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## G = D8×3- 1+2order 432 = 24·33

### Direct product of D8 and 3- 1+2

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

Series: Derived Chief Lower central Upper central

 Derived series C1 — C12 — D8×3- 1+2
 Chief series C1 — C2 — C6 — C12 — C3×C12 — C4×3- 1+2 — D4×3- 1+2 — D8×3- 1+2
 Lower central C1 — C2 — C12 — D8×3- 1+2
 Upper central C1 — C6 — C4×3- 1+2 — D8×3- 1+2

Generators and relations for D8×3- 1+2
G = < a,b,c,d | a8=b2=c9=d3=1, bab=a-1, ac=ca, ad=da, bc=cb, bd=db, dcd-1=c4 >

Subgroups: 190 in 88 conjugacy classes, 49 normal (20 characteristic)
C1, C2, C2, C3, C3, C4, C22, C6, C6, C8, D4, C9, C32, C12, C12, C2×C6, D8, C18, C18, C3×C6, C3×C6, C24, C24, C3×D4, C3×D4, 3- 1+2, C36, C2×C18, C3×C12, C62, C3×D8, C3×D8, C2×3- 1+2, C2×3- 1+2, C72, D4×C9, C3×C24, D4×C32, C4×3- 1+2, C22×3- 1+2, C9×D8, C32×D8, C8×3- 1+2, D4×3- 1+2, D8×3- 1+2
Quotients: C1, C2, C3, C22, C6, D4, C32, C2×C6, D8, C3×C6, C3×D4, 3- 1+2, C62, C3×D8, C2×3- 1+2, D4×C32, C22×3- 1+2, C32×D8, D4×3- 1+2, D8×3- 1+2

Smallest permutation representation of D8×3- 1+2
On 72 points
Generators in S72
(1 21 51 15 42 66 63 31)(2 22 52 16 43 67 55 32)(3 23 53 17 44 68 56 33)(4 24 54 18 45 69 57 34)(5 25 46 10 37 70 58 35)(6 26 47 11 38 71 59 36)(7 27 48 12 39 72 60 28)(8 19 49 13 40 64 61 29)(9 20 50 14 41 65 62 30)
(1 31)(2 32)(3 33)(4 34)(5 35)(6 36)(7 28)(8 29)(9 30)(10 37)(11 38)(12 39)(13 40)(14 41)(15 42)(16 43)(17 44)(18 45)(19 61)(20 62)(21 63)(22 55)(23 56)(24 57)(25 58)(26 59)(27 60)(46 70)(47 71)(48 72)(49 64)(50 65)(51 66)(52 67)(53 68)(54 69)
(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)(37 43 40)(38 41 44)(46 52 49)(47 50 53)(55 61 58)(56 59 62)(64 70 67)(65 68 71)

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

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

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

77 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 3 3 3 6 6 type + + + + + image C1 C2 C2 C3 C3 C6 C6 C6 C6 D4 D8 C3×D4 C3×D4 C3×D8 C3×D8 3- 1+2 C2×3- 1+2 C2×3- 1+2 D4×3- 1+2 D8×3- 1+2 kernel D8×3- 1+2 C8×3- 1+2 D4×3- 1+2 C9×D8 C32×D8 C72 D4×C9 C3×C24 D4×C32 C2×3- 1+2 3- 1+2 C18 C3×C6 C9 C32 D8 C8 D4 C2 C1 # reps 1 1 2 6 2 6 12 2 4 1 2 6 2 12 4 2 2 4 2 4

Matrix representation of D8×3- 1+2 in GL5(𝔽73)

 57 16 0 0 0 57 57 0 0 0 0 0 72 0 0 0 0 0 72 0 0 0 0 0 72
,
 57 16 0 0 0 16 16 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1
,
 8 0 0 0 0 0 8 0 0 0 0 0 8 63 0 0 0 64 65 64 0 0 0 9 0
,
 64 0 0 0 0 0 64 0 0 0 0 0 1 0 0 0 0 8 64 0 0 0 72 0 8

G:=sub<GL(5,GF(73))| [57,57,0,0,0,16,57,0,0,0,0,0,72,0,0,0,0,0,72,0,0,0,0,0,72],[57,16,0,0,0,16,16,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1],[8,0,0,0,0,0,8,0,0,0,0,0,8,64,0,0,0,63,65,9,0,0,0,64,0],[64,0,0,0,0,0,64,0,0,0,0,0,1,8,72,0,0,0,64,0,0,0,0,0,8] >;

D8×3- 1+2 in GAP, Magma, Sage, TeX

D_8\times 3_-^{1+2}
% in TeX

G:=Group("D8xES-(3,1)");
// GroupNames label

G:=SmallGroup(432,217);
// by ID

G=gap.SmallGroup(432,217);
# by ID

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

G:=Group<a,b,c,d|a^8=b^2=c^9=d^3=1,b*a*b=a^-1,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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