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G = C3⋊S3⋊Q16order 288 = 25·32

The semidirect product of C3⋊S3 and Q16 acting via Q16/C4=C22

non-abelian, soluble, monomial

Aliases: C3⋊S3⋊Q16, C4.8S3≀C2, C32⋊Q162C2, C321(C2×Q16), (C3×C12).15D4, C3⋊Dic3.6C23, Dic3.D6.6C2, C322C8.7C22, C322Q8.2C22, (C3×C6).9(C2×D4), C2.12(C2×S3≀C2), (C2×C3⋊S3).33D4, C3⋊S33C8.2C2, (C4×C3⋊S3).34C22, SmallGroup(288,876)

Series: Derived Chief Lower central Upper central

C1C32C3⋊Dic3 — C3⋊S3⋊Q16
C1C32C3×C6C3⋊Dic3C322Q8C32⋊Q16 — C3⋊S3⋊Q16
C32C3×C6C3⋊Dic3 — C3⋊S3⋊Q16
C1C2C4

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

Subgroups: 464 in 98 conjugacy classes, 25 normal (11 characteristic)
C1, C2, C2 [×2], C3 [×2], C4, C4 [×5], C22, S3 [×4], C6 [×2], C8 [×2], C2×C4 [×3], Q8 [×6], C32, Dic3 [×6], C12 [×6], D6 [×2], C2×C8, Q16 [×4], C2×Q8 [×2], C3⋊S3 [×2], C3×C6, Dic6 [×6], C4×S3 [×6], C3×Q8 [×2], C2×Q16, C3×Dic3 [×4], C3⋊Dic3, C3×C12, C2×C3⋊S3, S3×Q8 [×2], C322C8 [×2], C6.D6 [×2], C322Q8 [×4], C3×Dic6 [×2], C4×C3⋊S3, C32⋊Q16 [×4], C3⋊S33C8, Dic3.D6 [×2], C3⋊S3⋊Q16
Quotients: C1, C2 [×7], C22 [×7], D4 [×2], C23, Q16 [×2], C2×D4, C2×Q16, S3≀C2, C2×S3≀C2, C3⋊S3⋊Q16

Character table of C3⋊S3⋊Q16

 class 12A2B2C3A3B4A4B4C4D4E4F6A6B8A8B8C8D12A12B12C12D12E12F
 size 1199442121212121844181818188824242424
ρ1111111111111111111111111    trivial
ρ2111111111-1-1111-1-1-1-11111-1-1    linear of order 2
ρ311-1-111-11-1-111111-11-1-1-11-11-1    linear of order 2
ρ411-1-111-11-11-1111-11-11-1-11-1-11    linear of order 2
ρ511-1-111-1-111-11111-11-1-1-1-11-11    linear of order 2
ρ611-1-111-1-11-11111-11-11-1-1-111-1    linear of order 2
ρ71111111-1-1-1-1111111111-1-1-1-1    linear of order 2
ρ81111111-1-111111-1-1-1-111-1-111    linear of order 2
ρ922-2-22220000-2220000220000    orthogonal lifted from D4
ρ10222222-20000-2220000-2-20000    orthogonal lifted from D4
ρ112-2-2222000000-2-2-222-2000000    symplectic lifted from Q16, Schur index 2
ρ122-22-222000000-2-222-2-2000000    symplectic lifted from Q16, Schur index 2
ρ132-2-2222000000-2-22-2-22000000    symplectic lifted from Q16, Schur index 2
ρ142-22-222000000-2-2-2-222000000    symplectic lifted from Q16, Schur index 2
ρ154400-21-400-2201-200002-100-11    orthogonal lifted from C2×S3≀C2
ρ164400-214002201-20000-2100-1-1    orthogonal lifted from S3≀C2
ρ1744001-24-2-2000-2100001-21100    orthogonal lifted from S3≀C2
ρ1844001-2-4-22000-210000-121-100    orthogonal lifted from C2×S3≀C2
ρ1944001-2-42-2000-210000-12-1100    orthogonal lifted from C2×S3≀C2
ρ2044001-2422000-2100001-2-1-100    orthogonal lifted from S3≀C2
ρ214400-21-4002-201-200002-1001-1    orthogonal lifted from C2×S3≀C2
ρ224400-21400-2-201-20000-210011    orthogonal lifted from S3≀C2
ρ238-800-42000000-240000000000    symplectic faithful, Schur index 2
ρ248-8002-40000004-20000000000    symplectic faithful, Schur index 2

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

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

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

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

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

100000
010000
0072100
0072000
000010
000001
,
100000
010000
001000
000100
0000072
0000172
,
100000
010000
000100
001000
000001
000010
,
0380000
48410000
0000072
0000720
0072000
0007200
,
24700000
22490000
000010
000001
001000
000100

G:=sub<GL(6,GF(73))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,72,72,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,72,72],[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],[0,48,0,0,0,0,38,41,0,0,0,0,0,0,0,0,72,0,0,0,0,0,0,72,0,0,0,72,0,0,0,0,72,0,0,0],[24,22,0,0,0,0,70,49,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,0,1,0,0] >;

C3⋊S3⋊Q16 in GAP, Magma, Sage, TeX

C_3\rtimes S_3\rtimes Q_{16}
% in TeX

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

G:=SmallGroup(288,876);
// by ID

G=gap.SmallGroup(288,876);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,3,112,141,120,422,219,100,675,346,80,2693,2028,362,797,1203]);
// 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=e*b*e^-1=a^-1,d*a*d^-1=b,e*a*e^-1=c*b*c=b^-1,c*d=d*c,c*e=e*c,e*d*e^-1=d^-1>;
// generators/relations

Export

Character table of C3⋊S3⋊Q16 in TeX

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