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## G = D8×He3order 432 = 24·33

### Direct product of D8 and He3

direct product, metabelian, nilpotent (class 3), monomial

Series: Derived Chief Lower central Upper central

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

Generators and relations for D8×He3
G = < a,b,c,d,e | a8=b2=c3=d3=e3=1, bab=a-1, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, cd=dc, ece-1=cd-1, de=ed >

Subgroups: 361 in 121 conjugacy classes, 49 normal (15 characteristic)
C1, C2, C2, C3, C3, C4, C22, C6, C6, C8, D4, C32, C12, C12, C2×C6, D8, C3×C6, C3×C6, C24, C24, C3×D4, C3×D4, He3, C3×C12, C62, C3×D8, C3×D8, C2×He3, C2×He3, C3×C24, D4×C32, C4×He3, C22×He3, C32×D8, C8×He3, D4×He3, D8×He3
Quotients: C1, C2, C3, C22, C6, D4, C32, C2×C6, D8, C3×C6, C3×D4, He3, C62, C3×D8, C2×He3, D4×C32, C22×He3, C32×D8, D4×He3, D8×He3

Smallest permutation representation of D8×He3
On 72 points
Generators in S72
(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 16)(10 15)(11 14)(12 13)(17 24)(18 23)(19 22)(20 21)(25 30)(26 29)(27 28)(31 32)(33 34)(35 40)(36 39)(37 38)(41 44)(42 43)(45 48)(46 47)(49 50)(51 56)(52 55)(53 54)(57 64)(58 63)(59 62)(60 61)(65 68)(66 67)(69 72)(70 71)
(9 71 21)(10 72 22)(11 65 23)(12 66 24)(13 67 17)(14 68 18)(15 69 19)(16 70 20)(33 42 49)(34 43 50)(35 44 51)(36 45 52)(37 46 53)(38 47 54)(39 48 55)(40 41 56)
(1 28 61)(2 29 62)(3 30 63)(4 31 64)(5 32 57)(6 25 58)(7 26 59)(8 27 60)(9 71 21)(10 72 22)(11 65 23)(12 66 24)(13 67 17)(14 68 18)(15 69 19)(16 70 20)(33 49 42)(34 50 43)(35 51 44)(36 52 45)(37 53 46)(38 54 47)(39 55 48)(40 56 41)
(1 50 71)(2 51 72)(3 52 65)(4 53 66)(5 54 67)(6 55 68)(7 56 69)(8 49 70)(9 61 34)(10 62 35)(11 63 36)(12 64 37)(13 57 38)(14 58 39)(15 59 40)(16 60 33)(17 32 47)(18 25 48)(19 26 41)(20 27 42)(21 28 43)(22 29 44)(23 30 45)(24 31 46)

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

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

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

77 conjugacy classes

 class 1 2A 2B 2C 3A 3B 3C ··· 3J 4 6A 6B 6C ··· 6J 6K 6L 6M 6N 6O ··· 6AD 8A 8B 12A 12B 12C ··· 12J 24A 24B 24C 24D 24E ··· 24T order 1 2 2 2 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 4 4 1 1 3 ··· 3 2 1 1 3 ··· 3 4 4 4 4 12 ··· 12 2 2 2 2 6 ··· 6 2 2 2 2 6 ··· 6

77 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 3 3 3 6 6 type + + + + + image C1 C2 C2 C3 C6 C6 D4 D8 C3×D4 C3×D8 He3 C2×He3 C2×He3 D4×He3 D8×He3 kernel D8×He3 C8×He3 D4×He3 C32×D8 C3×C24 D4×C32 C2×He3 He3 C3×C6 C32 D8 C8 D4 C2 C1 # reps 1 1 2 8 8 16 1 2 8 16 2 2 4 2 4

Matrix representation of D8×He3 in GL5(𝔽73)

 0 41 0 0 0 16 41 0 0 0 0 0 72 0 0 0 0 0 72 0 0 0 0 0 72
,
 0 41 0 0 0 57 0 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1
,
 64 0 0 0 0 0 64 0 0 0 0 0 1 0 0 0 0 0 64 0 0 0 0 0 8
,
 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 0 64 0 0 0 0 0 64 0 0 64 0 0

G:=sub<GL(5,GF(73))| [0,16,0,0,0,41,41,0,0,0,0,0,72,0,0,0,0,0,72,0,0,0,0,0,72],[0,57,0,0,0,41,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1],[64,0,0,0,0,0,64,0,0,0,0,0,1,0,0,0,0,0,64,0,0,0,0,0,8],[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],[1,0,0,0,0,0,1,0,0,0,0,0,0,0,64,0,0,64,0,0,0,0,0,64,0] >;

D8×He3 in GAP, Magma, Sage, TeX

D_8\times {\rm He}_3
% in TeX

G:=Group("D8xHe3");
// GroupNames label

G:=SmallGroup(432,216);
// by ID

G=gap.SmallGroup(432,216);
# by ID

G:=PCGroup([7,-2,-2,-3,-3,-2,-3,-2,533,605,8824,4421,242]);
// Polycyclic

G:=Group<a,b,c,d,e|a^8=b^2=c^3=d^3=e^3=1,b*a*b=a^-1,a*c=c*a,a*d=d*a,a*e=e*a,b*c=c*b,b*d=d*b,b*e=e*b,c*d=d*c,e*c*e^-1=c*d^-1,d*e=e*d>;
// generators/relations

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