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## G = C33⋊18SD16order 432 = 24·33

### 10th semidirect product of C33 and SD16 acting via SD16/C4=C22

Series: Derived Chief Lower central Upper central

 Derived series C1 — C32×C12 — C33⋊18SD16
 Chief series C1 — C3 — C32 — C33 — C32×C6 — C32×C12 — C3×C12⋊S3 — C33⋊18SD16
 Lower central C33 — C32×C6 — C32×C12 — C33⋊18SD16
 Upper central C1 — C2 — C4

Generators and relations for C3318SD16
G = < a,b,c,d,e | a3=b3=c3=d8=e2=1, ab=ba, ac=ca, ad=da, eae=a-1, bc=cb, dbd-1=b-1, be=eb, dcd-1=ece=c-1, ede=d3 >

Subgroups: 632 in 122 conjugacy classes, 31 normal (all characteristic)
C1, C2, C2, C3, C3, C4, C4, C22, S3, C6, C6, C8, D4, Q8, C32, C32, Dic3, C12, C12, D6, C2×C6, SD16, C3×S3, C3⋊S3, C3×C6, C3×C6, C3⋊C8, C24, Dic6, D12, C3×D4, C3×Q8, C33, C3×Dic3, C3⋊Dic3, C3×C12, C3×C12, S3×C6, C2×C3⋊S3, C24⋊C2, D4.S3, Q82S3, C3×C3⋊S3, C32×C6, C3×C3⋊C8, C324C8, C3×Dic6, C3×D12, C324Q8, C12⋊S3, C3×C3⋊Dic3, C32×C12, C6×C3⋊S3, Dic6⋊S3, D12.S3, C325SD16, C3×C324C8, C3×C324Q8, C3×C12⋊S3, C3318SD16
Quotients: C1, C2, C22, S3, D4, D6, SD16, D12, C3⋊D4, S32, C24⋊C2, D4.S3, Q82S3, D6⋊S3, C3⋊D12, C324D6, Dic6⋊S3, D12.S3, C325SD16, C339D4, C3318SD16

Smallest permutation representation of C3318SD16
On 48 points
Generators in S48
(1 15 37)(2 16 38)(3 9 39)(4 10 40)(5 11 33)(6 12 34)(7 13 35)(8 14 36)(17 31 45)(18 32 46)(19 25 47)(20 26 48)(21 27 41)(22 28 42)(23 29 43)(24 30 44)
(1 15 37)(2 38 16)(3 9 39)(4 40 10)(5 11 33)(6 34 12)(7 13 35)(8 36 14)(17 45 31)(18 32 46)(19 47 25)(20 26 48)(21 41 27)(22 28 42)(23 43 29)(24 30 44)
(1 37 15)(2 16 38)(3 39 9)(4 10 40)(5 33 11)(6 12 34)(7 35 13)(8 14 36)(17 45 31)(18 32 46)(19 47 25)(20 26 48)(21 41 27)(22 28 42)(23 43 29)(24 30 44)
(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 17)(2 20)(3 23)(4 18)(5 21)(6 24)(7 19)(8 22)(9 43)(10 46)(11 41)(12 44)(13 47)(14 42)(15 45)(16 48)(25 35)(26 38)(27 33)(28 36)(29 39)(30 34)(31 37)(32 40)

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

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

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

45 conjugacy classes

 class 1 2A 2B 3A 3B 3C 3D ··· 3H 4A 4B 6A 6B 6C 6D ··· 6H 6I 6J 8A 8B 12A 12B 12C ··· 12N 12O 12P 24A 24B 24C 24D order 1 2 2 3 3 3 3 ··· 3 4 4 6 6 6 6 ··· 6 6 6 8 8 12 12 12 ··· 12 12 12 24 24 24 24 size 1 1 36 2 2 2 4 ··· 4 2 36 2 2 2 4 ··· 4 36 36 18 18 2 2 4 ··· 4 36 36 18 18 18 18

45 irreducible representations

 dim 1 1 1 1 2 2 2 2 2 2 2 2 2 4 4 4 4 4 4 4 4 4 4 4 type + + + + + + + + + + + - + - + - + image C1 C2 C2 C2 S3 S3 S3 D4 D6 SD16 D12 C3⋊D4 C24⋊C2 S32 D4.S3 Q8⋊2S3 D6⋊S3 C3⋊D12 C32⋊4D6 Dic6⋊S3 D12.S3 C32⋊5SD16 C33⋊9D4 C33⋊18SD16 kernel C33⋊18SD16 C3×C32⋊4C8 C3×C32⋊4Q8 C3×C12⋊S3 C32⋊4C8 C32⋊4Q8 C12⋊S3 C32×C6 C3×C12 C33 C3×C6 C3×C6 C32 C12 C32 C32 C6 C6 C4 C3 C3 C3 C2 C1 # reps 1 1 1 1 1 1 1 1 3 2 2 4 4 3 1 1 1 2 2 2 2 2 2 4

Matrix representation of C3318SD16 in GL8(𝔽73)

 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 72 0 0 0 0 0 0 1 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 72 72 0 0 0 0 0 0 1 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 72 0 0 0 0 0 0 1 72 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1
,
 63 31 0 0 0 0 0 0 0 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 14 5 0 0 0 0 0 0 19 59 0 0 0 0 0 0 0 0 48 11 0 0 0 0 0 0 36 25
,
 42 30 0 0 0 0 0 0 41 31 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 10 41 0 0 0 0 0 0 51 63 0 0 0 0 0 0 0 0 30 60 0 0 0 0 0 0 13 43

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],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,72,1,0,0,0,0,0,0,72,0],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,72,72,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[63,0,0,0,0,0,0,0,31,22,0,0,0,0,0,0,0,0,72,0,0,0,0,0,0,0,0,72,0,0,0,0,0,0,0,0,14,19,0,0,0,0,0,0,5,59,0,0,0,0,0,0,0,0,48,36,0,0,0,0,0,0,11,25],[42,41,0,0,0,0,0,0,30,31,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,10,51,0,0,0,0,0,0,41,63,0,0,0,0,0,0,0,0,30,13,0,0,0,0,0,0,60,43] >;

C3318SD16 in GAP, Magma, Sage, TeX

C_3^3\rtimes_{18}{\rm SD}_{16}
% in TeX

G:=Group("C3^3:18SD16");
// GroupNames label

G:=SmallGroup(432,458);
// by ID

G=gap.SmallGroup(432,458);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-3,-3,-3,85,64,254,135,58,1124,571,2028,14118]);
// Polycyclic

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

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