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G = C3⋊S3.2Q16order 288 = 25·32

1st non-split extension by C3⋊S3 of Q16 acting via Q16/C4=C22

non-abelian, soluble, monomial

Aliases: C4.12S3≀C2, (C3×C12).4D4, C3⋊S3.2Q16, C322Q81C4, C3⋊S3.3SD16, C322(Q8⋊C4), Dic3.D6.2C2, C12.29D6.2C2, C2.6(S32⋊C4), (C2×C3⋊S3).6D4, C4⋊(C32⋊C4).1C2, (C4×C3⋊S3).4C22, C3⋊Dic3.6(C2×C4), (C3×C6).5(C22⋊C4), SmallGroup(288,378)

Series: Derived Chief Lower central Upper central

C1C32C3⋊Dic3 — C3⋊S3.2Q16
C1C32C3×C6C3⋊Dic3C4×C3⋊S3Dic3.D6 — C3⋊S3.2Q16
C32C3×C6C3⋊Dic3 — C3⋊S3.2Q16
C1C2C4

Generators and relations for C3⋊S3.2Q16
 G = < a,b,c,d,e | a3=b3=c2=d8=1, e2=d4, ab=ba, cac=dbd-1=a-1, dad-1=cbc=ebe-1=b-1, ae=ea, cd=dc, ce=ec, ede-1=cd-1 >

Subgroups: 368 in 72 conjugacy classes, 17 normal (15 characteristic)
C1, C2, C2 [×2], C3 [×2], C4, C4 [×4], C22, S3 [×4], C6 [×2], C8, C2×C4 [×3], Q8 [×3], C32, Dic3 [×4], C12 [×4], D6 [×2], C4⋊C4, C2×C8, C2×Q8, C3⋊S3 [×2], C3×C6, C3⋊C8, C24, Dic6 [×3], C4×S3 [×4], C3×Q8, Q8⋊C4, C3×Dic3 [×2], C3⋊Dic3, C3×C12, C32⋊C4, C2×C3⋊S3, S3×C8, S3×Q8, C3×C3⋊C8, C6.D6, C322Q8 [×2], C3×Dic6, C4×C3⋊S3, C2×C32⋊C4, C12.29D6, C4⋊(C32⋊C4), Dic3.D6, C3⋊S3.2Q16
Quotients: C1, C2 [×3], C4 [×2], C22, C2×C4, D4 [×2], C22⋊C4, SD16, Q16, Q8⋊C4, S3≀C2, S32⋊C4, C3⋊S3.2Q16

Character table of C3⋊S3.2Q16

 class 12A2B2C3A3B4A4B4C4D4E4F6A6B8A8B8C8D12A12B12C12D12E24A24B24C24D
 size 11994421212183636446666448242412121212
ρ1111111111111111111111111111    trivial
ρ21111111-1-111111-1-1-1-1111-1-1-1-1-1-1    linear of order 2
ρ31111111111-1-111-1-1-1-111111-1-1-1-1    linear of order 2
ρ41111111-1-11-1-1111111111-1-11111    linear of order 2
ρ511-1-111-1-111-ii11-i-iii-1-1-11-1-i-iii    linear of order 4
ρ611-1-111-11-11-ii11ii-i-i-1-1-1-11ii-i-i    linear of order 4
ρ711-1-111-1-111i-i11ii-i-i-1-1-11-1ii-i-i    linear of order 4
ρ811-1-111-11-11i-i11-i-iii-1-1-1-11-i-iii    linear of order 4
ρ9222222-200-200220000-2-2-2000000    orthogonal lifted from D4
ρ1022-2-222200-200220000222000000    orthogonal lifted from D4
ρ112-22-222000000-2-22-22-200000-22-22    symplectic lifted from Q16, Schur index 2
ρ122-22-222000000-2-2-22-22000002-22-2    symplectic lifted from Q16, Schur index 2
ρ132-2-2222000000-2-2--2-2-2--200000-2--2--2-2    complex lifted from SD16
ρ142-2-2222000000-2-2-2--2--2-200000--2-2-2--2    complex lifted from SD16
ρ154400-214-2-20001-20000-2-21110000    orthogonal lifted from S3≀C2
ρ164400-214220001-20000-2-21-1-10000    orthogonal lifted from S3≀C2
ρ174400-21-42-20001-2000022-11-10000    orthogonal lifted from S32⋊C4
ρ1844001-2400000-21222211-200-1-1-1-1    orthogonal lifted from S3≀C2
ρ1944001-2400000-21-2-2-2-211-2001111    orthogonal lifted from S3≀C2
ρ204400-21-4-220001-2000022-1-110000    orthogonal lifted from S32⋊C4
ρ2144001-2-400000-212i2i-2i-2i-1-1200-i-iii    complex lifted from S32⋊C4
ρ2244001-2-400000-21-2i-2i2i2i-1-1200ii-i-i    complex lifted from S32⋊C4
ρ234-4001-20000002-18387858-3i3i000ζ83ζ87ζ85ζ8    complex faithful
ρ244-4001-20000002-188587833i-3i000ζ8ζ85ζ87ζ83    complex faithful
ρ254-4001-20000002-18783885-3i3i000ζ87ζ83ζ8ζ85    complex faithful
ρ264-4001-20000002-185883873i-3i000ζ85ζ8ζ83ζ87    complex faithful
ρ278-800-42000000-240000000000000    symplectic faithful, Schur index 2

Smallest permutation representation of C3⋊S3.2Q16
On 48 points
Generators in S48
(1 28 38)(2 29 39)(3 30 40)(4 31 33)(5 32 34)(6 25 35)(7 26 36)(8 27 37)(9 24 48)(10 41 17)(11 18 42)(12 43 19)(13 20 44)(14 45 21)(15 22 46)(16 47 23)
(1 38 28)(2 39 29)(3 40 30)(4 33 31)(5 34 32)(6 35 25)(7 36 26)(8 37 27)(9 24 48)(10 41 17)(11 18 42)(12 43 19)(13 20 44)(14 45 21)(15 22 46)(16 47 23)
(1 5)(2 6)(3 7)(4 8)(9 13)(10 14)(11 15)(12 16)(17 45)(18 46)(19 47)(20 48)(21 41)(22 42)(23 43)(24 44)(25 39)(26 40)(27 33)(28 34)(29 35)(30 36)(31 37)(32 38)
(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 16 5 12)(2 11 6 15)(3 14 7 10)(4 9 8 13)(17 40 21 36)(18 25 22 29)(19 38 23 34)(20 31 24 27)(26 41 30 45)(28 47 32 43)(33 48 37 44)(35 46 39 42)

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

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

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

Matrix representation of C3⋊S3.2Q16 in GL6(𝔽73)

100000
010000
0072100
0072000
0000721
0000720
,
100000
010000
0007200
0017200
0000721
0000720
,
100000
010000
000100
001000
000001
000010
,
0320000
57320000
001000
000100
000001
000010
,
12610000
6610000
000010
000001
001000
000100

G:=sub<GL(6,GF(73))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,72,72,0,0,0,0,1,0,0,0,0,0,0,0,72,72,0,0,0,0,1,0],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,72,72,0,0,0,0,0,0,72,72,0,0,0,0,1,0],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0],[0,57,0,0,0,0,32,32,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,1,0],[12,6,0,0,0,0,61,61,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,0,1,0,0] >;

C3⋊S3.2Q16 in GAP, Magma, Sage, TeX

C_3\rtimes S_3._2Q_{16}
% in TeX

G:=Group("C3:S3.2Q16");
// GroupNames label

G:=SmallGroup(288,378);
// by ID

G=gap.SmallGroup(288,378);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,3,112,85,64,422,100,675,80,2693,2028,691,797,2372]);
// Polycyclic

G:=Group<a,b,c,d,e|a^3=b^3=c^2=d^8=1,e^2=d^4,a*b=b*a,c*a*c=d*b*d^-1=a^-1,d*a*d^-1=c*b*c=e*b*e^-1=b^-1,a*e=e*a,c*d=d*c,c*e=e*c,e*d*e^-1=c*d^-1>;
// generators/relations

Export

Character table of C3⋊S3.2Q16 in TeX

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