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G = He39SD16order 432 = 24·33

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

non-abelian, supersoluble, monomial

Aliases: He39SD16, He34C86C2, He34Q82C2, (C3×C12).17D6, (D4×He3).2C2, D4.(He3⋊C2), (C2×He3).35D4, (D4×C32).4S3, C2.5(He37D4), C326(D4.S3), C6.41(C327D4), (C4×He3).13C22, C3.2(C329SD16), C12.45(C2×C3⋊S3), (C3×D4).7(C3⋊S3), C4.2(C2×He3⋊C2), (C3×C6).36(C3⋊D4), SmallGroup(432,193)

Series: Derived Chief Lower central Upper central

C1C3C4×He3 — He39SD16
C1C3C32He3C2×He3C4×He3He34Q8 — He39SD16
He3C2×He3C4×He3 — He39SD16
C1C6C12C3×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 2A2B3A3B3C3D3E3F4A4B6A6B6C6D6E6F6G6H6I···6P8A8B12A12B12C12D12E12F12G12H24A24B24C24D
order12233333344666666666···688121212121212121224242424
size1141166662361144666612···1218182212121212363618181818

41 irreducible representations

dim11112222233466
type+++++++-
imageC1C2C2C2S3D4D6SD16C3⋊D4He3⋊C2C2×He3⋊C2D4.S3He37D4He39SD16
kernelHe39SD16He34C8He34Q8D4×He3D4×C32C2×He3C3×C12He3C3×C6D4C4C32C2C1
# reps11114142844424

Matrix representation of He39SD16 in GL5(𝔽73)

10000
01000
00010
00001
00100
,
10000
01000
006400
000640
000064
,
10000
01000
00100
00080
000064
,
055000
412000
00100
00001
00010
,
10000
4872000
00100
00010
00001

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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