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G = C2×He3.2S3order 324 = 22·34

Direct product of C2 and He3.2S3

direct product, metabelian, supersoluble, monomial

Aliases: C2×He3.2S3, He3.4D6, C9⋊S34C6, (C3×C18)⋊3C6, (C2×He3).8S3, C32.7(S3×C6), C6.8(C32⋊C6), He3⋊C33C22, (C2×C9⋊S3)⋊3C3, (C3×C9)⋊4(C2×C6), (C3×C6).18(C3×S3), C3.4(C2×C32⋊C6), (C2×He3⋊C3)⋊2C2, SmallGroup(324,73)

Series: Derived Chief Lower central Upper central

C1C3×C9 — C2×He3.2S3
C1C3C32C3×C9He3⋊C3He3.2S3 — C2×He3.2S3
C3×C9 — C2×He3.2S3
C1C2

Generators and relations for C2×He3.2S3
 G = < a,b,c,d,e,f | a2=b3=c3=d3=f2=1, e3=c, ab=ba, ac=ca, ad=da, ae=ea, af=fa, bc=cb, dbd-1=bc-1, be=eb, fbf=b-1, cd=dc, ce=ec, fcf=c-1, ede-1=b-1cd, df=fd, fef=c-1e2 >

Subgroups: 448 in 56 conjugacy classes, 18 normal (14 characteristic)
C1, C2, C2, C3, C3, C22, S3, C6, C6, C9, C32, C32, D6, C2×C6, D9, C18, C3×S3, C3⋊S3, C3×C6, C3×C6, C3×C9, He3, He3, D18, S3×C6, C2×C3⋊S3, C32⋊C6, C9⋊S3, C3×C18, C2×He3, C2×He3, He3⋊C3, C2×C32⋊C6, C2×C9⋊S3, He3.2S3, C2×He3⋊C3, C2×He3.2S3
Quotients: C1, C2, C3, C22, S3, C6, D6, C2×C6, C3×S3, S3×C6, C32⋊C6, C2×C32⋊C6, He3.2S3, C2×He3.2S3

Character table of C2×He3.2S3

 class 12A2B2C3A3B3C3D3E3F6A6B6C6D6E6F6G6H6I6J9A9B9C18A18B18C
 size 112727269918182699181827272727666666
ρ111111111111111111111111111    trivial
ρ21-11-1111111-1-1-1-1-1-11-11-1111-1-1-1    linear of order 2
ρ31-1-11111111-1-1-1-1-1-1-11-11111-1-1-1    linear of order 2
ρ411-1-1111111111111-1-1-1-1111111    linear of order 2
ρ5111111ζ3ζ32ζ3ζ3211ζ3ζ32ζ3ζ32ζ32ζ3ζ3ζ32111111    linear of order 3
ρ61-11-111ζ3ζ32ζ3ζ32-1-1ζ65ζ6ζ65ζ6ζ32ζ65ζ3ζ6111-1-1-1    linear of order 6
ρ711-1-111ζ3ζ32ζ3ζ3211ζ3ζ32ζ3ζ32ζ6ζ65ζ65ζ6111111    linear of order 6
ρ81-1-1111ζ3ζ32ζ3ζ32-1-1ζ65ζ6ζ65ζ6ζ6ζ3ζ65ζ32111-1-1-1    linear of order 6
ρ91-1-1111ζ32ζ3ζ32ζ3-1-1ζ6ζ65ζ6ζ65ζ65ζ32ζ6ζ3111-1-1-1    linear of order 6
ρ101-11-111ζ32ζ3ζ32ζ3-1-1ζ6ζ65ζ6ζ65ζ3ζ6ζ32ζ65111-1-1-1    linear of order 6
ρ11111111ζ32ζ3ζ32ζ311ζ32ζ3ζ32ζ3ζ3ζ32ζ32ζ3111111    linear of order 3
ρ1211-1-111ζ32ζ3ζ32ζ311ζ32ζ3ζ32ζ3ζ65ζ6ζ6ζ65111111    linear of order 6
ρ132-2002222-1-1-2-2-2-2110000-1-1-1111    orthogonal lifted from D6
ρ1422002222-1-12222-1-10000-1-1-1-1-1-1    orthogonal lifted from S3
ρ15220022-1--3-1+-3ζ6ζ6522-1--3-1+-3ζ6ζ650000-1-1-1-1-1-1    complex lifted from C3×S3
ρ162-20022-1--3-1+-3ζ6ζ65-2-21+-31--3ζ32ζ30000-1-1-1111    complex lifted from S3×C6
ρ17220022-1+-3-1--3ζ65ζ622-1+-3-1--3ζ65ζ60000-1-1-1-1-1-1    complex lifted from C3×S3
ρ182-20022-1+-3-1--3ζ65ζ6-2-21--31+-3ζ3ζ320000-1-1-1111    complex lifted from S3×C6
ρ1966006-300006-300000000000000    orthogonal lifted from C32⋊C6
ρ206-6006-30000-6300000000000000    orthogonal lifted from C2×C32⋊C6
ρ216-600-3000003000000000989492998+2ζ979492ζ95+2ζ94929ζ989492+2ζ9ζ989794+2ζ929594929    orthogonal faithful
ρ226600-300000-3000000000989492998+2ζ979492ζ95+2ζ9492998+2ζ979492ζ95+2ζ949299894929    orthogonal lifted from He3.2S3
ρ236-600-300000300000000098+2ζ979492ζ95+2ζ949299894929ζ989794+2ζ929594929ζ989492+2ζ9    orthogonal faithful
ρ246600-300000-3000000000ζ95+2ζ94929989492998+2ζ979492989492998+2ζ979492ζ95+2ζ94929    orthogonal lifted from He3.2S3
ρ256600-300000-300000000098+2ζ979492ζ95+2ζ949299894929ζ95+2ζ94929989492998+2ζ979492    orthogonal lifted from He3.2S3
ρ266-600-3000003000000000ζ95+2ζ94929989492998+2ζ9794929594929ζ989492+2ζ9ζ989794+2ζ92    orthogonal faithful

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

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

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

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

Matrix representation of C2×He3.2S3 in GL6(𝔽19)

1800000
0180000
0018000
0001800
0000180
0000018
,
100000
010000
1111181800
001000
000001
77001818
,
1810000
1800000
800100
011181800
1200001
07001818
,
0000181
77001718
000080
100080
0010120
0001120
,
1220000
17140000
1827500
16014200
5000142
014001712
,
010000
100000
001000
1111181800
000010
77001818

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],[1,0,11,0,0,7,0,1,11,0,0,7,0,0,18,1,0,0,0,0,18,0,0,0,0,0,0,0,0,18,0,0,0,0,1,18],[18,18,8,0,12,0,1,0,0,11,0,7,0,0,0,18,0,0,0,0,1,18,0,0,0,0,0,0,0,18,0,0,0,0,1,18],[0,7,0,1,0,0,0,7,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,18,17,8,8,12,12,1,18,0,0,0,0],[12,17,18,16,5,0,2,14,2,0,0,14,0,0,7,14,0,0,0,0,5,2,0,0,0,0,0,0,14,17,0,0,0,0,2,12],[0,1,0,11,0,7,1,0,0,11,0,7,0,0,1,18,0,0,0,0,0,18,0,0,0,0,0,0,1,18,0,0,0,0,0,18] >;

C2×He3.2S3 in GAP, Magma, Sage, TeX

C_2\times {\rm He}_3._2S_3
% in TeX

G:=Group("C2xHe3.2S3");
// GroupNames label

G:=SmallGroup(324,73);
// by ID

G=gap.SmallGroup(324,73);
# by ID

G:=PCGroup([6,-2,-2,-3,-3,-3,-3,3171,303,453,2164,1096,7781]);
// Polycyclic

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

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Character table of C2×He3.2S3 in TeX

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