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## G = He3⋊8SD16order 432 = 24·33

### 1st semidirect product of He3 and SD16 acting via SD16/D4=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C3×C12 — He3⋊8SD16
 Chief series C1 — C3 — C32 — C3×C6 — C3×C12 — C4×He3 — He3⋊3Q8 — He3⋊8SD16
 Lower central C32 — C3×C6 — C3×C12 — He3⋊8SD16
 Upper central C1 — C2 — C4 — D4

Generators and relations for He38SD16
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, dbd-1=b-1, be=eb, cd=dc, ce=ec, ede=d3 >

Subgroups: 365 in 86 conjugacy classes, 26 normal (all characteristic)
C1, C2, C2, C3, C3, C4, C4, C22, C6, C6, C8, D4, Q8, C32, 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⋊Dic3, C3×C12, C3×C12, C62, D4.S3, C3×SD16, C2×He3, C2×He3, C3×C3⋊C8, C324C8, C3×Dic6, C324Q8, D4×C32, D4×C32, C32⋊C12, C4×He3, C22×He3, C3×D4.S3, C329SD16, He33C8, He33Q8, D4×He3, He38SD16
Quotients: C1, C2, C3, C22, S3, C6, D4, D6, C2×C6, SD16, C3×S3, C3⋊D4, C3×D4, S3×C6, D4.S3, C3×SD16, C32⋊C6, C3×C3⋊D4, C2×C32⋊C6, C3×D4.S3, He36D4, He38SD16

Smallest permutation representation of He38SD16
On 72 points
Generators in S72
(9 31 48)(10 41 32)(11 25 42)(12 43 26)(13 27 44)(14 45 28)(15 29 46)(16 47 30)(33 55 60)(34 61 56)(35 49 62)(36 63 50)(37 51 64)(38 57 52)(39 53 58)(40 59 54)
(1 67 24)(2 17 68)(3 69 18)(4 19 70)(5 71 20)(6 21 72)(7 65 22)(8 23 66)(9 48 31)(10 32 41)(11 42 25)(12 26 43)(13 44 27)(14 28 45)(15 46 29)(16 30 47)(33 55 60)(34 61 56)(35 49 62)(36 63 50)(37 51 64)(38 57 52)(39 53 58)(40 59 54)
(1 15 53)(2 16 54)(3 9 55)(4 10 56)(5 11 49)(6 12 50)(7 13 51)(8 14 52)(17 30 40)(18 31 33)(19 32 34)(20 25 35)(21 26 36)(22 27 37)(23 28 38)(24 29 39)(41 61 70)(42 62 71)(43 63 72)(44 64 65)(45 57 66)(46 58 67)(47 59 68)(48 60 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 4)(3 7)(6 8)(9 13)(10 16)(12 14)(17 19)(18 22)(21 23)(26 28)(27 31)(30 32)(33 37)(34 40)(36 38)(41 47)(43 45)(44 48)(50 52)(51 55)(54 56)(57 63)(59 61)(60 64)(65 69)(66 72)(68 70)

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

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

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

41 irreducible representations

 dim 1 1 1 1 1 1 1 1 12 2 2 2 2 2 2 2 2 2 2 4 4 6 6 6 type + + + + - + + + - + + image C1 C2 C2 C2 C3 C6 C6 C6 He3⋊8SD16 S3 D4 D6 SD16 C3×S3 C3⋊D4 C3×D4 S3×C6 C3×SD16 C3×C3⋊D4 D4.S3 C3×D4.S3 C32⋊C6 C2×C32⋊C6 He3⋊6D4 kernel He3⋊8SD16 He3⋊3C8 He3⋊3Q8 D4×He3 C32⋊9SD16 C32⋊4C8 C32⋊4Q8 D4×C32 C1 D4×C32 C2×He3 C3×C12 He3 C3×D4 C3×C6 C3×C6 C12 C32 C6 C32 C3 D4 C4 C2 # reps 1 1 1 1 2 2 2 2 1 1 1 1 2 2 2 2 2 4 4 1 2 1 1 2

Matrix representation of He38SD16 in GL10(𝔽73)

 0 72 0 0 0 0 0 0 0 0 1 72 0 0 0 0 0 0 0 0 0 0 0 72 0 0 0 0 0 0 0 0 1 72 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 64 64 0 0 0 0 0 0 0 0 9 0 8 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 16 34 0 0 8 0 0 0 0 0 6 16 16 65 8 64
,
 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 64 0 0 0 0 0 0 0 0 0 0 64 0 0 0 0 0 0 0 0 0 0 64 0 0 0 0 0 0 0 55 0 0 8 0 0 0 0 0 0 16 34 0 0 8 0 0 0 0 0 15 71 57 0 0 8
,
 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 64 63 0 0 0 0 0 0 0 0 0 9 1 0 0 0 0 0 0 0 0 65 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 8 2 16 65 9 63 0 0 0 0 0 12 0 0 0 64
,
 0 0 0 12 0 0 0 0 0 0 0 0 12 0 0 0 0 0 0 0 0 67 0 12 0 0 0 0 0 0 67 0 12 0 0 0 0 0 0 0 0 0 0 0 58 0 0 2 0 0 0 0 0 0 0 58 0 42 29 0 0 0 0 0 13 58 11 47 42 2 0 0 0 0 33 0 0 15 0 0 0 0 0 0 68 35 0 0 15 0 0 0 0 0 49 38 12 56 0 62
,
 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 72 0 0 0 0 0 0 0 0 1 0 72 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 15 0 0 72 0 0 0 0 0 0 11 69 0 0 72 0 0 0 0 0 24 26 62 0 0 72

G:=sub<GL(10,GF(73))| [0,1,0,0,0,0,0,0,0,0,72,72,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,72,72,0,0,0,0,0,0,0,0,0,0,1,64,9,0,16,6,0,0,0,0,0,64,0,0,34,16,0,0,0,0,0,0,8,0,0,16,0,0,0,0,0,0,0,1,0,65,0,0,0,0,0,0,0,0,8,8,0,0,0,0,0,0,0,0,0,64],[1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,64,0,0,55,16,15,0,0,0,0,0,64,0,0,34,71,0,0,0,0,0,0,64,0,0,57,0,0,0,0,0,0,0,8,0,0,0,0,0,0,0,0,0,0,8,0,0,0,0,0,0,0,0,0,0,8],[1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,64,0,0,0,8,0,0,0,0,0,63,9,65,0,2,12,0,0,0,0,0,1,0,0,16,0,0,0,0,0,0,0,0,0,65,0,0,0,0,0,0,0,0,1,9,0,0,0,0,0,0,0,0,0,63,64],[0,0,0,67,0,0,0,0,0,0,0,0,67,0,0,0,0,0,0,0,0,12,0,12,0,0,0,0,0,0,12,0,12,0,0,0,0,0,0,0,0,0,0,0,58,0,13,33,68,49,0,0,0,0,0,58,58,0,35,38,0,0,0,0,0,0,11,0,0,12,0,0,0,0,2,42,47,15,0,56,0,0,0,0,0,29,42,0,15,0,0,0,0,0,0,0,2,0,0,62],[1,0,1,0,0,0,0,0,0,0,0,1,0,1,0,0,0,0,0,0,0,0,72,0,0,0,0,0,0,0,0,0,0,72,0,0,0,0,0,0,0,0,0,0,1,0,0,15,11,24,0,0,0,0,0,1,0,0,69,26,0,0,0,0,0,0,1,0,0,62,0,0,0,0,0,0,0,72,0,0,0,0,0,0,0,0,0,0,72,0,0,0,0,0,0,0,0,0,0,72] >;

He38SD16 in GAP, Magma, Sage, TeX

{\rm He}_3\rtimes_8{\rm SD}_{16}
% in TeX

G:=Group("He3:8SD16");
// GroupNames label

G:=SmallGroup(432,152);
// by ID

G=gap.SmallGroup(432,152);
# by ID

G:=PCGroup([7,-2,-2,-3,-2,-2,-3,-3,168,197,1011,514,80,4037,2035,14118]);
// 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,d*b*d^-1=b^-1,b*e=e*b,c*d=d*c,c*e=e*c,e*d*e=d^3>;
// generators/relations

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