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G = He3⋊D8order 432 = 24·33

The semidirect product of He3 and D8 acting via D8/C2=D4

non-abelian, soluble

Aliases: He3⋊D8, C6.2S3≀C2, C3.(C32⋊D8), He32C81C2, He32D41C2, C2.4(He3⋊D4), (C2×He3).2D4, He33C4.2C22, SmallGroup(432,235)

Series: Derived Chief Lower central Upper central

C1C3He3He33C4 — He3⋊D8
C1C3He3C2×He3He33C4He32D4 — He3⋊D8
He3C2×He3He33C4 — He3⋊D8
C1C2

Generators and relations for He3⋊D8
 G = < a,b,c,d,e | a3=b3=c3=d8=e2=1, eae=ab=ba, cac-1=dcd-1=ab-1, dad-1=bc-1, bc=cb, bd=db, ebe=b-1, ece=c-1, ede=d-1 >

Subgroups: 679 in 65 conjugacy classes, 11 normal (9 characteristic)
C1, C2, C2 [×2], C3, C3 [×2], C4, C22 [×2], S3 [×4], C6, C6 [×4], C8, D4 [×2], C32 [×2], Dic3 [×2], C12, D6 [×4], C2×C6 [×2], D8, C3×S3 [×2], C3⋊S3 [×2], C3×C6 [×2], C24, D12 [×2], C3⋊D4 [×2], He3, C3×Dic3 [×2], S3×C6 [×2], C2×C3⋊S3 [×2], D24, C32⋊C6 [×2], C2×He3, C3⋊D12 [×2], He33C4, C2×C32⋊C6 [×2], He32C8, He32D4 [×2], He3⋊D8
Quotients: C1, C2 [×3], C22, D4, D8, S3≀C2, C32⋊D8, He3⋊D4, He3⋊D8

Character table of He3⋊D8

 class 12A2B2C3A3B3C46A6B6C6D6E6F6G8A8B12A12B24A24B24C24D
 size 113636212121821212363636361818181818181818
ρ111111111111111111111111    trivial
ρ211-111111111-1-111-1-111-1-1-1-1    linear of order 2
ρ311-1-11111111-1-1-1-111111111    linear of order 2
ρ4111-1111111111-1-1-1-111-1-1-1-1    linear of order 2
ρ52200222-2222000000-2-20000    orthogonal lifted from D4
ρ62-2002220-2-2-20000-22002-22-2    orthogonal lifted from D8
ρ72-2002220-2-2-200002-200-22-22    orthogonal lifted from D8
ρ844024-2104-2100-1-100000000    orthogonal lifted from S3≀C2
ρ9440-24-2104-21001100000000    orthogonal lifted from S3≀C2
ρ10442041-2041-2-1-10000000000    orthogonal lifted from S3≀C2
ρ1144-2041-2041-2110000000000    orthogonal lifted from S3≀C2
ρ124-40041-20-4-12--3-30000000000    complex lifted from C32⋊D8
ρ134-4004-210-42-100-3--300000000    complex lifted from C32⋊D8
ρ144-4004-210-42-100--3-300000000    complex lifted from C32⋊D8
ρ154-40041-20-4-12-3--30000000000    complex lifted from C32⋊D8
ρ166600-300-2-30000002211-1-1-1-1    orthogonal lifted from He3⋊D4
ρ176600-300-2-3000000-2-2111111    orthogonal lifted from He3⋊D4
ρ186600-3002-300000000-1-1-333-3    orthogonal lifted from He3⋊D4
ρ196600-3002-300000000-1-13-3-33    orthogonal lifted from He3⋊D4
ρ206-600-30003000000-22-33ζ83ζ3838ζ3ζ87ζ328785ζ32ζ87ζ385ζ385ζ83ζ328ζ328    orthogonal faithful
ρ216-600-300030000002-2-33ζ83ζ328ζ328ζ87ζ385ζ385ζ87ζ328785ζ32ζ83ζ3838ζ3    orthogonal faithful
ρ226-600-30003000000-223-3ζ87ζ385ζ385ζ83ζ328ζ328ζ83ζ3838ζ3ζ87ζ328785ζ32    orthogonal faithful
ρ236-600-300030000002-23-3ζ87ζ328785ζ32ζ83ζ3838ζ3ζ83ζ328ζ328ζ87ζ385ζ385    orthogonal faithful

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

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

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

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

Matrix representation of He3⋊D8 in GL10(𝔽73)

1000000000
0100000000
3057721000000
106720000000
0000010000
000072720000
000000727200
0000001000
0000000010
0000000001
,
1000000000
0100000000
0010000000
0001000000
0000010000
000072720000
0000000100
000000727200
0000000001
000000007272
,
72100000000
72000000000
14110000000
463601000000
0000001000
0000000100
0000000010
0000000001
0000100000
0000010000
,
48645634000000
72203917000000
35575756000000
5022721000000
0000244824482448
0000254925492549
0000244825492449
0000254924492448
0000244824492549
0000254924482449
,
0100000000
1000000000
37594360000000
9521330000000
0000100000
000072720000
0000000010
000000007272
0000001000
000000727200

G:=sub<GL(10,GF(73))| [1,0,30,10,0,0,0,0,0,0,0,1,57,6,0,0,0,0,0,0,0,0,72,72,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,72,0,0,0,0,0,0,0,0,1,72,0,0,0,0,0,0,0,0,0,0,72,1,0,0,0,0,0,0,0,0,72,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,72,0,0,0,0,0,0,0,0,1,72,0,0,0,0,0,0,0,0,0,0,0,72,0,0,0,0,0,0,0,0,1,72,0,0,0,0,0,0,0,0,0,0,0,72,0,0,0,0,0,0,0,0,1,72],[72,72,1,46,0,0,0,0,0,0,1,0,41,36,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0],[48,72,35,50,0,0,0,0,0,0,64,20,57,22,0,0,0,0,0,0,56,39,57,7,0,0,0,0,0,0,34,17,56,21,0,0,0,0,0,0,0,0,0,0,24,25,24,25,24,25,0,0,0,0,48,49,48,49,48,49,0,0,0,0,24,25,25,24,24,24,0,0,0,0,48,49,49,49,49,48,0,0,0,0,24,25,24,24,25,24,0,0,0,0,48,49,49,48,49,49],[0,1,37,9,0,0,0,0,0,0,1,0,59,52,0,0,0,0,0,0,0,0,43,13,0,0,0,0,0,0,0,0,60,30,0,0,0,0,0,0,0,0,0,0,1,72,0,0,0,0,0,0,0,0,0,72,0,0,0,0,0,0,0,0,0,0,0,0,1,72,0,0,0,0,0,0,0,0,0,72,0,0,0,0,0,0,1,72,0,0,0,0,0,0,0,0,0,72,0,0] >;

He3⋊D8 in GAP, Magma, Sage, TeX

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

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

G:=SmallGroup(432,235);
// by ID

G=gap.SmallGroup(432,235);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-3,3,-3,85,254,135,58,1124,851,298,348,1027,537,14118,7069]);
// Polycyclic

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

Export

Character table of He3⋊D8 in TeX

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