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## G = C3×C32⋊9SD16order 432 = 24·33

### Direct product of C3 and C32⋊9SD16

Series: Derived Chief Lower central Upper central

 Derived series C1 — C3×C12 — C3×C32⋊9SD16
 Chief series C1 — C3 — C32 — C3×C6 — C3×C12 — C32×C12 — C3×C32⋊4Q8 — C3×C32⋊9SD16
 Lower central C32 — C3×C6 — C3×C12 — C3×C32⋊9SD16
 Upper central C1 — C6 — C12 — C3×D4

Generators and relations for C3×C329SD16
G = < a,b,c,d,e | a3=b3=c3=d8=e2=1, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, dbd-1=b-1, be=eb, dcd-1=c-1, ce=ec, ede=d3 >

Subgroups: 500 in 184 conjugacy classes, 54 normal (22 characteristic)
C1, C2, C2, C3, C3, C3, C4, C4, C22, C6, C6, C6, C8, D4, Q8, C32, C32, C32, Dic3, C12, C12, C12, C2×C6, SD16, C3×C6, C3×C6, C3×C6, C3⋊C8, C24, Dic6, C3×D4, C3×D4, C3×D4, C3×Q8, C33, C3×Dic3, C3⋊Dic3, C3×C12, C3×C12, C3×C12, C62, D4.S3, C3×SD16, C32×C6, C32×C6, C3×C3⋊C8, C324C8, C3×Dic6, C324Q8, D4×C32, D4×C32, D4×C32, C3×C3⋊Dic3, C32×C12, C3×C62, C3×D4.S3, C329SD16, C3×C324C8, C3×C324Q8, D4×C33, C3×C329SD16
Quotients: C1, C2, C3, C22, S3, C6, D4, D6, C2×C6, SD16, C3×S3, C3⋊S3, C3⋊D4, C3×D4, S3×C6, C2×C3⋊S3, D4.S3, C3×SD16, C3×C3⋊S3, C3×C3⋊D4, C327D4, C6×C3⋊S3, C3×D4.S3, C329SD16, C3×C327D4, C3×C329SD16

Smallest permutation representation of C3×C329SD16
On 72 points
Generators in S72
(1 27 61)(2 28 62)(3 29 63)(4 30 64)(5 31 57)(6 32 58)(7 25 59)(8 26 60)(9 35 43)(10 36 44)(11 37 45)(12 38 46)(13 39 47)(14 40 48)(15 33 41)(16 34 42)(17 68 51)(18 69 52)(19 70 53)(20 71 54)(21 72 55)(22 65 56)(23 66 49)(24 67 50)
(1 24 47)(2 48 17)(3 18 41)(4 42 19)(5 20 43)(6 44 21)(7 22 45)(8 46 23)(9 31 71)(10 72 32)(11 25 65)(12 66 26)(13 27 67)(14 68 28)(15 29 69)(16 70 30)(33 63 52)(34 53 64)(35 57 54)(36 55 58)(37 59 56)(38 49 60)(39 61 50)(40 51 62)
(1 61 27)(2 28 62)(3 63 29)(4 30 64)(5 57 31)(6 32 58)(7 59 25)(8 26 60)(9 43 35)(10 36 44)(11 45 37)(12 38 46)(13 47 39)(14 40 48)(15 41 33)(16 34 42)(17 68 51)(18 52 69)(19 70 53)(20 54 71)(21 72 55)(22 56 65)(23 66 49)(24 50 67)
(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)(49 50 51 52 53 54 55 56)(57 58 59 60 61 62 63 64)(65 66 67 68 69 70 71 72)
(2 4)(3 7)(6 8)(10 12)(11 15)(14 16)(17 19)(18 22)(21 23)(25 29)(26 32)(28 30)(33 37)(34 40)(36 38)(41 45)(42 48)(44 46)(49 55)(51 53)(52 56)(58 60)(59 63)(62 64)(65 69)(66 72)(68 70)

G:=sub<Sym(72)| (1,27,61)(2,28,62)(3,29,63)(4,30,64)(5,31,57)(6,32,58)(7,25,59)(8,26,60)(9,35,43)(10,36,44)(11,37,45)(12,38,46)(13,39,47)(14,40,48)(15,33,41)(16,34,42)(17,68,51)(18,69,52)(19,70,53)(20,71,54)(21,72,55)(22,65,56)(23,66,49)(24,67,50), (1,24,47)(2,48,17)(3,18,41)(4,42,19)(5,20,43)(6,44,21)(7,22,45)(8,46,23)(9,31,71)(10,72,32)(11,25,65)(12,66,26)(13,27,67)(14,68,28)(15,29,69)(16,70,30)(33,63,52)(34,53,64)(35,57,54)(36,55,58)(37,59,56)(38,49,60)(39,61,50)(40,51,62), (1,61,27)(2,28,62)(3,63,29)(4,30,64)(5,57,31)(6,32,58)(7,59,25)(8,26,60)(9,43,35)(10,36,44)(11,45,37)(12,38,46)(13,47,39)(14,40,48)(15,41,33)(16,34,42)(17,68,51)(18,52,69)(19,70,53)(20,54,71)(21,72,55)(22,56,65)(23,66,49)(24,50,67), (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)(49,50,51,52,53,54,55,56)(57,58,59,60,61,62,63,64)(65,66,67,68,69,70,71,72), (2,4)(3,7)(6,8)(10,12)(11,15)(14,16)(17,19)(18,22)(21,23)(25,29)(26,32)(28,30)(33,37)(34,40)(36,38)(41,45)(42,48)(44,46)(49,55)(51,53)(52,56)(58,60)(59,63)(62,64)(65,69)(66,72)(68,70)>;

G:=Group( (1,27,61)(2,28,62)(3,29,63)(4,30,64)(5,31,57)(6,32,58)(7,25,59)(8,26,60)(9,35,43)(10,36,44)(11,37,45)(12,38,46)(13,39,47)(14,40,48)(15,33,41)(16,34,42)(17,68,51)(18,69,52)(19,70,53)(20,71,54)(21,72,55)(22,65,56)(23,66,49)(24,67,50), (1,24,47)(2,48,17)(3,18,41)(4,42,19)(5,20,43)(6,44,21)(7,22,45)(8,46,23)(9,31,71)(10,72,32)(11,25,65)(12,66,26)(13,27,67)(14,68,28)(15,29,69)(16,70,30)(33,63,52)(34,53,64)(35,57,54)(36,55,58)(37,59,56)(38,49,60)(39,61,50)(40,51,62), (1,61,27)(2,28,62)(3,63,29)(4,30,64)(5,57,31)(6,32,58)(7,59,25)(8,26,60)(9,43,35)(10,36,44)(11,45,37)(12,38,46)(13,47,39)(14,40,48)(15,41,33)(16,34,42)(17,68,51)(18,52,69)(19,70,53)(20,54,71)(21,72,55)(22,56,65)(23,66,49)(24,50,67), (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)(49,50,51,52,53,54,55,56)(57,58,59,60,61,62,63,64)(65,66,67,68,69,70,71,72), (2,4)(3,7)(6,8)(10,12)(11,15)(14,16)(17,19)(18,22)(21,23)(25,29)(26,32)(28,30)(33,37)(34,40)(36,38)(41,45)(42,48)(44,46)(49,55)(51,53)(52,56)(58,60)(59,63)(62,64)(65,69)(66,72)(68,70) );

G=PermutationGroup([[(1,27,61),(2,28,62),(3,29,63),(4,30,64),(5,31,57),(6,32,58),(7,25,59),(8,26,60),(9,35,43),(10,36,44),(11,37,45),(12,38,46),(13,39,47),(14,40,48),(15,33,41),(16,34,42),(17,68,51),(18,69,52),(19,70,53),(20,71,54),(21,72,55),(22,65,56),(23,66,49),(24,67,50)], [(1,24,47),(2,48,17),(3,18,41),(4,42,19),(5,20,43),(6,44,21),(7,22,45),(8,46,23),(9,31,71),(10,72,32),(11,25,65),(12,66,26),(13,27,67),(14,68,28),(15,29,69),(16,70,30),(33,63,52),(34,53,64),(35,57,54),(36,55,58),(37,59,56),(38,49,60),(39,61,50),(40,51,62)], [(1,61,27),(2,28,62),(3,63,29),(4,30,64),(5,57,31),(6,32,58),(7,59,25),(8,26,60),(9,43,35),(10,36,44),(11,45,37),(12,38,46),(13,47,39),(14,40,48),(15,41,33),(16,34,42),(17,68,51),(18,52,69),(19,70,53),(20,54,71),(21,72,55),(22,56,65),(23,66,49),(24,50,67)], [(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),(49,50,51,52,53,54,55,56),(57,58,59,60,61,62,63,64),(65,66,67,68,69,70,71,72)], [(2,4),(3,7),(6,8),(10,12),(11,15),(14,16),(17,19),(18,22),(21,23),(25,29),(26,32),(28,30),(33,37),(34,40),(36,38),(41,45),(42,48),(44,46),(49,55),(51,53),(52,56),(58,60),(59,63),(62,64),(65,69),(66,72),(68,70)]])

81 conjugacy classes

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

81 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 4 4 type + + + + + + + - image C1 C2 C2 C2 C3 C6 C6 C6 S3 D4 D6 SD16 C3×S3 C3⋊D4 C3×D4 S3×C6 C3×SD16 C3×C3⋊D4 D4.S3 C3×D4.S3 kernel C3×C32⋊9SD16 C3×C32⋊4C8 C3×C32⋊4Q8 D4×C33 C32⋊9SD16 C32⋊4C8 C32⋊4Q8 D4×C32 D4×C32 C32×C6 C3×C12 C33 C3×D4 C3×C6 C3×C6 C12 C32 C6 C32 C3 # reps 1 1 1 1 2 2 2 2 4 1 4 2 8 8 2 8 4 16 4 8

Matrix representation of C3×C329SD16 in GL6(𝔽73)

 8 0 0 0 0 0 0 8 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
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 64 0 0 0 0 0 0 8 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 64 0 0 0 0 0 0 8 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
,
 0 1 0 0 0 0 72 0 0 0 0 0 0 0 0 1 0 0 0 0 72 0 0 0 0 0 0 0 61 60 0 0 0 0 28 0
,
 1 0 0 0 0 0 0 72 0 0 0 0 0 0 1 0 0 0 0 0 0 72 0 0 0 0 0 0 1 0 0 0 0 0 44 72

G:=sub<GL(6,GF(73))| [8,0,0,0,0,0,0,8,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,64,0,0,0,0,0,0,8,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[64,0,0,0,0,0,0,8,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[0,72,0,0,0,0,1,0,0,0,0,0,0,0,0,72,0,0,0,0,1,0,0,0,0,0,0,0,61,28,0,0,0,0,60,0],[1,0,0,0,0,0,0,72,0,0,0,0,0,0,1,0,0,0,0,0,0,72,0,0,0,0,0,0,1,44,0,0,0,0,0,72] >;

C3×C329SD16 in GAP, Magma, Sage, TeX

C_3\times C_3^2\rtimes_9{\rm SD}_{16}
% in TeX

G:=Group("C3xC3^2:9SD16");
// GroupNames label

G:=SmallGroup(432,492);
// by ID

G=gap.SmallGroup(432,492);
# by ID

G:=PCGroup([7,-2,-2,-3,-2,-2,-3,-3,168,197,1011,514,80,4037,14118]);
// 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,a*e=e*a,b*c=c*b,d*b*d^-1=b^-1,b*e=e*b,d*c*d^-1=c^-1,c*e=e*c,e*d*e=d^3>;
// generators/relations

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