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G = C33⋊9Q16  order 432 = 24·33

6th semidirect product of C33 and Q16 acting via Q16/C4=C22

metabelian, supersoluble, monomial

Aliases: C33⋊9Q16, C32⋊6Dic12, C12.35S32, (C3×C6).46D12, (C3×C12).126D6, C32⋊4C8.3S3, (C32×C6).49D4, C6.9(D6⋊S3), C32⋊4Q8.5S3, C3⋊3(C32⋊3Q16), C3⋊1(C32⋊2Q16), C32⋊7(C3⋊Q16), C2.5(C33⋊9D4), C6.39(C3⋊D12), C4.3(C32⋊4D6), (C32×C12).22C22, (C3×C6).67(C3⋊D4), (C3×C32⋊4C8).3C2, (C3×C32⋊4Q8).4C2, SmallGroup(432,459)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C32×C12 — C33⋊9Q16
Chief series
C1 — C3 — C32 — C33 — C32×C6 — C32×C12 — C3×C32⋊4Q8 — C33⋊9Q16
Lower central
C33 — C32×C6 — C32×C12 — C33⋊9Q16
Upper central
C1 — C2 — C4

Generators and relations for C33⋊9Q16
 G = < a,b,c,d,e | a3=b3=c3=d8=1, e2=d4, ab=ba, ac=ca, ad=da, eae-1=a-1, bc=cb, dbd-1=ebe-1=b-1, dcd-1=c-1, ce=ec, ede-1=d-1 >

Subgroups: 456 in 110 conjugacy classes, 31 normal (15 characteristic)
C1, C2, C3, C3, C3, C4, C4, C6, C6, C6, C8, Q8, C32, C32, C32, Dic3, C12, C12, C12, Q16, C3×C6, C3×C6, C3×C6, C3⋊C8, C24, Dic6, C3×Q8, C33, C3×Dic3, C3⋊Dic3, C3×C12, C3×C12, C3×C12, Dic12, C3⋊Q16, C32×C6, C3×C3⋊C8, C32⋊4C8, C3×Dic6, C32⋊4Q8, C3×C3⋊Dic3, C32×C12, C32⋊2Q16, C32⋊3Q16, C3×C32⋊4C8, C3×C32⋊4Q8, C33⋊9Q16
Quotients: C1, C2, C22, S3, D4, D6, Q16, D12, C3⋊D4, S32, Dic12, C3⋊Q16, D6⋊S3, C3⋊D12, C32⋊4D6, C32⋊2Q16, C32⋊3Q16, C33⋊9D4, C33⋊9Q16

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

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

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

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

45 conjugacy classes

class 1  2 3A3B3C3D···3H4A4B4C6A6B6C6D···6H8A8B12A12B12C···12N12O12P12Q12R24A24B24C24D
order123333···34446666···688121212···121212121224242424
size112224···4236362224···41818224···43636363618181818

45 irreducible representations

dim11122222222444444444
type+++++++-+-+--+-
imageC1C2C2S3S3D4D6Q16D12C3⋊D4Dic12S32C3⋊Q16D6⋊S3C3⋊D12C32⋊4D6C32⋊2Q16C32⋊3Q16C33⋊9D4C33⋊9Q16
kernelC33⋊9Q16C3×C32⋊4C8C3×C32⋊4Q8C32⋊4C8C32⋊4Q8C32×C6C3×C12C33C3×C6C3×C6C32C12C32C6C6C4C3C3C2C1
# reps11212132244321222424

Matrix representation of C33⋊9Q16 ►in GL8(𝔽73)

10000000
01000000
000720000
001720000
00001000
00000100
00000010
00000001
,
10000000
01000000
00100000
00010000
00001000
00000100
000000072
000000172
,
10000000
01000000
00100000
00010000
000007200
000017200
00000010
00000001
,
1059000000
022000000
007200000
000720000
0000616300
0000511200
0000001959
000000554
,
48000000
769000000
000720000
007200000
0000436000
0000133000
0000002210
0000003251

G:=sub<GL(8,GF(73))| [1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,72,72,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,72,72],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,72,72,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[10,0,0,0,0,0,0,0,59,22,0,0,0,0,0,0,0,0,72,0,0,0,0,0,0,0,0,72,0,0,0,0,0,0,0,0,61,51,0,0,0,0,0,0,63,12,0,0,0,0,0,0,0,0,19,5,0,0,0,0,0,0,59,54],[4,7,0,0,0,0,0,0,8,69,0,0,0,0,0,0,0,0,0,72,0,0,0,0,0,0,72,0,0,0,0,0,0,0,0,0,43,13,0,0,0,0,0,0,60,30,0,0,0,0,0,0,0,0,22,32,0,0,0,0,0,0,10,51] >;
 

C33⋊9Q16 in GAP, Magma, Sage, TeX 

C_3^3\rtimes_9Q_{16}
 
% in TeX
 
G:=Group("C3^3:9Q16");
 
// GroupNames label
 
G:=SmallGroup(432,459);
 
// by ID
 
G=gap.SmallGroup(432,459);
 
# by ID
 
G:=PCGroup([7,-2,-2,-2,-2,-3,-3,-3,56,85,64,254,135,58,1124,571,2028,14118]);
 
// 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,a*d=d*a,e*a*e^-1=a^-1,b*c=c*b,d*b*d^-1=e*b*e^-1=b^-1,d*c*d^-1=c^-1,c*e=e*c,e*d*e^-1=d^-1>;
 
// generators/relations
 

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