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G = C244S3order 96 = 25·3

1st semidirect product of C24 and S3 acting via S3/C3=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C244S3, C23.30D6, (C2×C6)⋊8D4, C33C22≀C2, (C23×C6)⋊3C2, C6.63(C2×D4), C224(C3⋊D4), (C2×C6).61C23, C6.D413C2, (C22×S3)⋊2C22, (C2×Dic3)⋊3C22, C22.66(C22×S3), (C22×C6).42C22, (C2×C3⋊D4)⋊8C2, C2.26(C2×C3⋊D4), SmallGroup(96,160)

Series: Derived Chief Lower central Upper central

C1C2×C6 — C244S3
C1C3C6C2×C6C22×S3C2×C3⋊D4 — C244S3
C3C2×C6 — C244S3
C1C22C24

Generators and relations for C244S3
 G = < a,b,c,d,e,f | a2=b2=c2=d2=e3=f2=1, ab=ba, ac=ca, faf=ad=da, ae=ea, fbf=bc=cb, bd=db, be=eb, cd=dc, ce=ec, cf=fc, de=ed, df=fd, fef=e-1 >

Subgroups: 290 in 130 conjugacy classes, 41 normal (8 characteristic)
C1, C2 [×3], C2 [×7], C3, C4 [×3], C22, C22 [×6], C22 [×17], S3, C6 [×3], C6 [×6], C2×C4 [×3], D4 [×6], C23 [×3], C23 [×7], Dic3 [×3], D6 [×3], C2×C6, C2×C6 [×6], C2×C6 [×14], C22⋊C4 [×3], C2×D4 [×3], C24, C2×Dic3 [×3], C3⋊D4 [×6], C22×S3, C22×C6 [×3], C22×C6 [×6], C22≀C2, C6.D4 [×3], C2×C3⋊D4 [×3], C23×C6, C244S3
Quotients: C1, C2 [×7], C22 [×7], S3, D4 [×6], C23, D6 [×3], C2×D4 [×3], C3⋊D4 [×6], C22×S3, C22≀C2, C2×C3⋊D4 [×3], C244S3

Character table of C244S3

 class 12A2B2C2D2E2F2G2H2I2J34A4B4C6A6B6C6D6E6F6G6H6I6J6K6L6M6N6O
 size 1111222222122121212222222222222222
ρ1111111111111111111111111111111    trivial
ρ21111-1-111-1-1-111-1111111-1-1-1-1-1-11-1-11    linear of order 2
ρ31111-1-1-1-111111-1-1-1-11-1-1-111-1111-1-11    linear of order 2
ρ4111111-1-1-1-1-1111-1-1-11-1-11-1-11-1-11111    linear of order 2
ρ51111111111-11-1-1-1111111111111111    linear of order 2
ρ61111-1-111-1-111-11-111111-1-1-1-1-1-11-1-11    linear of order 2
ρ71111-1-1-1-111-11-111-1-11-1-1-111-1111-1-11    linear of order 2
ρ8111111-1-1-1-111-1-11-1-11-1-11-1-11-1-11111    linear of order 2
ρ922-2-200-220002000-2-2-222000000200-2    orthogonal lifted from D4
ρ102-22-2-2200000200000200200200-2-2-2-2    orthogonal lifted from D4
ρ1122222222220-1000-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1    orthogonal lifted from S3
ρ122222-2-222-2-20-1000-1-1-1-1-1111111-111-1    orthogonal lifted from D6
ρ132-2-220000-220200000-2000-2-2022-2002    orthogonal lifted from D4
ρ142-2-2200002-20200000-2000220-2-2-2002    orthogonal lifted from D4
ρ15222222-2-2-2-20-100011-111-111-111-1-1-1-1    orthogonal lifted from D6
ρ162222-2-2-2-2220-100011-1111-1-11-1-1-111-1    orthogonal lifted from D6
ρ1722-2-2002-2000200022-2-2-2000000200-2    orthogonal lifted from D4
ρ182-22-22-200000200000200-200-200-222-2    orthogonal lifted from D4
ρ192-2-220000-220-1000-3--31--3-3--311-3-1-11-3--3-1    complex lifted from C3⋊D4
ρ2022-2-200-22000-1000111-1-1-3--3-3--3--3-3-1-3--31    complex lifted from C3⋊D4
ρ212-2-2200002-20-1000-3--31--3-3-3-1-1--3111--3-3-1    complex lifted from C3⋊D4
ρ222-22-2-2200000-1000-3--3-1-3--3-1-3--3-1--3-31111    complex lifted from C3⋊D4
ρ2322-2-2002-2000-1000-1-1111-3-3--3--3-3--3-1-3--31    complex lifted from C3⋊D4
ρ242-22-2-2200000-1000--3-3-1--3-3-1--3-3-1-3--31111    complex lifted from C3⋊D4
ρ252-22-22-200000-1000-3--3-1-3--31--3-31-3--31-1-11    complex lifted from C3⋊D4
ρ2622-2-2002-2000-1000-1-1111--3--3-3-3--3-3-1--3-31    complex lifted from C3⋊D4
ρ272-22-22-200000-1000--3-3-1--3-31-3--31--3-31-1-11    complex lifted from C3⋊D4
ρ282-2-2200002-20-1000--3-31-3--3--3-1-1-3111-3--3-1    complex lifted from C3⋊D4
ρ2922-2-200-22000-1000111-1-1--3-3--3-3-3--3-1--3-31    complex lifted from C3⋊D4
ρ302-2-220000-220-1000--3-31-3--3-311--3-1-11--3-3-1    complex lifted from C3⋊D4

Permutation representations of C244S3
On 24 points - transitive group 24T116
Generators in S24
(13 16)(14 17)(15 18)(19 22)(20 23)(21 24)
(1 4)(2 5)(3 6)(7 10)(8 11)(9 12)(13 22)(14 23)(15 24)(16 19)(17 20)(18 21)
(1 7)(2 8)(3 9)(4 10)(5 11)(6 12)(13 19)(14 20)(15 21)(16 22)(17 23)(18 24)
(1 4)(2 5)(3 6)(7 10)(8 11)(9 12)(13 16)(14 17)(15 18)(19 22)(20 23)(21 24)
(1 2 3)(4 5 6)(7 8 9)(10 11 12)(13 14 15)(16 17 18)(19 20 21)(22 23 24)
(1 14)(2 13)(3 15)(4 17)(5 16)(6 18)(7 20)(8 19)(9 21)(10 23)(11 22)(12 24)

G:=sub<Sym(24)| (13,16)(14,17)(15,18)(19,22)(20,23)(21,24), (1,4)(2,5)(3,6)(7,10)(8,11)(9,12)(13,22)(14,23)(15,24)(16,19)(17,20)(18,21), (1,7)(2,8)(3,9)(4,10)(5,11)(6,12)(13,19)(14,20)(15,21)(16,22)(17,23)(18,24), (1,4)(2,5)(3,6)(7,10)(8,11)(9,12)(13,16)(14,17)(15,18)(19,22)(20,23)(21,24), (1,2,3)(4,5,6)(7,8,9)(10,11,12)(13,14,15)(16,17,18)(19,20,21)(22,23,24), (1,14)(2,13)(3,15)(4,17)(5,16)(6,18)(7,20)(8,19)(9,21)(10,23)(11,22)(12,24)>;

G:=Group( (13,16)(14,17)(15,18)(19,22)(20,23)(21,24), (1,4)(2,5)(3,6)(7,10)(8,11)(9,12)(13,22)(14,23)(15,24)(16,19)(17,20)(18,21), (1,7)(2,8)(3,9)(4,10)(5,11)(6,12)(13,19)(14,20)(15,21)(16,22)(17,23)(18,24), (1,4)(2,5)(3,6)(7,10)(8,11)(9,12)(13,16)(14,17)(15,18)(19,22)(20,23)(21,24), (1,2,3)(4,5,6)(7,8,9)(10,11,12)(13,14,15)(16,17,18)(19,20,21)(22,23,24), (1,14)(2,13)(3,15)(4,17)(5,16)(6,18)(7,20)(8,19)(9,21)(10,23)(11,22)(12,24) );

G=PermutationGroup([(13,16),(14,17),(15,18),(19,22),(20,23),(21,24)], [(1,4),(2,5),(3,6),(7,10),(8,11),(9,12),(13,22),(14,23),(15,24),(16,19),(17,20),(18,21)], [(1,7),(2,8),(3,9),(4,10),(5,11),(6,12),(13,19),(14,20),(15,21),(16,22),(17,23),(18,24)], [(1,4),(2,5),(3,6),(7,10),(8,11),(9,12),(13,16),(14,17),(15,18),(19,22),(20,23),(21,24)], [(1,2,3),(4,5,6),(7,8,9),(10,11,12),(13,14,15),(16,17,18),(19,20,21),(22,23,24)], [(1,14),(2,13),(3,15),(4,17),(5,16),(6,18),(7,20),(8,19),(9,21),(10,23),(11,22),(12,24)])

G:=TransitiveGroup(24,116);

C244S3 is a maximal subgroup of
C246D6  C24.38D6  C24.41D6  C24.42D6  C24.67D6  S3×C22≀C2  C247D6  C24.45D6  C24.46D6  C249D6  C24.47D6  C24.83D6  D4×C3⋊D4  C2412D6  C24.52D6  C24.53D6  C244D9  C23.D18  (C22×S3)⋊A4  C624D4  C625D4  C6224D4  C24⋊D9  (C2×C6)⋊4S4  (C2×C6)⋊S4  (C2×C30)⋊D4  (C2×C6)⋊8D20  C245D15  C24⋊D15
C244S3 is a maximal quotient of
C24.73D6  C24.76D6  (C2×C6)⋊8D8  (C3×D4).31D4  C24.31D6  C24.32D6  (C3×Q8)⋊13D4  (C2×C6)⋊8Q16  C22.52(S3×Q8)  (C22×Q8)⋊9S3  (C3×D4)⋊14D4  (C3×D4).32D4  2+ 1+46S3  2+ 1+4.4S3  2+ 1+4.5S3  2+ 1+47S3  2- 1+44S3  2- 1+4.2S3  C25.4S3  C244D9  C624D4  C625D4  C6224D4  (C2×C30)⋊D4  (C2×C6)⋊8D20  C245D15

Matrix representation of C244S3 in GL4(𝔽13) generated by

12000
01200
00120
0001
,
1000
01200
00120
00012
,
12000
01200
0010
0001
,
1000
0100
00120
00012
,
9000
0300
0030
0009
,
0100
1000
0001
0010
G:=sub<GL(4,GF(13))| [12,0,0,0,0,12,0,0,0,0,12,0,0,0,0,1],[1,0,0,0,0,12,0,0,0,0,12,0,0,0,0,12],[12,0,0,0,0,12,0,0,0,0,1,0,0,0,0,1],[1,0,0,0,0,1,0,0,0,0,12,0,0,0,0,12],[9,0,0,0,0,3,0,0,0,0,3,0,0,0,0,9],[0,1,0,0,1,0,0,0,0,0,0,1,0,0,1,0] >;

C244S3 in GAP, Magma, Sage, TeX

C_2^4\rtimes_4S_3
% in TeX

G:=Group("C2^4:4S3");
// GroupNames label

G:=SmallGroup(96,160);
// by ID

G=gap.SmallGroup(96,160);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-2,-3,217,218,2309]);
// Polycyclic

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

Export

Character table of C244S3 in TeX

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