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G = SD16×He3order 432 = 24·33

Direct product of SD16 and He3

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

Series: Derived Chief Lower central Upper central

 Derived series C1 — C12 — SD16×He3
 Chief series C1 — C2 — C6 — C12 — C3×C12 — C4×He3 — Q8×He3 — SD16×He3
 Lower central C1 — C2 — C12 — SD16×He3
 Upper central C1 — C6 — C4×He3 — SD16×He3

Generators and relations for SD16×He3
G = < a,b,c,d,e | a8=b2=c3=d3=e3=1, bab=a3, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, cd=dc, ece-1=cd-1, de=ed >

Subgroups: 285 in 110 conjugacy classes, 49 normal (21 characteristic)
C1, C2, C2, C3, C3, C4, C4, C22, C6, C6, C8, D4, Q8, C32, C12, C12, C2×C6, SD16, C3×C6, C3×C6, C24, C24, C3×D4, C3×D4, C3×Q8, C3×Q8, He3, C3×C12, C3×C12, C62, C3×SD16, C3×SD16, C2×He3, C2×He3, C3×C24, D4×C32, Q8×C32, C4×He3, C4×He3, C22×He3, C32×SD16, C8×He3, D4×He3, Q8×He3, SD16×He3
Quotients: C1, C2, C3, C22, C6, D4, C32, C2×C6, SD16, C3×C6, C3×D4, He3, C62, C3×SD16, C2×He3, D4×C32, C22×He3, C32×SD16, D4×He3, SD16×He3

Smallest permutation representation of SD16×He3
On 72 points
Generators in S72
(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)(18 20)(19 23)(22 24)(25 27)(26 30)(29 31)(33 37)(34 40)(36 38)(41 47)(43 45)(44 48)(49 55)(51 53)(52 56)(58 60)(59 63)(62 64)(65 69)(66 72)(68 70)
(9 67 17)(10 68 18)(11 69 19)(12 70 20)(13 71 21)(14 72 22)(15 65 23)(16 66 24)(33 44 52)(34 45 53)(35 46 54)(36 47 55)(37 48 56)(38 41 49)(39 42 50)(40 43 51)
(1 32 61)(2 25 62)(3 26 63)(4 27 64)(5 28 57)(6 29 58)(7 30 59)(8 31 60)(9 67 17)(10 68 18)(11 69 19)(12 70 20)(13 71 21)(14 72 22)(15 65 23)(16 66 24)(33 52 44)(34 53 45)(35 54 46)(36 55 47)(37 56 48)(38 49 41)(39 50 42)(40 51 43)
(1 54 67)(2 55 68)(3 56 69)(4 49 70)(5 50 71)(6 51 72)(7 52 65)(8 53 66)(9 61 35)(10 62 36)(11 63 37)(12 64 38)(13 57 39)(14 58 40)(15 59 33)(16 60 34)(17 32 46)(18 25 47)(19 26 48)(20 27 41)(21 28 42)(22 29 43)(23 30 44)(24 31 45)

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

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

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

77 conjugacy classes

 class 1 2A 2B 3A 3B 3C ··· 3J 4A 4B 6A 6B 6C ··· 6J 6K 6L 6M ··· 6T 8A 8B 12A 12B 12C 12D 12E ··· 12L 12M ··· 12T 24A 24B 24C 24D 24E ··· 24T order 1 2 2 3 3 3 ··· 3 4 4 6 6 6 ··· 6 6 6 6 ··· 6 8 8 12 12 12 12 12 ··· 12 12 ··· 12 24 24 24 24 24 ··· 24 size 1 1 4 1 1 3 ··· 3 2 4 1 1 3 ··· 3 4 4 12 ··· 12 2 2 2 2 4 4 6 ··· 6 12 ··· 12 2 2 2 2 6 ··· 6

77 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 2 2 3 3 3 3 6 6 type + + + + + image C1 C2 C2 C2 C3 C6 C6 C6 D4 SD16 C3×D4 C3×SD16 He3 C2×He3 C2×He3 C2×He3 D4×He3 SD16×He3 kernel SD16×He3 C8×He3 D4×He3 Q8×He3 C32×SD16 C3×C24 D4×C32 Q8×C32 C2×He3 He3 C3×C6 C32 SD16 C8 D4 Q8 C2 C1 # reps 1 1 1 1 8 8 8 8 1 2 8 16 2 2 2 2 2 4

Matrix representation of SD16×He3 in GL5(𝔽73)

 0 61 0 0 0 6 61 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1
,
 1 0 0 0 0 1 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 1 0 0 0 0 0 8 0 0 0 0 0 64
,
 1 0 0 0 0 0 1 0 0 0 0 0 8 0 0 0 0 0 8 0 0 0 0 0 8
,
 8 0 0 0 0 0 8 0 0 0 0 0 0 64 0 0 0 0 0 64 0 0 64 0 0

G:=sub<GL(5,GF(73))| [0,6,0,0,0,61,61,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1],[1,1,0,0,0,0,72,0,0,0,0,0,72,0,0,0,0,0,72,0,0,0,0,0,72],[8,0,0,0,0,0,8,0,0,0,0,0,1,0,0,0,0,0,8,0,0,0,0,0,64],[1,0,0,0,0,0,1,0,0,0,0,0,8,0,0,0,0,0,8,0,0,0,0,0,8],[8,0,0,0,0,0,8,0,0,0,0,0,0,0,64,0,0,64,0,0,0,0,0,64,0] >;

SD16×He3 in GAP, Magma, Sage, TeX

{\rm SD}_{16}\times {\rm He}_3
% in TeX

G:=Group("SD16xHe3");
// GroupNames label

G:=SmallGroup(432,219);
// by ID

G=gap.SmallGroup(432,219);
# by ID

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

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

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