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G = He35D8order 432 = 24·33

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

non-abelian, supersoluble, monomial

Aliases: He35D8, C324D24, (C3×C24)⋊2S3, (C8×He3)⋊2C2, C24.5(C3⋊S3), He35D47C2, (C3×C12).47D6, (C3×C6).24D12, C81(He3⋊C2), (C2×He3).23D4, C3.2(C325D8), C2.4(He35D4), C6.28(C12⋊S3), (C4×He3).36C22, C12.80(C2×C3⋊S3), C4.9(C2×He3⋊C2), SmallGroup(432,176)

Series: Derived Chief Lower central Upper central

C1C3C4×He3 — He35D8
C1C3C32He3C2×He3C4×He3He35D4 — He35D8
He3C2×He3C4×He3 — He35D8
C1C6C12C24

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 2A2B2C3A3B3C3D3E3F 4 6A6B6C6D6E6F6G6H6I6J8A8B12A12B12C···12J24A24B24C24D24E···24T
order12223333334666666666688121212···122424242424···24
size11363611666621166663636363622226···622226···6

53 irreducible representations

dim1112222223366
type+++++++++
imageC1C2C2S3D4D6D8D12D24He3⋊C2C2×He3⋊C2He35D4He35D8
kernelHe35D8C8×He3He35D4C3×C24C2×He3C3×C12He3C3×C6C32C8C4C2C1
# reps11241428164424

Matrix representation of He35D8 in GL5(𝔽73)

10000
01000
007210
007200
00701
,
10000
01000
006400
000640
000064
,
01000
7272000
0065648
0065072
005608
,
555000
6850000
00100
00010
00001
,
1868000
5055000
001720
000720
00071

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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