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G = C2×He3.3S3order 324 = 22·34

Direct product of C2 and He3.3S3

direct product, non-abelian, supersoluble, monomial

Aliases: C2×He3.3S3, He3.5D6, 3- 1+22D6, (C3×C9)⋊9D6, (C3×C18)⋊5S3, (C2×He3).9S3, He3.C32C22, C6.8(He3⋊C2), (C2×3- 1+2)⋊2S3, (C3×C6).6(C3⋊S3), C32.2(C2×C3⋊S3), (C2×He3.C3)⋊1C2, C3.3(C2×He3⋊C2), SmallGroup(324,78)

Series: Derived Chief Lower central Upper central

C1C32He3.C3 — C2×He3.3S3
C1C3C32C3×C9He3.C3He3.3S3 — C2×He3.3S3
He3.C3 — C2×He3.3S3
C1C2

Generators and relations for C2×He3.3S3
 G = < a,b,c,d,e,f | a2=b3=c3=d3=f2=1, e3=fcf=c-1, ab=ba, ac=ca, ad=da, ae=ea, af=fa, bc=cb, dbd-1=fbf=bc-1, be=eb, cd=dc, ce=ec, ede-1=b-1cd, fdf=d-1, fef=ce2 >

Subgroups: 436 in 70 conjugacy classes, 19 normal (13 characteristic)
C1, C2, C2 [×2], C3, C3 [×2], C22, S3 [×4], C6, C6 [×4], C9 [×3], C32, C32, D6 [×2], C2×C6, D9 [×6], C18 [×3], C3×S3 [×2], C3⋊S3 [×2], C3×C6, C3×C6, C3×C9, He3, 3- 1+2 [×2], D18 [×3], S3×C6, C2×C3⋊S3, C3×D9 [×2], C32⋊C6 [×2], C9⋊C6 [×4], C3×C18, C2×He3, C2×3- 1+2 [×2], He3.C3, C6×D9, C2×C32⋊C6, C2×C9⋊C6 [×2], He3.3S3 [×2], C2×He3.C3, C2×He3.3S3
Quotients: C1, C2 [×3], C22, S3 [×4], D6 [×4], C3⋊S3, C2×C3⋊S3, He3⋊C2, C2×He3⋊C2, He3.3S3, C2×He3.3S3

Character table of C2×He3.3S3

 class 12A2B2C3A3B3C3D6A6B6C6D6E6F6G6H9A9B9C9D9E18A18B18C18D18E
 size 11272723318233182727272766618186661818
ρ111111111111111111111111111    trivial
ρ21-1-111111-1-1-1-1-111-111111-1-1-1-1-1    linear of order 2
ρ31-11-11111-1-1-1-11-1-1111111-1-1-1-1-1    linear of order 2
ρ411-1-111111111-1-1-1-11111111111    linear of order 2
ρ52200222-1222-10000222-1-1222-1-1    orthogonal lifted from S3
ρ62-2002222-2-2-2-20000-1-1-1-1-111111    orthogonal lifted from D6
ρ72200222-1222-10000-1-1-12-1-1-1-1-12    orthogonal lifted from S3
ρ82-200222-1-2-2-210000-1-1-1-12111-21    orthogonal lifted from D6
ρ92-200222-1-2-2-210000-1-1-12-11111-2    orthogonal lifted from D6
ρ102200222-1222-10000-1-1-1-12-1-1-12-1    orthogonal lifted from S3
ρ112-200222-1-2-2-210000222-1-1-2-2-211    orthogonal lifted from D6
ρ122200222222220000-1-1-1-1-1-1-1-1-1-1    orthogonal lifted from S3
ρ133-3-113-3+3-3/2-3-3-3/20-33+3-3/23-3-3/20ζ6ζ3ζ32ζ650000000000    complex lifted from C2×He3⋊C2
ρ1433113-3-3-3/2-3+3-3/203-3+3-3/2-3-3-3/20ζ3ζ32ζ3ζ320000000000    complex lifted from He3⋊C2
ρ153-31-13-3+3-3/2-3-3-3/20-33+3-3/23-3-3/20ζ32ζ65ζ6ζ30000000000    complex lifted from C2×He3⋊C2
ρ163-31-13-3-3-3/2-3+3-3/20-33-3-3/23+3-3/20ζ3ζ6ζ65ζ320000000000    complex lifted from C2×He3⋊C2
ρ1733-1-13-3+3-3/2-3-3-3/203-3-3-3/2-3+3-3/20ζ6ζ65ζ6ζ650000000000    complex lifted from He3⋊C2
ρ1833113-3+3-3/2-3-3-3/203-3-3-3/2-3+3-3/20ζ32ζ3ζ32ζ30000000000    complex lifted from He3⋊C2
ρ193-3-113-3-3-3/2-3+3-3/20-33-3-3/23+3-3/20ζ65ζ32ζ3ζ60000000000    complex lifted from C2×He3⋊C2
ρ2033-1-13-3-3-3/2-3+3-3/203-3+3-3/2-3-3-3/20ζ65ζ6ζ65ζ60000000000    complex lifted from He3⋊C2
ρ216-600-300030000000ζ989794+2ζ92ζ989492+2ζ995949290098+2ζ9794929894929ζ95+2ζ9492900    orthogonal faithful
ρ226600-3000-300000009594929ζ989794+2ζ92ζ989492+2ζ900ζ989794+2ζ92ζ989492+2ζ9959492900    orthogonal lifted from He3.3S3
ρ236-600-3000300000009594929ζ989794+2ζ92ζ989492+2ζ900ζ95+2ζ9492998+2ζ979492989492900    orthogonal faithful
ρ246-600-300030000000ζ989492+2ζ99594929ζ989794+2ζ92009894929ζ95+2ζ9492998+2ζ97949200    orthogonal faithful
ρ256600-3000-30000000ζ989794+2ζ92ζ989492+2ζ9959492900ζ989492+2ζ99594929ζ989794+2ζ9200    orthogonal lifted from He3.3S3
ρ266600-3000-30000000ζ989492+2ζ99594929ζ989794+2ζ92009594929ζ989794+2ζ92ζ989492+2ζ900    orthogonal lifted from He3.3S3

Smallest permutation representation of C2×He3.3S3
On 54 points
Generators in S54
(1 21)(2 22)(3 23)(4 24)(5 25)(6 26)(7 27)(8 19)(9 20)(10 34)(11 35)(12 36)(13 28)(14 29)(15 30)(16 31)(17 32)(18 33)(37 51)(38 52)(39 53)(40 54)(41 46)(42 47)(43 48)(44 49)(45 50)
(1 35 45)(2 36 37)(3 28 38)(4 29 39)(5 30 40)(6 31 41)(7 32 42)(8 33 43)(9 34 44)(10 49 20)(11 50 21)(12 51 22)(13 52 23)(14 53 24)(15 54 25)(16 46 26)(17 47 27)(18 48 19)
(1 7 4)(2 8 5)(3 9 6)(10 16 13)(11 17 14)(12 18 15)(19 25 22)(20 26 23)(21 27 24)(28 34 31)(29 35 32)(30 36 33)(37 43 40)(38 44 41)(39 45 42)(46 52 49)(47 53 50)(48 54 51)
(2 37 30)(3 28 44)(5 40 33)(6 31 38)(8 43 36)(9 34 41)(10 46 20)(11 17 14)(12 19 48)(13 49 23)(15 22 51)(16 52 26)(18 25 54)(29 35 32)(39 42 45)(47 50 53)
(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 21)(2 20)(3 19)(4 27)(5 26)(6 25)(7 24)(8 23)(9 22)(10 30)(11 29)(12 28)(13 36)(14 35)(15 34)(16 33)(17 32)(18 31)(37 46)(38 54)(39 53)(40 52)(41 51)(42 50)(43 49)(44 48)(45 47)

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

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

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

Matrix representation of C2×He3.3S3 in GL6(𝔽19)

1800000
0180000
0018000
0001800
0000180
0000018
,
001000
000100
000010
000001
100000
010000
,
1810000
1800000
0018100
0018000
0000181
0000180
,
100000
010000
0001800
0011800
0000181
0000180
,
1313151
1641641816
1511313
1816164164
1315113
1641816164
,
0180000
1800000
0018100
000100
000010
0000118

G:=sub<GL(6,GF(19))| [18,0,0,0,0,0,0,18,0,0,0,0,0,0,18,0,0,0,0,0,0,18,0,0,0,0,0,0,18,0,0,0,0,0,0,18],[0,0,0,0,1,0,0,0,0,0,0,1,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0],[18,18,0,0,0,0,1,0,0,0,0,0,0,0,18,18,0,0,0,0,1,0,0,0,0,0,0,0,18,18,0,0,0,0,1,0],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,18,18,0,0,0,0,0,0,18,18,0,0,0,0,1,0],[1,16,15,18,1,16,3,4,1,16,3,4,1,16,1,16,15,18,3,4,3,4,1,16,15,18,1,16,1,16,1,16,3,4,3,4],[0,18,0,0,0,0,18,0,0,0,0,0,0,0,18,0,0,0,0,0,1,1,0,0,0,0,0,0,1,1,0,0,0,0,0,18] >;

C2×He3.3S3 in GAP, Magma, Sage, TeX

C_2\times {\rm He}_3._3S_3
% in TeX

G:=Group("C2xHe3.3S3");
// GroupNames label

G:=SmallGroup(324,78);
// by ID

G=gap.SmallGroup(324,78);
# by ID

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

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

Export

Character table of C2×He3.3S3 in TeX

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