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G = C2xC33:Q8order 432 = 24·33

Direct product of C2 and C33:Q8

direct product, non-abelian, soluble, monomial

Aliases: C2xC33:Q8, C6:PSU3(F2), C33:3(C2xQ8), C3:S3:2Dic6, (C3xC6):2Dic6, C32:C4.7D6, (C32xC6):2Q8, C32:3(C2xDic6), C3:2(C2xPSU3(F2)), C33:C4.2C22, (C3xC3:S3):3Q8, (C2xC3:S3).23D6, (C2xC32:C4).6S3, (C6xC32:C4).7C2, (C3xC3:S3).8C23, C3:S3.5(C22xS3), (C6xC3:S3).37C22, (C2xC33:C4).7C2, (C3xC32:C4).8C22, SmallGroup(432,758)

Series: Derived Chief Lower central Upper central

C1C32C3xC3:S3 — C2xC33:Q8
C1C3C33C3xC3:S3C33:C4C33:Q8 — C2xC33:Q8
C33C3xC3:S3 — C2xC33:Q8
C1C2

Generators and relations for C2xC33:Q8
 G = < a,b,c,d,e,f | a2=b3=c3=d3=e4=1, f2=e2, ab=ba, ac=ca, ad=da, ae=ea, af=fa, bc=cb, bd=db, ebe-1=c, fbf-1=bc-1, cd=dc, ece-1=b-1, fcf-1=b-1c-1, de=ed, fdf-1=d-1, fef-1=e-1 >

Subgroups: 688 in 90 conjugacy classes, 31 normal (17 characteristic)
C1, C2, C2, C3, C3, C4, C22, S3, C6, C6, C2xC4, Q8, C32, C32, Dic3, C12, D6, C2xC6, C2xQ8, C3xS3, C3:S3, C3xC6, C3xC6, Dic6, C2xDic3, C2xC12, C33, C32:C4, C32:C4, S3xC6, C2xC3:S3, C2xDic6, C3xC3:S3, C32xC6, PSU3(F2), C2xC32:C4, C2xC32:C4, C3xC32:C4, C33:C4, C6xC3:S3, C2xPSU3(F2), C33:Q8, C6xC32:C4, C2xC33:C4, C2xC33:Q8
Quotients: C1, C2, C22, S3, Q8, C23, D6, C2xQ8, Dic6, C22xS3, C2xDic6, PSU3(F2), C2xPSU3(F2), C33:Q8, C2xC33:Q8

Character table of C2xC33:Q8

 class 12A2B2C3A3B3C3D4A4B4C4D4E4F6A6B6C6D6E6F12A12B12C12D
 size 119928881818545454542888181818181818
ρ1111111111111111111111111    trivial
ρ21111111111-1-1-1-11111111111    linear of order 2
ρ31-1-111111-1111-1-1-1-1-1-1-11-11-11    linear of order 2
ρ41-1-111111-11-1-111-1-1-1-1-11-11-11    linear of order 2
ρ51-1-1111111-11-11-1-1-1-1-1-111-11-1    linear of order 2
ρ61-1-1111111-1-11-11-1-1-1-1-111-11-1    linear of order 2
ρ711111111-1-11-1-11111111-1-1-1-1    linear of order 2
ρ811111111-1-1-111-1111111-1-1-1-1    linear of order 2
ρ92222-1-12-1-2-20000-12-1-1-1-11111    orthogonal lifted from D6
ρ102-2-22-1-12-1-2200001-2111-11-11-1    orthogonal lifted from D6
ρ112-2-22-1-12-12-200001-2111-1-11-11    orthogonal lifted from D6
ρ122222-1-12-1220000-12-1-1-1-1-1-1-1-1    orthogonal lifted from S3
ρ132-22-22222000000-2-2-2-22-20000    symplectic lifted from Q8, Schur index 2
ρ1422-2-222220000002222-2-20000    symplectic lifted from Q8, Schur index 2
ρ1522-2-2-1-12-1000000-12-1-111-3-333    symplectic lifted from Dic6, Schur index 2
ρ1622-2-2-1-12-1000000-12-1-11133-3-3    symplectic lifted from Dic6, Schur index 2
ρ172-22-2-1-12-10000001-211-11-333-3    symplectic lifted from Dic6, Schur index 2
ρ182-22-2-1-12-10000001-211-113-3-33    symplectic lifted from Dic6, Schur index 2
ρ198-8008-1-1-1000000-8111000000    orthogonal lifted from C2xPSU3(F2)
ρ2088008-1-1-10000008-1-1-1000000    orthogonal lifted from PSU3(F2)
ρ218800-41-3-3/2-11+3-3/2000000-4-11-3-3/21+3-3/2000000    complex lifted from C33:Q8
ρ228-800-41-3-3/2-11+3-3/200000041-1+3-3/2-1-3-3/2000000    complex faithful
ρ238-800-41+3-3/2-11-3-3/200000041-1-3-3/2-1+3-3/2000000    complex faithful
ρ248800-41+3-3/2-11-3-3/2000000-4-11+3-3/21-3-3/2000000    complex lifted from C33:Q8

Smallest permutation representation of C2xC33:Q8
On 48 points
Generators in S48
(1 11)(2 12)(3 9)(4 10)(5 24)(6 21)(7 22)(8 23)(13 41)(14 42)(15 43)(16 44)(17 26)(18 27)(19 28)(20 25)(29 45)(30 46)(31 47)(32 48)(33 37)(34 38)(35 39)(36 40)
(1 19 6)(3 8 17)(9 23 26)(11 28 21)(13 38 47)(14 48 39)(15 45 40)(16 37 46)(29 36 43)(30 44 33)(31 41 34)(32 35 42)
(2 7 20)(4 18 5)(10 27 24)(12 22 25)(13 47 38)(14 48 39)(15 40 45)(16 37 46)(29 43 36)(30 44 33)(31 34 41)(32 35 42)
(1 6 19)(2 7 20)(3 8 17)(4 5 18)(9 23 26)(10 24 27)(11 21 28)(12 22 25)(13 47 38)(14 48 39)(15 45 40)(16 46 37)(29 36 43)(30 33 44)(31 34 41)(32 35 42)
(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)
(1 40 3 38)(2 39 4 37)(5 46 7 48)(6 45 8 47)(9 34 11 36)(10 33 12 35)(13 19 15 17)(14 18 16 20)(21 29 23 31)(22 32 24 30)(25 42 27 44)(26 41 28 43)

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

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

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

Matrix representation of C2xC33:Q8 in GL8(F13)

120000000
012000000
001200000
000120000
000012000
000001200
000000120
000000012
,
90000000
03000000
00100000
00010000
00003000
00000900
00000090
120330943
,
10000000
01000000
00900000
00030000
00009000
00000300
00000090
99409043
,
30000000
03000000
00300000
00030000
00009000
00000900
00000090
11990009
,
00100000
00010000
01000000
10000000
1010121211128
00000010
00001000
777700012
,
000012000
000001200
000000120
3311121215
012000000
120000000
000120000
770066012

G:=sub<GL(8,GF(13))| [12,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,12],[9,0,0,0,0,0,0,12,0,3,0,0,0,0,0,0,0,0,1,0,0,0,0,3,0,0,0,1,0,0,0,3,0,0,0,0,3,0,0,0,0,0,0,0,0,9,0,9,0,0,0,0,0,0,9,4,0,0,0,0,0,0,0,3],[1,0,0,0,0,0,0,9,0,1,0,0,0,0,0,9,0,0,9,0,0,0,0,4,0,0,0,3,0,0,0,0,0,0,0,0,9,0,0,9,0,0,0,0,0,3,0,0,0,0,0,0,0,0,9,4,0,0,0,0,0,0,0,3],[3,0,0,0,0,0,0,1,0,3,0,0,0,0,0,1,0,0,3,0,0,0,0,9,0,0,0,3,0,0,0,9,0,0,0,0,9,0,0,0,0,0,0,0,0,9,0,0,0,0,0,0,0,0,9,0,0,0,0,0,0,0,0,9],[0,0,0,1,10,0,0,7,0,0,1,0,10,0,0,7,1,0,0,0,12,0,0,7,0,1,0,0,12,0,0,7,0,0,0,0,1,0,1,0,0,0,0,0,1,0,0,0,0,0,0,0,12,1,0,0,0,0,0,0,8,0,0,12],[0,0,0,3,0,12,0,7,0,0,0,3,12,0,0,7,0,0,0,1,0,0,0,0,0,0,0,1,0,0,12,0,12,0,0,12,0,0,0,6,0,12,0,12,0,0,0,6,0,0,12,1,0,0,0,0,0,0,0,5,0,0,0,12] >;

C2xC33:Q8 in GAP, Magma, Sage, TeX

C_2\times C_3^3\rtimes Q_8
% in TeX

G:=Group("C2xC3^3:Q8");
// GroupNames label

G:=SmallGroup(432,758);
// by ID

G=gap.SmallGroup(432,758);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-3,3,-3,56,141,64,2804,1691,165,2693,348,530,14118]);
// Polycyclic

G:=Group<a,b,c,d,e,f|a^2=b^3=c^3=d^3=e^4=1,f^2=e^2,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,a*f=f*a,b*c=c*b,b*d=d*b,e*b*e^-1=c,f*b*f^-1=b*c^-1,c*d=d*c,e*c*e^-1=b^-1,f*c*f^-1=b^-1*c^-1,d*e=e*d,f*d*f^-1=d^-1,f*e*f^-1=e^-1>;
// generators/relations

Export

Character table of C2xC33:Q8 in TeX

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