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G = C33⋊Q16order 432 = 24·33

2nd semidirect product of C33 and Q16 acting via Q16/C2=D4

non-abelian, soluble, monomial

Aliases: C332Q16, C6.18S3≀C2, C334C8.C2, C322Q8.S3, C32(C32⋊Q16), C3⋊Dic3.13D6, (C32×C6).12D4, C335Q8.2C2, C323(C3⋊Q16), C2.7(C33⋊D4), (C3×C6).18(C3⋊D4), (C3×C322Q8).1C2, (C3×C3⋊Dic3).10C22, SmallGroup(432,585)

Series: Derived Chief Lower central Upper central

C1C32C3×C3⋊Dic3 — C33⋊Q16
C1C3C33C32×C6C3×C3⋊Dic3C335Q8 — C33⋊Q16
C33C32×C6C3×C3⋊Dic3 — C33⋊Q16
C1C2

Generators and relations for C33⋊Q16
 G = < a,b,c,d,e | a3=b3=c3=d8=1, e2=d4, ab=ba, ac=ca, dad-1=b-1, eae-1=a-1, bc=cb, dbd-1=a, be=eb, dcd-1=c-1, ce=ec, ede-1=d-1 >

Subgroups: 412 in 72 conjugacy classes, 15 normal (all characteristic)
C1, C2, C3, C3 [×4], C4 [×3], C6, C6 [×4], C8, Q8 [×2], C32, C32 [×4], Dic3 [×6], C12 [×5], Q16, C3×C6, C3×C6 [×4], C3⋊C8, Dic6 [×3], C3×Q8, C33, C3×Dic3 [×8], C3⋊Dic3, C3⋊Dic3, C3×C12, C3⋊Q16, C32×C6, C322C8, C322Q8, C322Q8 [×2], C3×Dic6, C32×Dic3, C3×C3⋊Dic3, C3×C3⋊Dic3, C32⋊Q16, C334C8, C3×C322Q8, C335Q8, C33⋊Q16
Quotients: C1, C2 [×3], C22, S3, D4, D6, Q16, C3⋊D4, C3⋊Q16, S3≀C2, C32⋊Q16, C33⋊D4, C33⋊Q16

Character table of C33⋊Q16

 class 123A3B3C3D3E3F4A4B4C6A6B6C6D6E6F8A8B12A12B12C12D12E12F12G12H12I12J12K
 size 1124444812183624444854541212121212121212363636
ρ1111111111111111111111111111111    trivial
ρ211111111-11-111111111-1-1-1-1-1-1-1-1-11-1    linear of order 2
ρ31111111111-1111111-1-111111111-11-1    linear of order 2
ρ411111111-111111111-1-1-1-1-1-1-1-1-1-1111    linear of order 2
ρ5222222220-2022222200000000000-20    orthogonal lifted from D4
ρ622-12-12-1-1220-1-122-1-1002-1-12-1-1-1-10-10    orthogonal lifted from S3
ρ722-12-12-1-1-220-1-122-1-100-211-211110-10    orthogonal lifted from D6
ρ82-2222222000-2-2-2-2-2-22-200000000000    symplectic lifted from Q16, Schur index 2
ρ92-2222222000-2-2-2-2-2-2-2200000000000    symplectic lifted from Q16, Schur index 2
ρ1022-12-12-1-10-20-1-122-1-1000--3-30--3-3--3-3010    complex lifted from C3⋊D4
ρ1122-12-12-1-10-20-1-122-1-1000-3--30-3--3-3--3010    complex lifted from C3⋊D4
ρ12444-2111-2-20041-211-200111111-2-2000    orthogonal lifted from S3≀C2
ρ13444-2111-220041-211-200-1-1-1-1-1-122000    orthogonal lifted from S3≀C2
ρ144441-2-2-210024-21-2-210000000000-10-1    orthogonal lifted from S3≀C2
ρ154441-2-2-2100-24-21-2-210000000000101    orthogonal lifted from S3≀C2
ρ164-4-24-24-2-200022-4-4220000000000000    symplectic lifted from C3⋊Q16, Schur index 2
ρ174-44-2111-2000-4-12-1-1200333-3-3-300000    symplectic lifted from C32⋊Q16, Schur index 2
ρ184-44-2111-2000-4-12-1-1200-3-3-333300000    symplectic lifted from C32⋊Q16, Schur index 2
ρ194-441-2-2-21000-42-122-10000000000-303    symplectic lifted from C32⋊Q16, Schur index 2
ρ204-441-2-2-21000-42-122-1000000000030-3    symplectic lifted from C32⋊Q16, Schur index 2
ρ2144-2-2-1-3-3/21-1+3-3/21-200-2-1-3-3/2-21-1+3-3/21001ζ3ζ321ζ3ζ321--31+-3000    complex lifted from C33⋊D4
ρ2244-2-2-1+3-3/21-1-3-3/21-200-2-1+3-3/2-21-1-3-3/21001ζ32ζ31ζ32ζ31+-31--3000    complex lifted from C33⋊D4
ρ234-4-2-2-1+3-3/21-1-3-3/2100021-3-3/22-11+3-3/2-100-3ζ4ζ32+2ζ4ζ43ζ3+2ζ433ζ43ζ32+2ζ43ζ4ζ3+2ζ400000    complex faithful
ρ2444-2-2-1-3-3/21-1+3-3/21200-2-1-3-3/2-21-1+3-3/2100-1ζ65ζ6-1ζ65ζ6-1+-3-1--3000    complex lifted from C33⋊D4
ρ254-4-2-2-1-3-3/21-1+3-3/2100021+3-3/22-11-3-3/2-100-3ζ43ζ3+2ζ43ζ4ζ32+2ζ43ζ4ζ3+2ζ4ζ43ζ32+2ζ4300000    complex faithful
ρ2644-2-2-1+3-3/21-1-3-3/21200-2-1+3-3/2-21-1-3-3/2100-1ζ6ζ65-1ζ6ζ65-1--3-1+-3000    complex lifted from C33⋊D4
ρ274-4-2-2-1+3-3/21-1-3-3/2100021-3-3/22-11+3-3/2-1003ζ43ζ32+2ζ43ζ4ζ3+2ζ4-3ζ4ζ32+2ζ4ζ43ζ3+2ζ4300000    complex faithful
ρ284-4-2-2-1-3-3/21-1+3-3/2100021+3-3/22-11-3-3/2-1003ζ4ζ3+2ζ4ζ43ζ32+2ζ43-3ζ43ζ3+2ζ43ζ4ζ32+2ζ400000    complex faithful
ρ2988-422-42-1000-422-42-10000000000000    orthogonal lifted from C33⋊D4
ρ308-8-422-42-10004-2-24-210000000000000    symplectic faithful, Schur index 2

Smallest permutation representation of C33⋊Q16
On 48 points
Generators in S48
(1 11 46)(2 47 12)(3 48 13)(4 14 41)(5 15 42)(6 43 16)(7 44 9)(8 10 45)(17 34 28)(18 35 29)(19 30 36)(20 31 37)(21 38 32)(22 39 25)(23 26 40)(24 27 33)
(1 11 46)(2 12 47)(3 48 13)(4 41 14)(5 15 42)(6 16 43)(7 44 9)(8 45 10)(17 28 34)(18 35 29)(19 36 30)(20 31 37)(21 32 38)(22 39 25)(23 40 26)(24 27 33)
(1 46 11)(2 12 47)(3 48 13)(4 14 41)(5 42 15)(6 16 43)(7 44 9)(8 10 45)(17 28 34)(18 35 29)(19 30 36)(20 37 31)(21 32 38)(22 39 25)(23 26 40)(24 33 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 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)| (1,11,46)(2,47,12)(3,48,13)(4,14,41)(5,15,42)(6,43,16)(7,44,9)(8,10,45)(17,34,28)(18,35,29)(19,30,36)(20,31,37)(21,38,32)(22,39,25)(23,26,40)(24,27,33), (1,11,46)(2,12,47)(3,48,13)(4,41,14)(5,15,42)(6,16,43)(7,44,9)(8,45,10)(17,28,34)(18,35,29)(19,36,30)(20,31,37)(21,32,38)(22,39,25)(23,40,26)(24,27,33), (1,46,11)(2,12,47)(3,48,13)(4,14,41)(5,42,15)(6,16,43)(7,44,9)(8,10,45)(17,28,34)(18,35,29)(19,30,36)(20,37,31)(21,32,38)(22,39,25)(23,26,40)(24,33,27), (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), (1,23,5,19)(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( (1,11,46)(2,47,12)(3,48,13)(4,14,41)(5,15,42)(6,43,16)(7,44,9)(8,10,45)(17,34,28)(18,35,29)(19,30,36)(20,31,37)(21,38,32)(22,39,25)(23,26,40)(24,27,33), (1,11,46)(2,12,47)(3,48,13)(4,41,14)(5,15,42)(6,16,43)(7,44,9)(8,45,10)(17,28,34)(18,35,29)(19,36,30)(20,31,37)(21,32,38)(22,39,25)(23,40,26)(24,27,33), (1,46,11)(2,12,47)(3,48,13)(4,14,41)(5,42,15)(6,16,43)(7,44,9)(8,10,45)(17,28,34)(18,35,29)(19,30,36)(20,37,31)(21,32,38)(22,39,25)(23,26,40)(24,33,27), (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), (1,23,5,19)(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([(1,11,46),(2,47,12),(3,48,13),(4,14,41),(5,15,42),(6,43,16),(7,44,9),(8,10,45),(17,34,28),(18,35,29),(19,30,36),(20,31,37),(21,38,32),(22,39,25),(23,26,40),(24,27,33)], [(1,11,46),(2,12,47),(3,48,13),(4,41,14),(5,15,42),(6,16,43),(7,44,9),(8,45,10),(17,28,34),(18,35,29),(19,36,30),(20,31,37),(21,32,38),(22,39,25),(23,40,26),(24,27,33)], [(1,46,11),(2,12,47),(3,48,13),(4,14,41),(5,42,15),(6,16,43),(7,44,9),(8,10,45),(17,28,34),(18,35,29),(19,30,36),(20,37,31),(21,32,38),(22,39,25),(23,26,40),(24,33,27)], [(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)], [(1,23,5,19),(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 C33⋊Q16 in GL8(𝔽73)

10000000
01000000
00100000
00010000
000000072
000000720
000001720
000010072
,
10000000
01000000
00100000
00010000
000000072
000007210
000007200
000010072
,
072000000
172000000
000720000
001720000
00001000
00000100
00000010
00000001
,
000410000
004100000
0160410000
1604100000
000007200
000000072
000072000
000000720
,
4101300000
0410130000
1103200000
0110320000
00000010
00000001
00001000
00000100

G:=sub<GL(8,GF(73))| [1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,72,72,0,0,0,0,0,72,0,0,72],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,72,72,0,0,0,0,0,0,1,0,0,0,0,0,0,72,0,0,72],[0,1,0,0,0,0,0,0,72,72,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,72,72,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,41,0,41,0,0,0,0,41,0,41,0,0,0,0,0,0,0,0,0,0,0,72,0,0,0,0,0,72,0,0,0,0,0,0,0,0,0,0,72,0,0,0,0,0,72,0,0],[41,0,11,0,0,0,0,0,0,41,0,11,0,0,0,0,13,0,32,0,0,0,0,0,0,13,0,32,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0] >;

C33⋊Q16 in GAP, Magma, Sage, TeX

C_3^3\rtimes Q_{16}
% in TeX

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

G:=SmallGroup(432,585);
// by ID

G=gap.SmallGroup(432,585);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-3,3,-3,56,85,64,254,135,58,1684,571,298,677,1027,14118]);
// 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=b^-1,e*a*e^-1=a^-1,b*c=c*b,d*b*d^-1=a,b*e=e*b,d*c*d^-1=c^-1,c*e=e*c,e*d*e^-1=d^-1>;
// generators/relations

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Character table of C33⋊Q16 in TeX

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