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G = He3.3Dic3order 324 = 22·34

3rd non-split extension by He3 of Dic3 acting via Dic3/C2=S3

non-abelian, supersoluble, monomial

Aliases: He3.3Dic3, 3- 1+22Dic3, (C3×C18).7S3, (C3×C9)⋊5Dic3, He3.C31C4, (C2×He3).5S3, C6.3(He3⋊C2), C3.3(He33C4), C2.(He3.3S3), C32.2(C3⋊Dic3), (C2×3- 1+2).2S3, (C3×C6).2(C3⋊S3), (C2×He3.C3).1C2, SmallGroup(324,23)

Series: Derived Chief Lower central Upper central

C1C32He3.C3 — He3.3Dic3
C1C3C32C3×C9He3.C3C2×He3.C3 — He3.3Dic3
He3.C3 — He3.3Dic3
C1C2

Generators and relations for He3.3Dic3
 G = < a,b,c,d,e | a3=b3=c3=1, d6=ebe-1=b-1, e2=b-1d3, ab=ba, cac-1=eae-1=ab-1, ad=da, bc=cb, bd=db, dcd-1=ab-1c, ece-1=c-1, ede-1=bd5 >

3C3
9C3
27C4
3C6
9C6
3C9
3C9
3C9
3C32
9Dic3
27C12
27Dic3
3C3×C6
3C18
3C18
3C18
3Dic9
3Dic9
3Dic9
3C3⋊Dic3
9C3×Dic3
3C3×Dic9
3C9⋊C12
3C9⋊C12
3C32⋊C12

Character table of He3.3Dic3

 class 123A3B3C3D4A4B6A6B6C6D9A9B9C9D9E12A12B12C12D18A18B18C18D18E
 size 11233182727233186661818272727276661818
ρ111111111111111111111111111    trivial
ρ2111111-1-1111111111-1-1-1-111111    linear of order 2
ρ31-11111i-i-1-1-1-111111i-ii-i-1-1-1-1-1    linear of order 4
ρ41-11111-ii-1-1-1-111111-ii-ii-1-1-1-1-1    linear of order 4
ρ522222-100222-1-1-1-12-10000-1-1-12-1    orthogonal lifted from S3
ρ6222222002222-1-1-1-1-10000-1-1-1-1-1    orthogonal lifted from S3
ρ722222-100222-1-1-1-1-120000-1-1-1-12    orthogonal lifted from S3
ρ822222-100222-1222-1-10000222-1-1    orthogonal lifted from S3
ρ92-2222-100-2-2-21-1-1-1-1200001111-2    symplectic lifted from Dic3, Schur index 2
ρ102-2222-100-2-2-21222-1-10000-2-2-211    symplectic lifted from Dic3, Schur index 2
ρ112-2222-100-2-2-21-1-1-12-10000111-21    symplectic lifted from Dic3, Schur index 2
ρ122-2222200-2-2-2-2-1-1-1-1-1000011111    symplectic lifted from Dic3, Schur index 2
ρ13333-3-3-3/2-3+3-3/20-1-13-3-3-3/2-3+3-3/2000000ζ6ζ65ζ65ζ600000    complex lifted from He3⋊C2
ρ14333-3+3-3/2-3-3-3/20-1-13-3+3-3/2-3-3-3/2000000ζ65ζ6ζ6ζ6500000    complex lifted from He3⋊C2
ρ15333-3-3-3/2-3+3-3/20113-3-3-3/2-3+3-3/2000000ζ32ζ3ζ3ζ3200000    complex lifted from He3⋊C2
ρ16333-3+3-3/2-3-3-3/20113-3+3-3/2-3-3-3/2000000ζ3ζ32ζ32ζ300000    complex lifted from He3⋊C2
ρ173-33-3-3-3/2-3+3-3/20-ii-33+3-3/23-3-3/2000000ζ43ζ32ζ4ζ3ζ43ζ3ζ4ζ3200000    complex lifted from He33C4
ρ183-33-3+3-3/2-3-3-3/20-ii-33-3-3/23+3-3/2000000ζ43ζ3ζ4ζ32ζ43ζ32ζ4ζ300000    complex lifted from He33C4
ρ193-33-3-3-3/2-3+3-3/20i-i-33+3-3/23-3-3/2000000ζ4ζ32ζ43ζ3ζ4ζ3ζ43ζ3200000    complex lifted from He33C4
ρ203-33-3+3-3/2-3-3-3/20i-i-33-3-3/23+3-3/2000000ζ4ζ3ζ43ζ32ζ4ζ32ζ43ζ300000    complex lifted from He33C4
ρ2166-300000-3000ζ989794+2ζ92ζ989492+2ζ99594929000000ζ989794+2ζ92ζ989492+2ζ9959492900    orthogonal lifted from He3.3S3
ρ2266-300000-3000ζ989492+2ζ99594929ζ989794+2ζ92000000ζ989492+2ζ99594929ζ989794+2ζ9200    orthogonal lifted from He3.3S3
ρ2366-300000-30009594929ζ989794+2ζ92ζ989492+2ζ90000009594929ζ989794+2ζ92ζ989492+2ζ900    orthogonal lifted from He3.3S3
ρ246-6-3000003000ζ989794+2ζ92ζ989492+2ζ99594929000000ζ95+2ζ9492998+2ζ979492989492900    symplectic faithful, Schur index 2
ρ256-6-30000030009594929ζ989794+2ζ92ζ989492+2ζ90000009894929ζ95+2ζ9492998+2ζ97949200    symplectic faithful, Schur index 2
ρ266-6-3000003000ζ989492+2ζ99594929ζ989794+2ζ9200000098+2ζ9794929894929ζ95+2ζ9492900    symplectic faithful, Schur index 2

Smallest permutation representation of He3.3Dic3
On 108 points
Generators in S108
(1 26 106)(2 27 107)(3 28 108)(4 29 91)(5 30 92)(6 31 93)(7 32 94)(8 33 95)(9 34 96)(10 35 97)(11 36 98)(12 19 99)(13 20 100)(14 21 101)(15 22 102)(16 23 103)(17 24 104)(18 25 105)(37 90 64)(38 73 65)(39 74 66)(40 75 67)(41 76 68)(42 77 69)(43 78 70)(44 79 71)(45 80 72)(46 81 55)(47 82 56)(48 83 57)(49 84 58)(50 85 59)(51 86 60)(52 87 61)(53 88 62)(54 89 63)
(1 13 7)(2 14 8)(3 15 9)(4 16 10)(5 17 11)(6 18 12)(19 31 25)(20 32 26)(21 33 27)(22 34 28)(23 35 29)(24 36 30)(37 49 43)(38 50 44)(39 51 45)(40 52 46)(41 53 47)(42 54 48)(55 67 61)(56 68 62)(57 69 63)(58 70 64)(59 71 65)(60 72 66)(73 85 79)(74 86 80)(75 87 81)(76 88 82)(77 89 83)(78 90 84)(91 103 97)(92 104 98)(93 105 99)(94 106 100)(95 107 101)(96 108 102)
(2 27 101)(3 108 34)(5 30 104)(6 93 19)(8 33 107)(9 96 22)(11 36 92)(12 99 25)(14 21 95)(15 102 28)(17 24 98)(18 105 31)(20 32 26)(23 35 29)(37 78 70)(38 71 73)(39 45 51)(40 81 55)(41 56 76)(42 48 54)(43 84 58)(44 59 79)(46 87 61)(47 62 82)(49 90 64)(50 65 85)(52 75 67)(53 68 88)(57 69 63)(60 72 66)(91 97 103)(94 100 106)
(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)(73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90)(91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108)
(1 80 10 89)(2 79 11 88)(3 78 12 87)(4 77 13 86)(5 76 14 85)(6 75 15 84)(7 74 16 83)(8 73 17 82)(9 90 18 81)(19 67 28 58)(20 66 29 57)(21 65 30 56)(22 64 31 55)(23 63 32 72)(24 62 33 71)(25 61 34 70)(26 60 35 69)(27 59 36 68)(37 99 46 108)(38 98 47 107)(39 97 48 106)(40 96 49 105)(41 95 50 104)(42 94 51 103)(43 93 52 102)(44 92 53 101)(45 91 54 100)

G:=sub<Sym(108)| (1,26,106)(2,27,107)(3,28,108)(4,29,91)(5,30,92)(6,31,93)(7,32,94)(8,33,95)(9,34,96)(10,35,97)(11,36,98)(12,19,99)(13,20,100)(14,21,101)(15,22,102)(16,23,103)(17,24,104)(18,25,105)(37,90,64)(38,73,65)(39,74,66)(40,75,67)(41,76,68)(42,77,69)(43,78,70)(44,79,71)(45,80,72)(46,81,55)(47,82,56)(48,83,57)(49,84,58)(50,85,59)(51,86,60)(52,87,61)(53,88,62)(54,89,63), (1,13,7)(2,14,8)(3,15,9)(4,16,10)(5,17,11)(6,18,12)(19,31,25)(20,32,26)(21,33,27)(22,34,28)(23,35,29)(24,36,30)(37,49,43)(38,50,44)(39,51,45)(40,52,46)(41,53,47)(42,54,48)(55,67,61)(56,68,62)(57,69,63)(58,70,64)(59,71,65)(60,72,66)(73,85,79)(74,86,80)(75,87,81)(76,88,82)(77,89,83)(78,90,84)(91,103,97)(92,104,98)(93,105,99)(94,106,100)(95,107,101)(96,108,102), (2,27,101)(3,108,34)(5,30,104)(6,93,19)(8,33,107)(9,96,22)(11,36,92)(12,99,25)(14,21,95)(15,102,28)(17,24,98)(18,105,31)(20,32,26)(23,35,29)(37,78,70)(38,71,73)(39,45,51)(40,81,55)(41,56,76)(42,48,54)(43,84,58)(44,59,79)(46,87,61)(47,62,82)(49,90,64)(50,65,85)(52,75,67)(53,68,88)(57,69,63)(60,72,66)(91,97,103)(94,100,106), (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)(73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90)(91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108), (1,80,10,89)(2,79,11,88)(3,78,12,87)(4,77,13,86)(5,76,14,85)(6,75,15,84)(7,74,16,83)(8,73,17,82)(9,90,18,81)(19,67,28,58)(20,66,29,57)(21,65,30,56)(22,64,31,55)(23,63,32,72)(24,62,33,71)(25,61,34,70)(26,60,35,69)(27,59,36,68)(37,99,46,108)(38,98,47,107)(39,97,48,106)(40,96,49,105)(41,95,50,104)(42,94,51,103)(43,93,52,102)(44,92,53,101)(45,91,54,100)>;

G:=Group( (1,26,106)(2,27,107)(3,28,108)(4,29,91)(5,30,92)(6,31,93)(7,32,94)(8,33,95)(9,34,96)(10,35,97)(11,36,98)(12,19,99)(13,20,100)(14,21,101)(15,22,102)(16,23,103)(17,24,104)(18,25,105)(37,90,64)(38,73,65)(39,74,66)(40,75,67)(41,76,68)(42,77,69)(43,78,70)(44,79,71)(45,80,72)(46,81,55)(47,82,56)(48,83,57)(49,84,58)(50,85,59)(51,86,60)(52,87,61)(53,88,62)(54,89,63), (1,13,7)(2,14,8)(3,15,9)(4,16,10)(5,17,11)(6,18,12)(19,31,25)(20,32,26)(21,33,27)(22,34,28)(23,35,29)(24,36,30)(37,49,43)(38,50,44)(39,51,45)(40,52,46)(41,53,47)(42,54,48)(55,67,61)(56,68,62)(57,69,63)(58,70,64)(59,71,65)(60,72,66)(73,85,79)(74,86,80)(75,87,81)(76,88,82)(77,89,83)(78,90,84)(91,103,97)(92,104,98)(93,105,99)(94,106,100)(95,107,101)(96,108,102), (2,27,101)(3,108,34)(5,30,104)(6,93,19)(8,33,107)(9,96,22)(11,36,92)(12,99,25)(14,21,95)(15,102,28)(17,24,98)(18,105,31)(20,32,26)(23,35,29)(37,78,70)(38,71,73)(39,45,51)(40,81,55)(41,56,76)(42,48,54)(43,84,58)(44,59,79)(46,87,61)(47,62,82)(49,90,64)(50,65,85)(52,75,67)(53,68,88)(57,69,63)(60,72,66)(91,97,103)(94,100,106), (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)(73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90)(91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108), (1,80,10,89)(2,79,11,88)(3,78,12,87)(4,77,13,86)(5,76,14,85)(6,75,15,84)(7,74,16,83)(8,73,17,82)(9,90,18,81)(19,67,28,58)(20,66,29,57)(21,65,30,56)(22,64,31,55)(23,63,32,72)(24,62,33,71)(25,61,34,70)(26,60,35,69)(27,59,36,68)(37,99,46,108)(38,98,47,107)(39,97,48,106)(40,96,49,105)(41,95,50,104)(42,94,51,103)(43,93,52,102)(44,92,53,101)(45,91,54,100) );

G=PermutationGroup([[(1,26,106),(2,27,107),(3,28,108),(4,29,91),(5,30,92),(6,31,93),(7,32,94),(8,33,95),(9,34,96),(10,35,97),(11,36,98),(12,19,99),(13,20,100),(14,21,101),(15,22,102),(16,23,103),(17,24,104),(18,25,105),(37,90,64),(38,73,65),(39,74,66),(40,75,67),(41,76,68),(42,77,69),(43,78,70),(44,79,71),(45,80,72),(46,81,55),(47,82,56),(48,83,57),(49,84,58),(50,85,59),(51,86,60),(52,87,61),(53,88,62),(54,89,63)], [(1,13,7),(2,14,8),(3,15,9),(4,16,10),(5,17,11),(6,18,12),(19,31,25),(20,32,26),(21,33,27),(22,34,28),(23,35,29),(24,36,30),(37,49,43),(38,50,44),(39,51,45),(40,52,46),(41,53,47),(42,54,48),(55,67,61),(56,68,62),(57,69,63),(58,70,64),(59,71,65),(60,72,66),(73,85,79),(74,86,80),(75,87,81),(76,88,82),(77,89,83),(78,90,84),(91,103,97),(92,104,98),(93,105,99),(94,106,100),(95,107,101),(96,108,102)], [(2,27,101),(3,108,34),(5,30,104),(6,93,19),(8,33,107),(9,96,22),(11,36,92),(12,99,25),(14,21,95),(15,102,28),(17,24,98),(18,105,31),(20,32,26),(23,35,29),(37,78,70),(38,71,73),(39,45,51),(40,81,55),(41,56,76),(42,48,54),(43,84,58),(44,59,79),(46,87,61),(47,62,82),(49,90,64),(50,65,85),(52,75,67),(53,68,88),(57,69,63),(60,72,66),(91,97,103),(94,100,106)], [(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),(73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90),(91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108)], [(1,80,10,89),(2,79,11,88),(3,78,12,87),(4,77,13,86),(5,76,14,85),(6,75,15,84),(7,74,16,83),(8,73,17,82),(9,90,18,81),(19,67,28,58),(20,66,29,57),(21,65,30,56),(22,64,31,55),(23,63,32,72),(24,62,33,71),(25,61,34,70),(26,60,35,69),(27,59,36,68),(37,99,46,108),(38,98,47,107),(39,97,48,106),(40,96,49,105),(41,95,50,104),(42,94,51,103),(43,93,52,102),(44,92,53,101),(45,91,54,100)]])

Matrix representation of He3.3Dic3 in GL9(𝔽37)

2600000000
0260000000
0026000000
000001000
000000100
000000010
000000001
000100000
000010000
,
100000000
010000000
001000000
000010000
00036360000
000000100
00000363600
000000001
00000003636
,
100000000
1100000000
27026000000
000100000
000010000
00000363600
000001000
000000001
00000003636
,
12340000000
42536000000
2210000000
000143420231434
0003171434317
000143414342023
0003173171434
000202314341434
0001434317317
,
3100000000
8031000000
29310000000
000010000
000100000
000001000
00000363600
00000003636
000000001

G:=sub<GL(9,GF(37))| [26,0,0,0,0,0,0,0,0,0,26,0,0,0,0,0,0,0,0,0,26,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0],[1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,36,0,0,0,0,0,0,0,1,36,0,0,0,0,0,0,0,0,0,0,36,0,0,0,0,0,0,0,1,36,0,0,0,0,0,0,0,0,0,0,36,0,0,0,0,0,0,0,1,36],[1,1,27,0,0,0,0,0,0,0,10,0,0,0,0,0,0,0,0,0,26,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,36,1,0,0,0,0,0,0,0,36,0,0,0,0,0,0,0,0,0,0,0,36,0,0,0,0,0,0,0,1,36],[12,4,2,0,0,0,0,0,0,34,25,21,0,0,0,0,0,0,0,36,0,0,0,0,0,0,0,0,0,0,14,3,14,3,20,14,0,0,0,34,17,34,17,23,34,0,0,0,20,14,14,3,14,3,0,0,0,23,34,34,17,34,17,0,0,0,14,3,20,14,14,3,0,0,0,34,17,23,34,34,17],[31,8,29,0,0,0,0,0,0,0,0,31,0,0,0,0,0,0,0,31,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,36,0,0,0,0,0,0,0,0,36,0,0,0,0,0,0,0,0,0,36,0,0,0,0,0,0,0,0,36,1] >;

He3.3Dic3 in GAP, Magma, Sage, TeX

{\rm He}_3._3{\rm Dic}_3
% in TeX

G:=Group("He3.3Dic3");
// GroupNames label

G:=SmallGroup(324,23);
// by ID

G=gap.SmallGroup(324,23);
# by ID

G:=PCGroup([6,-2,-2,-3,-3,-3,-3,12,146,5763,303,237,7564,1096,7781]);
// Polycyclic

G:=Group<a,b,c,d,e|a^3=b^3=c^3=1,d^6=e*b*e^-1=b^-1,e^2=b^-1*d^3,a*b=b*a,c*a*c^-1=e*a*e^-1=a*b^-1,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d^-1=a*b^-1*c,e*c*e^-1=c^-1,e*d*e^-1=b*d^5>;
// generators/relations

Export

Subgroup lattice of He3.3Dic3 in TeX
Character table of He3.3Dic3 in TeX

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