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G = C333Q16order 432 = 24·33

3rd semidirect product of C33 and Q16 acting via Q16/C2=D4

non-abelian, soluble, monomial

Aliases: C333Q16, C322Dic12, C6.11S3≀C2, (C3×C6).12D12, C31(C32⋊Q16), C3⋊Dic3.16D6, C322C8.2S3, (C32×C6).17D4, C335Q8.1C2, C2.5(C322D12), (C3×C322C8).2C2, (C3×C3⋊Dic3).3C22, SmallGroup(432,590)

Series: Derived Chief Lower central Upper central

C1C32C3×C3⋊Dic3 — C333Q16
C1C3C33C32×C6C3×C3⋊Dic3C335Q8 — C333Q16
C33C32×C6C3×C3⋊Dic3 — C333Q16
C1C2

Generators and relations for C333Q16
 G = < a,b,c,d,e | a3=b3=c3=d8=1, e2=d4, ab=ba, ac=ca, dad-1=eae-1=b, bc=cb, dbd-1=a-1, ebe-1=a, cd=dc, ece-1=c-1, ede-1=d-1 >

Subgroups: 464 in 72 conjugacy classes, 15 normal (13 characteristic)
C1, C2, C3, C3 [×4], C4 [×3], C6, C6 [×4], C8, Q8 [×2], C32, C32 [×4], Dic3 [×8], C12 [×3], Q16, C3×C6, C3×C6 [×4], C24, Dic6 [×4], C33, C3×Dic3 [×8], C3⋊Dic3, C3⋊Dic3 [×2], Dic12, C32×C6, C322C8, C322Q8 [×4], C3×C3⋊Dic3, C3×C3⋊Dic3 [×2], C32⋊Q16, C3×C322C8, C335Q8 [×2], C333Q16
Quotients: C1, C2 [×3], C22, S3, D4, D6, Q16, D12, Dic12, S3≀C2, C32⋊Q16, C322D12, C333Q16

Character table of C333Q16

 class 123A3B3C3D3E4A4B4C6A6B6C6D6E8A8B12A12B12C12D12E12F24A24B24C24D
 size 112448818363624488181818183636363618181818
ρ1111111111111111111111111111    trivial
ρ2111111111-111111-1-1111-1-11-1-1-1-1    linear of order 2
ρ311111111-1-1111111111-1-1-1-11111    linear of order 2
ρ411111111-1111111-1-111-111-1-1-1-1-1    linear of order 2
ρ522-122-1-1200-122-1-122-1-10000-1-1-1-1    orthogonal lifted from S3
ρ622-122-1-1200-122-1-1-2-2-1-100001111    orthogonal lifted from D6
ρ72222222-2002222200-2-200000000    orthogonal lifted from D4
ρ822-122-1-1-200-122-1-100110000-33-33    orthogonal lifted from D12
ρ922-122-1-1-200-122-1-1001100003-33-3    orthogonal lifted from D12
ρ102-222222000-2-2-2-2-22-2000000-2-222    symplectic lifted from Q16, Schur index 2
ρ112-222222000-2-2-2-2-2-2200000022-2-2    symplectic lifted from Q16, Schur index 2
ρ122-2-122-1-10001-2-211-223-30000ζ87ζ385ζ385ζ83ζ3838ζ3ζ87ζ328785ζ32ζ83ζ328ζ328    symplectic lifted from Dic12, Schur index 2
ρ132-2-122-1-10001-2-2112-2-330000ζ83ζ328ζ328ζ87ζ328785ζ32ζ83ζ3838ζ3ζ87ζ385ζ385    symplectic lifted from Dic12, Schur index 2
ρ142-2-122-1-10001-2-211-22-330000ζ83ζ3838ζ3ζ87ζ385ζ385ζ83ζ328ζ328ζ87ζ328785ζ32    symplectic lifted from Dic12, Schur index 2
ρ152-2-122-1-10001-2-2112-23-30000ζ87ζ328785ζ32ζ83ζ328ζ328ζ87ζ385ζ385ζ83ζ3838ζ3    symplectic lifted from Dic12, Schur index 2
ρ164441-21-200-241-21-2000001100000    orthogonal lifted from S3≀C2
ρ17444-21-210-204-21-21000010010000    orthogonal lifted from S3≀C2
ρ184441-21-200241-21-200000-1-100000    orthogonal lifted from S3≀C2
ρ19444-21-210204-21-210000-100-10000    orthogonal lifted from S3≀C2
ρ204-44-21-21000-42-12-10000-30030000    symplectic lifted from C32⋊Q16, Schur index 2
ρ214-44-21-21000-42-12-10000300-30000    symplectic lifted from C32⋊Q16, Schur index 2
ρ224-441-21-2000-4-12-1200000-3300000    symplectic lifted from C32⋊Q16, Schur index 2
ρ234-441-21-2000-4-12-12000003-300000    symplectic lifted from C32⋊Q16, Schur index 2
ρ2488-4-422-1000-4-422-1000000000000    orthogonal lifted from C322D12
ρ2588-42-4-12000-42-4-12000000000000    orthogonal lifted from C322D12
ρ268-8-42-4-120004-241-2000000000000    symplectic faithful, Schur index 2
ρ278-8-4-422-100044-2-21000000000000    symplectic faithful, Schur index 2

Smallest permutation representation of C333Q16
On 48 points
Generators in S48
(1 34 25)(3 27 36)(5 38 29)(7 31 40)(10 23 41)(12 43 17)(14 19 45)(16 47 21)
(2 26 35)(4 37 28)(6 30 39)(8 33 32)(9 22 48)(11 42 24)(13 18 44)(15 46 20)
(1 25 34)(2 26 35)(3 27 36)(4 28 37)(5 29 38)(6 30 39)(7 31 40)(8 32 33)(9 48 22)(10 41 23)(11 42 24)(12 43 17)(13 44 18)(14 45 19)(15 46 20)(16 47 21)
(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 15 5 11)(2 14 6 10)(3 13 7 9)(4 12 8 16)(17 32 21 28)(18 31 22 27)(19 30 23 26)(20 29 24 25)(33 47 37 43)(34 46 38 42)(35 45 39 41)(36 44 40 48)

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

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

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

Matrix representation of C333Q16 in GL6(𝔽73)

100000
010000
0000072
0000172
000101
00721721
,
100000
010000
0000721
0000720
00172172
001010
,
0720000
1720000
001000
000100
000010
000001
,
23680000
5180000
0007200
000001
001000
0000720
,
18530000
71550000
000100
001000
000001
000010

G:=sub<GL(6,GF(73))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,72,0,0,0,0,1,1,0,0,0,1,0,72,0,0,72,72,1,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,1,0,0,0,0,72,0,0,0,72,72,1,1,0,0,1,0,72,0],[0,1,0,0,0,0,72,72,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[23,5,0,0,0,0,68,18,0,0,0,0,0,0,0,0,1,0,0,0,72,0,0,0,0,0,0,0,0,72,0,0,0,1,0,0],[18,71,0,0,0,0,53,55,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0] >;

C333Q16 in GAP, Magma, Sage, TeX

C_3^3\rtimes_3Q_{16}
% in TeX

G:=Group("C3^3:3Q16");
// GroupNames label

G:=SmallGroup(432,590);
// by ID

G=gap.SmallGroup(432,590);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-3,3,-3,56,85,92,254,58,1684,1691,298,677,348,1027,14118]);
// Polycyclic

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

Export

Character table of C333Q16 in TeX

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