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## G = He3⋊5D8order 432 = 24·33

### 2nd semidirect product of He3 and D8 acting via D8/C8=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C3 — C4×He3 — He3⋊5D8
 Chief series C1 — C3 — C32 — He3 — C2×He3 — C4×He3 — He3⋊5D4 — He3⋊5D8
 Lower central He3 — C2×He3 — C4×He3 — He3⋊5D8
 Upper central C1 — C6 — C12 — C24

Generators and relations for He35D8
G = < a,b,c,d,e | a3=b3=c3=d8=e2=1, ab=ba, cac-1=ab-1, ad=da, eae=a-1, bc=cb, bd=db, be=eb, cd=dc, ece=c-1, ede=d-1 >

Subgroups: 721 in 121 conjugacy classes, 31 normal (13 characteristic)
C1, C2, C2 [×2], C3, C3 [×4], C4, C22 [×2], S3 [×8], C6, C6 [×6], C8, D4 [×2], C32 [×4], C12, C12 [×4], D6 [×8], C2×C6 [×2], D8, C3×S3 [×8], C3×C6 [×4], C24, C24 [×4], D12 [×8], C3×D4 [×2], He3, C3×C12 [×4], S3×C6 [×8], D24 [×4], C3×D8, He3⋊C2 [×2], C2×He3, C3×C24 [×4], C3×D12 [×8], C4×He3, C2×He3⋊C2 [×2], C3×D24 [×4], C8×He3, He35D4 [×2], He35D8
Quotients: C1, C2 [×3], C22, S3 [×4], D4, D6 [×4], D8, C3⋊S3, D12 [×4], C2×C3⋊S3, D24 [×4], He3⋊C2, C12⋊S3, C2×He3⋊C2, C325D8, He35D4, He35D8

Smallest permutation representation of He35D8
On 72 points
Generators in S72
(1 59 47)(2 60 48)(3 61 41)(4 62 42)(5 63 43)(6 64 44)(7 57 45)(8 58 46)(9 26 51)(10 27 52)(11 28 53)(12 29 54)(13 30 55)(14 31 56)(15 32 49)(16 25 50)(17 70 38)(18 71 39)(19 72 40)(20 65 33)(21 66 34)(22 67 35)(23 68 36)(24 69 37)
(1 25 18)(2 26 19)(3 27 20)(4 28 21)(5 29 22)(6 30 23)(7 31 24)(8 32 17)(9 40 48)(10 33 41)(11 34 42)(12 35 43)(13 36 44)(14 37 45)(15 38 46)(16 39 47)(49 70 58)(50 71 59)(51 72 60)(52 65 61)(53 66 62)(54 67 63)(55 68 64)(56 69 57)
(1 71 39)(2 72 40)(3 65 33)(4 66 34)(5 67 35)(6 68 36)(7 69 37)(8 70 38)(9 19 51)(10 20 52)(11 21 53)(12 22 54)(13 23 55)(14 24 56)(15 17 49)(16 18 50)(25 59 47)(26 60 48)(27 61 41)(28 62 42)(29 63 43)(30 64 44)(31 57 45)(32 58 46)
(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)
(1 8)(2 7)(3 6)(4 5)(9 56)(10 55)(11 54)(12 53)(13 52)(14 51)(15 50)(16 49)(17 18)(19 24)(20 23)(21 22)(25 32)(26 31)(27 30)(28 29)(33 68)(34 67)(35 66)(36 65)(37 72)(38 71)(39 70)(40 69)(41 64)(42 63)(43 62)(44 61)(45 60)(46 59)(47 58)(48 57)

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

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

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

53 conjugacy classes

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

53 irreducible representations

 dim 1 1 1 2 2 2 2 2 2 3 3 6 6 type + + + + + + + + + image C1 C2 C2 S3 D4 D6 D8 D12 D24 He3⋊C2 C2×He3⋊C2 He3⋊5D4 He3⋊5D8 kernel He3⋊5D8 C8×He3 He3⋊5D4 C3×C24 C2×He3 C3×C12 He3 C3×C6 C32 C8 C4 C2 C1 # reps 1 1 2 4 1 4 2 8 16 4 4 2 4

Matrix representation of He35D8 in GL5(𝔽73)

 1 0 0 0 0 0 1 0 0 0 0 0 72 1 0 0 0 72 0 0 0 0 7 0 1
,
 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
,
 0 1 0 0 0 72 72 0 0 0 0 0 65 64 8 0 0 65 0 72 0 0 56 0 8
,
 55 5 0 0 0 68 50 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1
,
 18 68 0 0 0 50 55 0 0 0 0 0 1 72 0 0 0 0 72 0 0 0 0 7 1

G:=sub<GL(5,GF(73))| [1,0,0,0,0,0,1,0,0,0,0,0,72,72,7,0,0,1,0,0,0,0,0,0,1],[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],[0,72,0,0,0,1,72,0,0,0,0,0,65,65,56,0,0,64,0,0,0,0,8,72,8],[55,68,0,0,0,5,50,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1],[18,50,0,0,0,68,55,0,0,0,0,0,1,0,0,0,0,72,72,7,0,0,0,0,1] >;

He35D8 in GAP, Magma, Sage, TeX

{\rm He}_3\rtimes_5D_8
% in TeX

G:=Group("He3:5D8");
// GroupNames label

G:=SmallGroup(432,176);
// by ID

G=gap.SmallGroup(432,176);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-3,-3,-3,85,92,254,58,1124,4037,537]);
// 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,a*d=d*a,e*a*e=a^-1,b*c=c*b,b*d=d*b,b*e=e*b,c*d=d*c,e*c*e=c^-1,e*d*e=d^-1>;
// generators/relations

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