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G = He3⋊9SD16order 432 = 24·33

2nd semidirect product of He3 and SD16 acting via SD16/D4=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C3 — C4×He3 — He3⋊9SD16
 Chief series C1 — C3 — C32 — He3 — C2×He3 — C4×He3 — He3⋊4Q8 — He3⋊9SD16
 Lower central He3 — C2×He3 — C4×He3 — He3⋊9SD16
 Upper central C1 — C6 — C12 — C3×D4

Generators and relations for He39SD16
G = < a,b,c,d,e | a3=b3=c3=d8=e2=1, ab=ba, cac-1=ab-1, dad-1=a-1, ae=ea, bc=cb, bd=db, be=eb, dcd-1=c-1, ce=ec, ede=d3 >

Subgroups: 405 in 110 conjugacy classes, 31 normal (15 characteristic)
C1, C2, C2, C3, C3, C4, C4, C22, C6, C6, C8, D4, Q8, C32, Dic3, C12, C12, C2×C6, SD16, C3×C6, C3×C6, C3⋊C8, C24, Dic6, C3×D4, C3×D4, C3×Q8, He3, C3×Dic3, C3×C12, C62, D4.S3, C3×SD16, C2×He3, C2×He3, C3×C3⋊C8, C3×Dic6, D4×C32, He33C4, C4×He3, C22×He3, C3×D4.S3, He34C8, He34Q8, D4×He3, He39SD16
Quotients: C1, C2, C22, S3, D4, D6, SD16, C3⋊S3, C3⋊D4, C2×C3⋊S3, D4.S3, He3⋊C2, C327D4, C2×He3⋊C2, C329SD16, He37D4, He39SD16

Smallest permutation representation of He39SD16
On 72 points
Generators in S72
(1 38 20)(2 21 39)(3 40 22)(4 23 33)(5 34 24)(6 17 35)(7 36 18)(8 19 37)(9 43 52)(10 53 44)(11 45 54)(12 55 46)(13 47 56)(14 49 48)(15 41 50)(16 51 42)(25 69 60)(26 61 70)(27 71 62)(28 63 72)(29 65 64)(30 57 66)(31 67 58)(32 59 68)
(1 41 27)(2 42 28)(3 43 29)(4 44 30)(5 45 31)(6 46 32)(7 47 25)(8 48 26)(9 64 22)(10 57 23)(11 58 24)(12 59 17)(13 60 18)(14 61 19)(15 62 20)(16 63 21)(33 53 66)(34 54 67)(35 55 68)(36 56 69)(37 49 70)(38 50 71)(39 51 72)(40 52 65)
(9 22 64)(10 57 23)(11 24 58)(12 59 17)(13 18 60)(14 61 19)(15 20 62)(16 63 21)(33 66 53)(34 54 67)(35 68 55)(36 56 69)(37 70 49)(38 50 71)(39 72 51)(40 52 65)
(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)(9 13)(10 16)(12 14)(17 19)(18 22)(21 23)(25 29)(26 32)(28 30)(33 39)(35 37)(36 40)(42 44)(43 47)(46 48)(49 55)(51 53)(52 56)(57 63)(59 61)(60 64)(65 69)(66 72)(68 70)

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

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

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

41 conjugacy classes

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

41 irreducible representations

 dim 1 1 1 1 2 2 2 2 2 3 3 4 6 6 type + + + + + + + - image C1 C2 C2 C2 S3 D4 D6 SD16 C3⋊D4 He3⋊C2 C2×He3⋊C2 D4.S3 He3⋊7D4 He3⋊9SD16 kernel He3⋊9SD16 He3⋊4C8 He3⋊4Q8 D4×He3 D4×C32 C2×He3 C3×C12 He3 C3×C6 D4 C4 C32 C2 C1 # reps 1 1 1 1 4 1 4 2 8 4 4 4 2 4

Matrix representation of He39SD16 in GL5(𝔽73)

 1 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 1 0 0 1 0 0
,
 1 0 0 0 0 0 1 0 0 0 0 0 64 0 0 0 0 0 64 0 0 0 0 0 64
,
 1 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 8 0 0 0 0 0 64
,
 0 55 0 0 0 4 12 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 1 0
,
 1 0 0 0 0 48 72 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1

G:=sub<GL(5,GF(73))| [1,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,1,0],[1,0,0,0,0,0,1,0,0,0,0,0,64,0,0,0,0,0,64,0,0,0,0,0,64],[1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,8,0,0,0,0,0,64],[0,4,0,0,0,55,12,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,1,0],[1,48,0,0,0,0,72,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1] >;

He39SD16 in GAP, Magma, Sage, TeX

{\rm He}_3\rtimes_9{\rm SD}_{16}
% in TeX

G:=Group("He3:9SD16");
// GroupNames label

G:=SmallGroup(432,193);
// by ID

G=gap.SmallGroup(432,193);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-3,-3,-3,56,85,254,135,58,1124,4037,537]);
// Polycyclic

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