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G = D8xHe3order 432 = 24·33

Direct product of D8 and He3

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

Aliases: D8xHe3, C12.18C62, D4:(C2xHe3), (C3xC24):3C6, C8:1(C2xHe3), (C32xD8):C3, C24.3(C3xC6), (C8xHe3):5C2, (D4xHe3):7C2, C32:6(C3xD8), C2.3(D4xHe3), (D4xC32):4C6, C3.2(C32xD8), (C2xHe3).41D4, (C3xD8).2C32, C6.29(D4xC32), C4.1(C22xHe3), (C4xHe3).50C22, (C3xD4).3(C3xC6), (C3xC6).32(C3xD4), (C3xC12).17(C2xC6), SmallGroup(432,216)

Series: Derived Chief Lower central Upper central

C1C12 — D8xHe3
C1C2C6C12C3xC12C4xHe3D4xHe3 — D8xHe3
C1C2C12 — D8xHe3
C1C6C4xHe3 — D8xHe3

Generators and relations for D8xHe3
 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, C2xC6, D8, C3xC6, C3xC6, C24, C24, C3xD4, C3xD4, He3, C3xC12, C62, C3xD8, C3xD8, C2xHe3, C2xHe3, C3xC24, D4xC32, C4xHe3, C22xHe3, C32xD8, C8xHe3, D4xHe3, D8xHe3
Quotients: C1, C2, C3, C22, C6, D4, C32, C2xC6, D8, C3xC6, C3xD4, He3, C62, C3xD8, C2xHe3, D4xC32, C22xHe3, C32xD8, D4xHe3, D8xHe3

Smallest permutation representation of D8xHe3
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 2A2B2C3A3B3C···3J 4 6A6B6C···6J6K6L6M6N6O···6AD8A8B12A12B12C···12J24A24B24C24D24E···24T
order1222333···34666···666666···688121212···122424242424···24
size1144113···32113···3444412···1222226···622226···6

77 irreducible representations

dim111111222233366
type+++++
imageC1C2C2C3C6C6D4D8C3xD4C3xD8He3C2xHe3C2xHe3D4xHe3D8xHe3
kernelD8xHe3C8xHe3D4xHe3C32xD8C3xC24D4xC32C2xHe3He3C3xC6C32D8C8D4C2C1
# reps11288161281622424

Matrix representation of D8xHe3 in GL5(F73)

041000
1641000
007200
000720
000072
,
041000
570000
00100
00010
00001
,
640000
064000
00100
000640
00008
,
10000
01000
006400
000640
000064
,
10000
01000
000640
000064
006400

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] >;

D8xHe3 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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