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G = C32:Q16:C2  order 288 = 25·32

1st semidirect product of C32:Q16 and C2 acting faithfully

non-abelian, soluble, monomial

Aliases: C4.15S3wrC2, C32:Q16:1C2, (C3xC12).13D4, Dic3.D6:8C2, D6.D6.4C2, C32:2SD16:3C2, C32:1(C8.C22), C3:Dic3.4C23, D6:S3.6C22, C32:M4(2):5C2, C32:2C8.1C22, C32:2Q8.1C22, (C3xC6).7(C2xD4), C2.10(C2xS3wrC2), (C2xC3:S3).31D4, (C4xC3:S3).32C22, SmallGroup(288,874)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C32 — C3:Dic3 — C32:Q16:C2
Chief series
C1 — C32 — C3xC6 — C3:Dic3 — D6:S3 — C32:2SD16 — C32:Q16:C2
Lower central
C32 — C3xC6 — C3:Dic3 — C32:Q16:C2
Upper central
C1 — C2 — C4

Generators and relations for C32:Q16:C2
 G = < a,b,c,d,e | a3=b3=c8=e2=1, d2=c4, ab=ba, cac-1=dad-1=b, eae=cbc-1=a-1, dbd-1=a, ebe=b-1, dcd-1=c-1, ece=c5, de=ed >

Subgroups: 496 in 99 conjugacy classes, 23 normal (15 characteristic)
C1, C2, C2, C3, C4, C4, C22, S3, C6, C8, C2xC4, D4, Q8, C32, Dic3, C12, D6, C2xC6, M4(2), SD16, Q16, C2xQ8, C4oD4, C3xS3, C3:S3, C3xC6, Dic6, C4xS3, D12, C3:D4, C2xC12, C3xQ8, C8.C22, C3xDic3, C3:Dic3, C3xC12, S3xC6, C2xC3:S3, C4oD12, S3xQ8, C32:2C8, C6.D6, D6:S3, C3:D12, C32:2Q8, C32:2Q8, C3xDic6, S3xC12, C4xC3:S3, C32:2SD16, C32:Q16, C32:M4(2), Dic3.D6, D6.D6, C32:Q16:C2
Quotients: C1, C2, C22, D4, C23, C2xD4, C8.C22, S3wrC2, C2xS3wrC2, C32:Q16:C2

Character table of C32:Q16:C2

 class 12A2B2C3A3B4A4B4C4D4E6A6B6C6D8A8B12A12B12C12D12E12F12G
 size 11121844212121218441212363644812122424
ρ1111111111111111111111111    trivial
ρ211-1-111-1-111111-1-1-11-1-1-111-11    linear of order 2
ρ3111-111-1-11-1111111-1-1-1-1-1-1-11    linear of order 2
ρ411-1111111-1111-1-1-1-1111-1-111    linear of order 2
ρ511-1-111-11-11111-1-11-1-1-1-1111-1    linear of order 2
ρ61111111-1-1111111-1-111111-1-1    linear of order 2
ρ711-11111-1-1-1111-1-111111-1-1-1-1    linear of order 2
ρ8111-111-11-1-111111-11-1-1-1-1-11-1    linear of order 2
ρ9220-2222000-22200002220000    orthogonal lifted from D4
ρ10220222-2000-2220000-2-2-20000    orthogonal lifted from D4
ρ1144001-2-42-2001-2000022-100-11    orthogonal lifted from C2xS3wrC2
ρ1244001-24-2-2001-20000-2-210011    orthogonal lifted from S3wrC2
ρ1344-20-21400-20-21110011-21100    orthogonal lifted from S3wrC2
ρ144420-2140020-21-1-10011-2-1-100    orthogonal lifted from S3wrC2
ρ1544001-2-4-22001-2000022-1001-1    orthogonal lifted from C2xS3wrC2
ρ1644-20-21-40020-211100-1-12-1-100    orthogonal lifted from C2xS3wrC2
ρ1744001-2422001-20000-2-2100-1-1    orthogonal lifted from S3wrC2
ρ184420-21-400-20-21-1-100-1-121100    orthogonal lifted from C2xS3wrC2
ρ194-4004400000-4-400000000000    symplectic lifted from C8.C22, Schur index 2
ρ204-400-21000002-1√-3-√-300-3i3i0√3-√300    complex faithful
ρ214-400-21000002-1√-3-√-3003i-3i0-√3√300    complex faithful
ρ224-400-21000002-1-√-3√-300-3i3i0-√3√300    complex faithful
ρ234-400-21000002-1-√-3√-3003i-3i0√3-√300    complex faithful
ρ248-8002-400000-2400000000000    symplectic faithful, Schur index 2

Smallest permutation representation of C32:Q16:C2
►On 48 points
Generators in S48
(2 14 46)(4 48 16)(6 10 42)(8 44 12)(17 32 33)(19 35 26)(21 28 37)(23 39 30)

(1 13 45)(3 47 15)(5 9 41)(7 43 11)(18 34 25)(20 27 36)(22 38 29)(24 31 40)

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

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

Matrix representation of C32:Q16:C2 ►in GL4(F73) generated by

1000
0100
007272
0010
,
727200
1000
0010
0001
,
00714
00766
431300
603000
,
00270
00027
27000
02700
,
1000
727200
0010
007272
G:=sub<GL(4,GF(73))| [1,0,0,0,0,1,0,0,0,0,72,1,0,0,72,0],[72,1,0,0,72,0,0,0,0,0,1,0,0,0,0,1],[0,0,43,60,0,0,13,30,7,7,0,0,14,66,0,0],[0,0,27,0,0,0,0,27,27,0,0,0,0,27,0,0],[1,72,0,0,0,72,0,0,0,0,1,72,0,0,0,72] >;
 

C32:Q16:C2 in GAP, Magma, Sage, TeX 

C_3^2\rtimes Q_{16}\rtimes C_2
 
% in TeX
 
G:=Group("C3^2:Q16:C2");
 
// GroupNames label
 
G:=SmallGroup(288,874);
 
// by ID
 
G=gap.SmallGroup(288,874);
 
# by ID
 
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,3,141,120,219,100,675,346,80,2693,2028,362,797,1203]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e|a^3=b^3=c^8=e^2=1,d^2=c^4,a*b=b*a,c*a*c^-1=d*a*d^-1=b,e*a*e=c*b*c^-1=a^-1,d*b*d^-1=a,e*b*e=b^-1,d*c*d^-1=c^-1,e*c*e=c^5,d*e=e*d>;
 
// generators/relations
 

Export

Character table of C32:Q16:C2 in TeX

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