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## G = He3⋊10SD16order 432 = 24·33

### 1st semidirect product of He3 and SD16 acting via SD16/Q8=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C3×C12 — He3⋊10SD16
 Chief series C1 — C3 — C32 — C3×C6 — C3×C12 — C4×He3 — He3⋊4D4 — He3⋊10SD16
 Lower central C32 — C3×C6 — C3×C12 — He3⋊10SD16
 Upper central C1 — C2 — C4 — Q8

Generators and relations for He310SD16
G = < a,b,c,d,e | a3=b3=c3=d8=e2=1, ab=ba, cac-1=ab-1, dad-1=eae=a-1, bc=cb, dbd-1=ebe=b-1, cd=dc, ce=ec, ede=d3 >

Subgroups: 429 in 82 conjugacy classes, 26 normal (all characteristic)
C1, C2, C2, C3, C3, C4, C4, C22, S3, C6, C6, C8, D4, Q8, C32, C32, C12, C12, D6, C2×C6, SD16, C3×S3, C3⋊S3, C3×C6, C3×C6, C3⋊C8, C24, D12, C3×D4, C3×Q8, C3×Q8, He3, C3×C12, C3×C12, S3×C6, C2×C3⋊S3, Q82S3, C3×SD16, C32⋊C6, C2×He3, C3×C3⋊C8, C324C8, C3×D12, C12⋊S3, Q8×C32, Q8×C32, C4×He3, C4×He3, C2×C32⋊C6, C3×Q82S3, C3211SD16, He33C8, He34D4, Q8×He3, He310SD16
Quotients: C1, C2, C3, C22, S3, C6, D4, D6, C2×C6, SD16, C3×S3, C3⋊D4, C3×D4, S3×C6, Q82S3, C3×SD16, C32⋊C6, C3×C3⋊D4, C2×C32⋊C6, C3×Q82S3, He36D4, He310SD16

Smallest permutation representation of He310SD16
On 72 points
Generators in S72
(9 29 24)(10 17 30)(11 31 18)(12 19 32)(13 25 20)(14 21 26)(15 27 22)(16 23 28)(33 41 56)(34 49 42)(35 43 50)(36 51 44)(37 45 52)(38 53 46)(39 47 54)(40 55 48)
(1 58 71)(2 72 59)(3 60 65)(4 66 61)(5 62 67)(6 68 63)(7 64 69)(8 70 57)(9 29 24)(10 17 30)(11 31 18)(12 19 32)(13 25 20)(14 21 26)(15 27 22)(16 23 28)(33 56 41)(34 42 49)(35 50 43)(36 44 51)(37 52 45)(38 46 53)(39 54 47)(40 48 55)
(1 37 20)(2 38 21)(3 39 22)(4 40 23)(5 33 24)(6 34 17)(7 35 18)(8 36 19)(9 62 56)(10 63 49)(11 64 50)(12 57 51)(13 58 52)(14 59 53)(15 60 54)(16 61 55)(25 71 45)(26 72 46)(27 65 47)(28 66 48)(29 67 41)(30 68 42)(31 69 43)(32 70 44)
(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 29)(10 32)(11 27)(12 30)(13 25)(14 28)(15 31)(16 26)(17 19)(18 22)(21 23)(34 36)(35 39)(38 40)(41 56)(42 51)(43 54)(44 49)(45 52)(46 55)(47 50)(48 53)(57 68)(58 71)(59 66)(60 69)(61 72)(62 67)(63 70)(64 65)

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

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

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

41 conjugacy classes

 class 1 2A 2B 3A 3B 3C 3D 3E 3F 4A 4B 6A 6B 6C 6D 6E 6F 6G 6H 8A 8B 12A 12B 12C 12D 12E 12F ··· 12P 24A 24B 24C 24D order 1 2 2 3 3 3 3 3 3 4 4 6 6 6 6 6 6 6 6 8 8 12 12 12 12 12 12 ··· 12 24 24 24 24 size 1 1 36 2 3 3 6 6 6 2 4 2 3 3 6 6 6 36 36 18 18 4 4 4 6 6 12 ··· 12 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⋊10SD16 S3 D4 D6 SD16 C3×S3 C3⋊D4 C3×D4 S3×C6 C3×SD16 C3×C3⋊D4 Q8⋊2S3 C3×Q8⋊2S3 C32⋊C6 C2×C32⋊C6 He3⋊6D4 kernel He3⋊10SD16 He3⋊3C8 He3⋊4D4 Q8×He3 C32⋊11SD16 C32⋊4C8 C12⋊S3 Q8×C32 C1 Q8×C32 C2×He3 C3×C12 He3 C3×Q8 C3×C6 C3×C6 C12 C32 C6 C32 C3 Q8 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 He310SD16 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 0 1 0 0 0 0 0 0 0 0 0 0 72 1 0 0 0 0 0 0 0 0 72 0 0 0 0 0 0 0 72 72 0 72 72 72 0 0 0 0 0 0 1 0 1 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 1 0 0 0 0 0 0 0 0 0 0 72 1 0 0 0 0 0 0 0 0 72 0 0 0 0 0 0 0 0 0 0 0 72 1 0 0 0 0 0 0 0 0 72 0 0 0 0 0 0 0 1 0 1 0 0 1 0 0 0 0 0 72 0 72 72 72
,
 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 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 72 1 0 0 0 0 72 72 72 72 71 72 0 0 0 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 1 0
,
 67 6 67 6 0 0 0 0 0 0 0 6 0 6 0 0 0 0 0 0 6 67 67 6 0 0 0 0 0 0 0 67 0 6 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 72 72 72 72 72 72
,
 1 72 0 0 0 0 0 0 0 0 0 72 0 0 0 0 0 0 0 0 0 0 72 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 72 72 72 72 72 72

G:=sub<GL(10,GF(73))| [0,1,0,0,0,0,0,0,0,0,72,72,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,72,72,0,0,0,0,0,0,0,0,0,0,1,0,0,0,72,0,0,0,0,0,0,1,0,0,72,0,0,0,0,0,0,0,72,72,0,1,0,0,0,0,0,0,1,0,72,0,0,0,0,0,0,0,0,0,72,1,0,0,0,0,0,0,0,0,72,0],[1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,72,72,0,0,1,0,0,0,0,0,1,0,0,0,0,72,0,0,0,0,0,0,72,72,1,0,0,0,0,0,0,0,1,0,0,72,0,0,0,0,0,0,0,0,0,72,0,0,0,0,0,0,0,0,1,72],[1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,72,0,1,0,0,0,0,0,0,0,72,0,0,0,0,0,0,1,0,0,72,0,0,0,0,0,0,0,1,0,72,0,0,0,0,0,0,0,0,72,71,1,1,0,0,0,0,0,0,1,72,0,0],[67,0,6,0,0,0,0,0,0,0,6,6,67,67,0,0,0,0,0,0,67,0,67,0,0,0,0,0,0,0,6,6,6,6,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,72,0,0,0,0,1,0,0,0,0,72,0,0,0,0,0,0,0,1,0,72,0,0,0,0,0,0,1,0,0,72,0,0,0,0,0,0,0,0,1,72,0,0,0,0,0,0,0,0,0,72],[1,0,0,0,0,0,0,0,0,0,72,72,0,0,0,0,0,0,0,0,0,0,72,0,0,0,0,0,0,0,0,0,1,1,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,72,0,0,0,0,1,0,0,0,0,72,0,0,0,0,0,0,0,1,0,72,0,0,0,0,0,0,1,0,0,72,0,0,0,0,0,0,0,0,1,72,0,0,0,0,0,0,0,0,0,72] >;

He310SD16 in GAP, Magma, Sage, TeX

{\rm He}_3\rtimes_{10}{\rm SD}_{16}
% in TeX

G:=Group("He3:10SD16");
// GroupNames label

G:=SmallGroup(432,161);
// by ID

G=gap.SmallGroup(432,161);
# by ID

G:=PCGroup([7,-2,-2,-3,-2,-2,-3,-3,197,176,1011,514,80,4037,2035,14118]);
// 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=e*a*e=a^-1,b*c=c*b,d*b*d^-1=e*b*e=b^-1,c*d=d*c,c*e=e*c,e*d*e=d^3>;
// generators/relations

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