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## G = SD16×3- 1+2order 432 = 24·33

### Direct product of SD16 and 3- 1+2

direct product, metacyclic, nilpotent (class 3), monomial

Series: Derived Chief Lower central Upper central

 Derived series C1 — C12 — SD16×3- 1+2
 Chief series C1 — C2 — C6 — C12 — C3×C12 — C4×3- 1+2 — Q8×3- 1+2 — SD16×3- 1+2
 Lower central C1 — C2 — C12 — SD16×3- 1+2
 Upper central C1 — C6 — C4×3- 1+2 — SD16×3- 1+2

Generators and relations for SD16×3- 1+2
G = < a,b,c,d | a8=b2=c9=d3=1, bab=a3, ac=ca, ad=da, bc=cb, bd=db, dcd-1=c4 >

Subgroups: 150 in 80 conjugacy classes, 49 normal (28 characteristic)
C1, C2, C2, C3, C3, C4, C4, C22, C6, C6, C8, D4, Q8, C9, C32, C12, C12, C2×C6, SD16, C18, C18, C3×C6, C3×C6, C24, C24, C3×D4, C3×D4, C3×Q8, C3×Q8, 3- 1+2, C36, C36, C2×C18, C3×C12, C3×C12, C62, C3×SD16, C3×SD16, C2×3- 1+2, C2×3- 1+2, C72, D4×C9, Q8×C9, C3×C24, D4×C32, Q8×C32, C4×3- 1+2, C4×3- 1+2, C22×3- 1+2, C9×SD16, C32×SD16, C8×3- 1+2, D4×3- 1+2, Q8×3- 1+2, SD16×3- 1+2
Quotients: C1, C2, C3, C22, C6, D4, C32, C2×C6, SD16, C3×C6, C3×D4, 3- 1+2, C62, C3×SD16, C2×3- 1+2, D4×C32, C22×3- 1+2, C32×SD16, D4×3- 1+2, SD16×3- 1+2

Smallest permutation representation of SD16×3- 1+2
On 72 points
Generators in S72
(1 66 63 15 48 21 43 31)(2 67 55 16 49 22 44 32)(3 68 56 17 50 23 45 33)(4 69 57 18 51 24 37 34)(5 70 58 10 52 25 38 35)(6 71 59 11 53 26 39 36)(7 72 60 12 54 27 40 28)(8 64 61 13 46 19 41 29)(9 65 62 14 47 20 42 30)
(10 70)(11 71)(12 72)(13 64)(14 65)(15 66)(16 67)(17 68)(18 69)(19 29)(20 30)(21 31)(22 32)(23 33)(24 34)(25 35)(26 36)(27 28)(37 57)(38 58)(39 59)(40 60)(41 61)(42 62)(43 63)(44 55)(45 56)
(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 8 5)(3 6 9)(10 16 13)(11 14 17)(19 25 22)(20 23 26)(29 35 32)(30 33 36)(38 44 41)(39 42 45)(46 52 49)(47 50 53)(55 61 58)(56 59 62)(64 70 67)(65 68 71)

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

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

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

77 conjugacy classes

 class 1 2A 2B 3A 3B 3C 3D 4A 4B 6A 6B 6C 6D 6E 6F 6G 6H 8A 8B 9A ··· 9F 12A 12B 12C 12D 12E 12F 12G 12H 18A ··· 18F 18G ··· 18L 24A 24B 24C 24D 24E 24F 24G 24H 36A ··· 36F 36G ··· 36L 72A ··· 72L order 1 2 2 3 3 3 3 4 4 6 6 6 6 6 6 6 6 8 8 9 ··· 9 12 12 12 12 12 12 12 12 18 ··· 18 18 ··· 18 24 24 24 24 24 24 24 24 36 ··· 36 36 ··· 36 72 ··· 72 size 1 1 4 1 1 3 3 2 4 1 1 3 3 4 4 12 12 2 2 3 ··· 3 2 2 4 4 6 6 12 12 3 ··· 3 12 ··· 12 2 2 2 2 6 6 6 6 6 ··· 6 12 ··· 12 6 ··· 6

77 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 3 3 3 3 6 6 type + + + + + image C1 C2 C2 C2 C3 C3 C6 C6 C6 C6 C6 C6 D4 SD16 C3×D4 C3×D4 C3×SD16 C3×SD16 3- 1+2 C2×3- 1+2 C2×3- 1+2 C2×3- 1+2 D4×3- 1+2 SD16×3- 1+2 kernel SD16×3- 1+2 C8×3- 1+2 D4×3- 1+2 Q8×3- 1+2 C9×SD16 C32×SD16 C72 D4×C9 Q8×C9 C3×C24 D4×C32 Q8×C32 C2×3- 1+2 3- 1+2 C18 C3×C6 C9 C32 SD16 C8 D4 Q8 C2 C1 # reps 1 1 1 1 6 2 6 6 6 2 2 2 1 2 6 2 12 4 2 2 2 2 2 4

Matrix representation of SD16×3- 1+2 in GL5(𝔽73)

 67 6 0 0 0 67 67 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1
,
 1 0 0 0 0 0 72 0 0 0 0 0 72 0 0 0 0 0 72 0 0 0 0 0 72
,
 8 0 0 0 0 0 8 0 0 0 0 0 64 16 0 0 0 68 9 1 0 0 0 72 0
,
 8 0 0 0 0 0 8 0 0 0 0 0 1 0 0 0 0 33 64 0 0 0 5 0 8

G:=sub<GL(5,GF(73))| [67,67,0,0,0,6,67,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1],[1,0,0,0,0,0,72,0,0,0,0,0,72,0,0,0,0,0,72,0,0,0,0,0,72],[8,0,0,0,0,0,8,0,0,0,0,0,64,68,0,0,0,16,9,72,0,0,0,1,0],[8,0,0,0,0,0,8,0,0,0,0,0,1,33,5,0,0,0,64,0,0,0,0,0,8] >;

SD16×3- 1+2 in GAP, Magma, Sage, TeX

{\rm SD}_{16}\times 3_-^{1+2}
% in TeX

G:=Group("SD16xES-(3,1)");
// GroupNames label

G:=SmallGroup(432,220);
// by ID

G=gap.SmallGroup(432,220);
# by ID

G:=PCGroup([7,-2,-2,-3,-3,-2,-3,-2,1512,533,394,605,8824,4421,242]);
// Polycyclic

G:=Group<a,b,c,d|a^8=b^2=c^9=d^3=1,b*a*b=a^3,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d^-1=c^4>;
// generators/relations

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