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G = He3⋊A4order 324 = 22·34

1st semidirect product of He3 and A4 acting via A4/C22=C3

metabelian, soluble, monomial

Aliases: He31A4, C62.2C32, C32⋊A42C3, (C2×C6).6He3, C32.2(C3×A4), C3.7(C32⋊A4), (C22×He3)⋊1C3, C222(He3⋊C3), (C3×C3.A4)⋊3C3, SmallGroup(324,54)

Series: Derived Chief Lower central Upper central

C1C62 — He3⋊A4
C1C22C2×C6C62C32⋊A4 — He3⋊A4
C22C2×C6C62 — He3⋊A4
C1C3C32He3

Generators and relations for He3⋊A4
 G = < a,b,c,d,e,f | a3=b3=c3=d2=e2=f3=1, ab=ba, cac-1=faf-1=ab-1, ad=da, ae=ea, bc=cb, bd=db, be=eb, bf=fb, cd=dc, ce=ec, fcf-1=a-1b-1c, fdf-1=de=ed, fef-1=d >

3C2
3C3
9C3
36C3
36C3
3C6
9C6
9C6
9C6
9C6
3C32
12C32
12C32
12C9
3C2×C6
9A4
9C2×C6
9A4
3C3×C6
3C3×C6
3C3×C6
3C3×C6
4He3
4C3×C9
4He3
3C62
3C3×A4
3C3×A4
3C3.A4
3C2×He3
4He3⋊C3

Character table of He3⋊A4

 class 123A3B3C3D3E3F3G3H3I3J6A6B6C6D6E6F6G6H6I6J9A9B9C9D9E9F
 size 13113399363636363399999999121212121212
ρ11111111111111111111111111111    trivial
ρ2111111ζ3ζ321ζ31ζ3211ζ32ζ3ζ3ζ31ζ32ζ321ζ32ζ32ζ3ζ3ζ3ζ32    linear of order 3
ρ3111111ζ3ζ32ζ31ζ32111ζ32ζ3ζ3ζ31ζ32ζ321ζ3ζ3ζ32ζ32ζ32ζ3    linear of order 3
ρ4111111ζ3ζ32ζ32ζ32ζ3ζ311ζ32ζ3ζ3ζ31ζ32ζ321111111    linear of order 3
ρ511111111ζ3ζ32ζ32ζ31111111111ζ32ζ32ζ3ζ3ζ3ζ32    linear of order 3
ρ611111111ζ32ζ3ζ3ζ321111111111ζ3ζ3ζ32ζ32ζ32ζ3    linear of order 3
ρ7111111ζ32ζ3ζ321ζ3111ζ3ζ32ζ32ζ321ζ3ζ31ζ32ζ32ζ3ζ3ζ3ζ32    linear of order 3
ρ8111111ζ32ζ31ζ321ζ311ζ3ζ32ζ32ζ321ζ3ζ31ζ3ζ3ζ32ζ32ζ32ζ3    linear of order 3
ρ9111111ζ32ζ3ζ3ζ3ζ32ζ3211ζ3ζ32ζ32ζ321ζ3ζ31111111    linear of order 3
ρ103-13333330000-1-1-1-1-1-1-1-1-1-1000000    orthogonal lifted from A4
ρ113-13333-3+3-3/2-3-3-3/20000-1-1ζ6ζ65ζ65ζ65-1ζ6ζ6-1000000    complex lifted from C3×A4
ρ123-133-3+3-3/2-3-3-3/2000000-1-1-1--32-1+-3-1--3ζ6-1+-32ζ65000000    complex lifted from C32⋊A4
ρ133-133-3-3-3/2-3+3-3/2000000-1-12-1+-32-1--3ζ65-1+-3-1--3ζ6000000    complex lifted from C32⋊A4
ρ143333-3+3-3/2-3-3-3/2000000330000-3-3-3/200-3+3-3/2000000    complex lifted from He3
ρ153-13333-3-3-3/2-3+3-3/20000-1-1ζ65ζ6ζ6ζ6-1ζ65ζ65-1000000    complex lifted from C3×A4
ρ163333-3-3-3/2-3+3-3/2000000330000-3+3-3/200-3-3-3/2000000    complex lifted from He3
ρ173-133-3+3-3/2-3-3-3/2000000-1-1-1+-3-1+-3-1--32ζ62-1--3ζ65000000    complex lifted from C32⋊A4
ρ183-133-3-3-3/2-3+3-3/2000000-1-1-1+-32-1--3-1+-3ζ65-1--32ζ6000000    complex lifted from C32⋊A4
ρ193-133-3+3-3/2-3-3-3/2000000-1-12-1--32-1+-3ζ6-1--3-1+-3ζ65000000    complex lifted from C32⋊A4
ρ203-133-3-3-3/2-3+3-3/2000000-1-1-1--3-1--3-1+-32ζ652-1+-3ζ6000000    complex lifted from C32⋊A4
ρ2133-3+3-3/2-3-3-3/200000000-3+3-3/2-3-3-3/200000000ζ98+2ζ95ζ95+2ζ929794949ζ97+2ζ99892    complex lifted from He3⋊C3
ρ2233-3-3-3/2-3+3-3/200000000-3-3-3/2-3+3-3/200000000ζ97+2ζ9949ζ98+2ζ959892ζ95+2ζ929794    complex lifted from He3⋊C3
ρ2333-3-3-3/2-3+3-3/200000000-3-3-3/2-3+3-3/2000000009794ζ97+2ζ99892ζ95+2ζ92ζ98+2ζ95949    complex lifted from He3⋊C3
ρ2433-3-3-3/2-3+3-3/200000000-3-3-3/2-3+3-3/2000000009499794ζ95+2ζ92ζ98+2ζ959892ζ97+2ζ9    complex lifted from He3⋊C3
ρ2533-3+3-3/2-3-3-3/200000000-3+3-3/2-3-3-3/200000000ζ95+2ζ929892ζ97+2ζ99794949ζ98+2ζ95    complex lifted from He3⋊C3
ρ2633-3+3-3/2-3-3-3/200000000-3+3-3/2-3-3-3/2000000009892ζ98+2ζ95949ζ97+2ζ99794ζ95+2ζ92    complex lifted from He3⋊C3
ρ279-3-9-9-3/2-9+9-3/2000000003+3-3/23-3-3/200000000000000    complex faithful
ρ289-3-9+9-3/2-9-9-3/2000000003-3-3/23+3-3/200000000000000    complex faithful

Smallest permutation representation of He3⋊A4
On 54 points
Generators in S54
(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)
(1 10 22)(2 11 23)(3 12 24)(4 16 26)(5 17 27)(6 18 25)(7 43 49)(8 44 50)(9 45 51)(13 53 41)(14 54 42)(15 52 40)(19 36 48)(20 34 46)(21 35 47)(28 38 32)(29 39 33)(30 37 31)
(2 11 23)(3 24 12)(4 26 16)(6 18 25)(7 45 44)(8 49 9)(13 52 54)(14 41 15)(19 34 21)(20 47 48)(28 39 30)(29 31 32)(33 37 38)(35 36 46)(40 42 53)(43 51 50)
(1 17)(2 18)(3 16)(4 24)(5 22)(6 23)(7 53)(8 54)(9 52)(10 27)(11 25)(12 26)(13 49)(14 50)(15 51)(40 45)(41 43)(42 44)
(7 53)(8 54)(9 52)(13 49)(14 50)(15 51)(19 38)(20 39)(21 37)(28 48)(29 46)(30 47)(31 35)(32 36)(33 34)(40 45)(41 43)(42 44)
(1 9 29)(2 43 31)(3 50 38)(4 42 48)(5 15 34)(6 53 21)(7 37 23)(8 32 12)(10 45 39)(11 49 30)(13 47 25)(14 19 16)(17 52 46)(18 41 35)(20 27 40)(22 51 33)(24 44 28)(26 54 36)

G:=sub<Sym(54)| (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), (1,10,22)(2,11,23)(3,12,24)(4,16,26)(5,17,27)(6,18,25)(7,43,49)(8,44,50)(9,45,51)(13,53,41)(14,54,42)(15,52,40)(19,36,48)(20,34,46)(21,35,47)(28,38,32)(29,39,33)(30,37,31), (2,11,23)(3,24,12)(4,26,16)(6,18,25)(7,45,44)(8,49,9)(13,52,54)(14,41,15)(19,34,21)(20,47,48)(28,39,30)(29,31,32)(33,37,38)(35,36,46)(40,42,53)(43,51,50), (1,17)(2,18)(3,16)(4,24)(5,22)(6,23)(7,53)(8,54)(9,52)(10,27)(11,25)(12,26)(13,49)(14,50)(15,51)(40,45)(41,43)(42,44), (7,53)(8,54)(9,52)(13,49)(14,50)(15,51)(19,38)(20,39)(21,37)(28,48)(29,46)(30,47)(31,35)(32,36)(33,34)(40,45)(41,43)(42,44), (1,9,29)(2,43,31)(3,50,38)(4,42,48)(5,15,34)(6,53,21)(7,37,23)(8,32,12)(10,45,39)(11,49,30)(13,47,25)(14,19,16)(17,52,46)(18,41,35)(20,27,40)(22,51,33)(24,44,28)(26,54,36)>;

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), (1,10,22)(2,11,23)(3,12,24)(4,16,26)(5,17,27)(6,18,25)(7,43,49)(8,44,50)(9,45,51)(13,53,41)(14,54,42)(15,52,40)(19,36,48)(20,34,46)(21,35,47)(28,38,32)(29,39,33)(30,37,31), (2,11,23)(3,24,12)(4,26,16)(6,18,25)(7,45,44)(8,49,9)(13,52,54)(14,41,15)(19,34,21)(20,47,48)(28,39,30)(29,31,32)(33,37,38)(35,36,46)(40,42,53)(43,51,50), (1,17)(2,18)(3,16)(4,24)(5,22)(6,23)(7,53)(8,54)(9,52)(10,27)(11,25)(12,26)(13,49)(14,50)(15,51)(40,45)(41,43)(42,44), (7,53)(8,54)(9,52)(13,49)(14,50)(15,51)(19,38)(20,39)(21,37)(28,48)(29,46)(30,47)(31,35)(32,36)(33,34)(40,45)(41,43)(42,44), (1,9,29)(2,43,31)(3,50,38)(4,42,48)(5,15,34)(6,53,21)(7,37,23)(8,32,12)(10,45,39)(11,49,30)(13,47,25)(14,19,16)(17,52,46)(18,41,35)(20,27,40)(22,51,33)(24,44,28)(26,54,36) );

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)], [(1,10,22),(2,11,23),(3,12,24),(4,16,26),(5,17,27),(6,18,25),(7,43,49),(8,44,50),(9,45,51),(13,53,41),(14,54,42),(15,52,40),(19,36,48),(20,34,46),(21,35,47),(28,38,32),(29,39,33),(30,37,31)], [(2,11,23),(3,24,12),(4,26,16),(6,18,25),(7,45,44),(8,49,9),(13,52,54),(14,41,15),(19,34,21),(20,47,48),(28,39,30),(29,31,32),(33,37,38),(35,36,46),(40,42,53),(43,51,50)], [(1,17),(2,18),(3,16),(4,24),(5,22),(6,23),(7,53),(8,54),(9,52),(10,27),(11,25),(12,26),(13,49),(14,50),(15,51),(40,45),(41,43),(42,44)], [(7,53),(8,54),(9,52),(13,49),(14,50),(15,51),(19,38),(20,39),(21,37),(28,48),(29,46),(30,47),(31,35),(32,36),(33,34),(40,45),(41,43),(42,44)], [(1,9,29),(2,43,31),(3,50,38),(4,42,48),(5,15,34),(6,53,21),(7,37,23),(8,32,12),(10,45,39),(11,49,30),(13,47,25),(14,19,16),(17,52,46),(18,41,35),(20,27,40),(22,51,33),(24,44,28),(26,54,36)]])

Matrix representation of He3⋊A4 in GL6(𝔽19)

010000
001000
100000
000100
000010
000001
,
1100000
0110000
0011000
000100
000010
000001
,
100000
070000
0011000
000100
000010
000001
,
100000
010000
001000
0001800
0001410
0000018
,
100000
010000
001000
000100
0005180
0009018
,
131515000
101510000
131310000
0005170
0008141
0003100

G:=sub<GL(6,GF(19))| [0,0,1,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[11,0,0,0,0,0,0,11,0,0,0,0,0,0,11,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,7,0,0,0,0,0,0,11,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,18,14,0,0,0,0,0,1,0,0,0,0,0,0,18],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,5,9,0,0,0,0,18,0,0,0,0,0,0,18],[13,10,13,0,0,0,15,15,13,0,0,0,15,10,10,0,0,0,0,0,0,5,8,3,0,0,0,17,14,10,0,0,0,0,1,0] >;

He3⋊A4 in GAP, Magma, Sage, TeX

{\rm He}_3\rtimes A_4
% in TeX

G:=Group("He3:A4");
// GroupNames label

G:=SmallGroup(324,54);
// by ID

G=gap.SmallGroup(324,54);
# by ID

G:=PCGroup([6,-3,-3,-3,-3,-2,2,145,650,224,4864,8753]);
// Polycyclic

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

Export

Subgroup lattice of He3⋊A4 in TeX
Character table of He3⋊A4 in TeX

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