
ρ164401-2440-21-20-200000-2-21110011001    orthogonal lifted from S3≀C2
ρ17440-214401-20202000011-2-20-1-100-1-10    orthogonal lifted from S3≀C2
ρ184401-2440-2120200000-2-211-100-1-100-1    orthogonal lifted from S3≀C2
ρ19440-21-4-401-20-2i02i0000-1-1220ii00-i-i0    complex lifted from C3⋊S3.Q8
ρ204401-2-4-40-21-2i02i0000022-1-1i00i-i00-i    complex lifted from C3⋊S3.Q8
ρ214401-2-4-40-212i0-2i0000022-1-1-i00-ii00i    complex lifted from C3⋊S3.Q8
ρ22440-21-4-401-202i0-2i0000-1-1220-i-i00ii0    complex lifted from C3⋊S3.Q8
ρ234-40-214i-4i0-1200000000i-i2i-2i08ζ3885ζ3850087ζ38783ζ3830    complex faithful
ρ244-40-21-4i4i0-1200000000-ii-2i2i083ζ38387ζ3870085ζ3858ζ380    complex faithful
ρ254-40-21-4i4i0-1200000000-ii-2i2i087ζ38783ζ383008ζ3885ζ3850    complex faithful
ρ264-401-24i-4i02-100000000-2i2i-ii85ζ385008ζ3887ζ3870083ζ383    complex faithful
ρ274-401-2-4i4i02-1000000002i-2ii-i87ζ3870083ζ38385ζ385008ζ38    complex faithful
ρ284-401-2-4i4i02-1000000002i-2ii-i83ζ3830087ζ3878ζ380085ζ385    complex faithful
ρ294-401-24i-4i02-100000000-2i2i-ii8ζ380085ζ38583ζ3830087ζ387    complex faithful
ρ304-40-214i-4i0-1200000000i-i2i-2i085ζ3858ζ380083ζ38387ζ3870    complex faithful

Smallest permutation representation of C4.19S3≀C2
On 48 points
Generators in S48
(1 7 5 3)(2 8 6 4)(9 46 13 42)(10 47 14 43)(11 48 15 44)(12 41 16 45)(17 19 21 23)(18 20 22 24)(25 37 29 33)(26 38 30 34)(27 39 31 35)(28 40 32 36)
(2 47 16)(4 10 41)(6 43 12)(8 14 45)(17 38 28)(19 30 40)(21 34 32)(23 26 36)
(1 46 15)(3 9 48)(5 42 11)(7 13 44)(18 29 39)(20 33 31)(22 25 35)(24 37 27)
(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)
(1 20 7 22 5 24 3 18)(2 23 8 17 6 19 4 21)(9 29 46 33 13 25 42 37)(10 32 47 36 14 28 43 40)(11 27 48 39 15 31 44 35)(12 30 41 34 16 26 45 38)

G:=sub<Sym(48)| (1,7,5,3)(2,8,6,4)(9,46,13,42)(10,47,14,43)(11,48,15,44)(12,41,16,45)(17,19,21,23)(18,20,22,24)(25,37,29,33)(26,38,30,34)(27,39,31,35)(28,40,32,36), (2,47,16)(4,10,41)(6,43,12)(8,14,45)(17,38,28)(19,30,40)(21,34,32)(23,26,36), (1,46,15)(3,9,48)(5,42,11)(7,13,44)(18,29,39)(20,33,31)(22,25,35)(24,37,27), (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), (1,20,7,22,5,24,3,18)(2,23,8,17,6,19,4,21)(9,29,46,33,13,25,42,37)(10,32,47,36,14,28,43,40)(11,27,48,39,15,31,44,35)(12,30,41,34,16,26,45,38)>;

G:=Group( (1,7,5,3)(2,8,6,4)(9,46,13,42)(10,47,14,43)(11,48,15,44)(12,41,16,45)(17,19,21,23)(18,20,22,24)(25,37,29,33)(26,38,30,34)(27,39,31,35)(28,40,32,36), (2,47,16)(4,10,41)(6,43,12)(8,14,45)(17,38,28)(19,30,40)(21,34,32)(23,26,36), (1,46,15)(3,9,48)(5,42,11)(7,13,44)(18,29,39)(20,33,31)(22,25,35)(24,37,27), (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), (1,20,7,22,5,24,3,18)(2,23,8,17,6,19,4,21)(9,29,46,33,13,25,42,37)(10,32,47,36,14,28,43,40)(11,27,48,39,15,31,44,35)(12,30,41,34,16,26,45,38) );

G=PermutationGroup([(1,7,5,3),(2,8,6,4),(9,46,13,42),(10,47,14,43),(11,48,15,44),(12,41,16,45),(17,19,21,23),(18,20,22,24),(25,37,29,33),(26,38,30,34),(27,39,31,35),(28,40,32,36)], [(2,47,16),(4,10,41),(6,43,12),(8,14,45),(17,38,28),(19,30,40),(21,34,32),(23,26,36)], [(1,46,15),(3,9,48),(5,42,11),(7,13,44),(18,29,39),(20,33,31),(22,25,35),(24,37,27)], [(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)], [(1,20,7,22,5,24,3,18),(2,23,8,17,6,19,4,21),(9,29,46,33,13,25,42,37),(10,32,47,36,14,28,43,40),(11,27,48,39,15,31,44,35),(12,30,41,34,16,26,45,38)])

Matrix representation of C4.19S3≀C2 in GL4(𝔽5) generated by

3000
0300
0030
0003
,
2020
0201
4020
0302
,
2020
0204
4020
0202
,
0003
0040
0300
1000
,
0300
1000
0001
0030
G:=sub<GL(4,GF(5))| [3,0,0,0,0,3,0,0,0,0,3,0,0,0,0,3],[2,0,4,0,0,2,0,3,2,0,2,0,0,1,0,2],[2,0,4,0,0,2,0,2,2,0,2,0,0,4,0,2],[0,0,0,1,0,0,3,0,0,4,0,0,3,0,0,0],[0,1,0,0,3,0,0,0,0,0,0,3,0,0,1,0] >;

C4.19S3≀C2 in GAP, Magma, Sage, TeX

C_4._{19}S_3\wr C_2
% in TeX

G:=Group("C4.19S3wrC2");
// GroupNames label

G:=SmallGroup(288,381);
// by ID

G=gap.SmallGroup(288,381);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,3,56,85,64,422,219,100,80,2693,2028,691,797,2372]);
// Polycyclic

G:=Group<a,b,c,d,e|a^4=b^3=c^3=1,d^4=a^2,e^2=a,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,b*c=c*b,d*b*d^-1=c,e*b*e^-1=d*c*d^-1=b^-1,c*e=e*c,e*d*e^-1=d^3>;
// generators/relations

Export

Subgroup lattice of C4.19S3≀C2 in TeX
Character table of C4.19S3≀C2 in TeX

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