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G = He34D8order 432 = 24·33

1st semidirect product of He3 and D8 acting via D8/C8=C2

metabelian, supersoluble, monomial

Aliases: He34D8, C323D24, C325D8⋊C3, (C3×C24)⋊1C6, (C3×C24)⋊1S3, C24.4(C3×S3), (C8×He3)⋊1C2, C6.6(C3×D12), C3.2(C3×D24), C12⋊S31C6, C322(C3×D8), C12.69(S3×C6), C81(C32⋊C6), (C3×C12).42D6, (C3×C6).15D12, He34D410C2, (C2×He3).19D4, C2.5(He34D4), (C4×He3).34C22, (C3×C6).8(C3×D4), (C3×C12).10(C2×C6), C4.10(C2×C32⋊C6), SmallGroup(432,118)

Series: Derived Chief Lower central Upper central

C1C3×C12 — He34D8
C1C3C32C3×C6C3×C12C4×He3He34D4 — He34D8
C32C3×C6C3×C12 — He34D8
C1C2C4C8

Generators and relations for He34D8
 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, ebe=b-1, cd=dc, ce=ec, ede=d-1 >

Subgroups: 601 in 85 conjugacy classes, 26 normal (22 characteristic)
C1, C2, C2 [×2], C3, C3 [×3], C4, C22 [×2], S3 [×4], C6, C6 [×5], C8, D4 [×2], C32 [×2], C32, C12, C12 [×3], D6 [×4], C2×C6 [×2], D8, C3×S3 [×2], C3⋊S3 [×2], C3×C6 [×2], C3×C6, C24, C24 [×3], D12 [×4], C3×D4 [×2], He3, C3×C12 [×2], C3×C12, S3×C6 [×2], C2×C3⋊S3 [×2], D24 [×2], C3×D8, C32⋊C6 [×2], C2×He3, C3×C24 [×2], C3×C24, C3×D12 [×2], C12⋊S3 [×2], C4×He3, C2×C32⋊C6 [×2], C3×D24, C325D8, C8×He3, He34D4 [×2], He34D8
Quotients: C1, C2 [×3], C3, C22, S3, C6 [×3], D4, D6, C2×C6, D8, C3×S3, D12, C3×D4, S3×C6, D24, C3×D8, C32⋊C6, C3×D12, C2×C32⋊C6, C3×D24, He34D4, He34D8

Smallest permutation representation of He34D8
On 72 points
Generators in S72
(1 61 22)(2 62 23)(3 63 24)(4 64 17)(5 57 18)(6 58 19)(7 59 20)(8 60 21)(9 52 37)(10 53 38)(11 54 39)(12 55 40)(13 56 33)(14 49 34)(15 50 35)(16 51 36)(25 48 70)(26 41 71)(27 42 72)(28 43 65)(29 44 66)(30 45 67)(31 46 68)(32 47 69)
(1 15 45)(2 16 46)(3 9 47)(4 10 48)(5 11 41)(6 12 42)(7 13 43)(8 14 44)(17 38 25)(18 39 26)(19 40 27)(20 33 28)(21 34 29)(22 35 30)(23 36 31)(24 37 32)(49 66 60)(50 67 61)(51 68 62)(52 69 63)(53 70 64)(54 71 57)(55 72 58)(56 65 59)
(17 25 38)(18 26 39)(19 27 40)(20 28 33)(21 29 34)(22 30 35)(23 31 36)(24 32 37)(49 66 60)(50 67 61)(51 68 62)(52 69 63)(53 70 64)(54 71 57)(55 72 58)(56 65 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)
(1 7)(2 6)(3 5)(9 41)(10 48)(11 47)(12 46)(13 45)(14 44)(15 43)(16 42)(17 64)(18 63)(19 62)(20 61)(21 60)(22 59)(23 58)(24 57)(25 53)(26 52)(27 51)(28 50)(29 49)(30 56)(31 55)(32 54)(33 67)(34 66)(35 65)(36 72)(37 71)(38 70)(39 69)(40 68)

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

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

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

53 conjugacy classes

class 1 2A2B2C3A3B3C3D3E3F 4 6A6B6C6D6E6F6G6H6I6J8A8B12A12B12C···12J24A24B24C24D24E···24T
order12223333334666666666688121212···122424242424···24
size11363623366622336663636363622226···622226···6

53 irreducible representations

dim1111112222222222226666
type+++++++++++++
imageC1C2C2C3C6C6S3D4D6D8C3×S3D12C3×D4S3×C6D24C3×D8C3×D12C3×D24C32⋊C6C2×C32⋊C6He34D4He34D8
kernelHe34D8C8×He3He34D4C325D8C3×C24C12⋊S3C3×C24C2×He3C3×C12He3C24C3×C6C3×C6C12C32C32C6C3C8C4C2C1
# reps1122241112222244481124

Matrix representation of He34D8 in GL8(𝔽73)

01000000
7272000000
00001000
00000100
00000010
00000001
00100000
00010000
,
10000000
01000000
00010000
0072720000
00000100
0000727200
00000001
0000007272
,
80000000
08000000
00100000
00010000
0000727200
00001000
00000001
0000007272
,
5068000000
555000000
007200000
000720000
000072000
000007200
000000720
000000072
,
6659000000
667000000
00100000
0072720000
00000010
0000007272
00001000
0000727200

G:=sub<GL(8,GF(73))| [0,72,0,0,0,0,0,0,1,72,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,72,0,0,0,0,0,0,1,72,0,0,0,0,0,0,0,0,0,72,0,0,0,0,0,0,1,72,0,0,0,0,0,0,0,0,0,72,0,0,0,0,0,0,1,72],[8,0,0,0,0,0,0,0,0,8,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,72,1,0,0,0,0,0,0,72,0,0,0,0,0,0,0,0,0,0,72,0,0,0,0,0,0,1,72],[50,5,0,0,0,0,0,0,68,55,0,0,0,0,0,0,0,0,72,0,0,0,0,0,0,0,0,72,0,0,0,0,0,0,0,0,72,0,0,0,0,0,0,0,0,72,0,0,0,0,0,0,0,0,72,0,0,0,0,0,0,0,0,72],[66,66,0,0,0,0,0,0,59,7,0,0,0,0,0,0,0,0,1,72,0,0,0,0,0,0,0,72,0,0,0,0,0,0,0,0,0,0,1,72,0,0,0,0,0,0,0,72,0,0,0,0,1,72,0,0,0,0,0,0,0,72,0,0] >;

He34D8 in GAP, Magma, Sage, TeX

{\rm He}_3\rtimes_4D_8
% in TeX

G:=Group("He3:4D8");
// GroupNames label

G:=SmallGroup(432,118);
// by ID

G=gap.SmallGroup(432,118);
# by ID

G:=PCGroup([7,-2,-2,-3,-2,-2,-3,-3,197,260,1011,80,4037,2035,14118]);
// 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,a*d=d*a,e*a*e=a^-1,b*c=c*b,b*d=d*b,e*b*e=b^-1,c*d=d*c,c*e=e*c,e*d*e=d^-1>;
// generators/relations

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