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G = C32:6C4wrC2  order 288 = 25·32

The semidirect product of C32 and C4wrC2 acting via C4wrC2/D4=C4

metabelian, soluble, monomial

Aliases: C32:6C4wrC2, D4:2(C32:C4), (D4xC32):2C4, C32:4Q8:1C4, C3:Dic3.52D4, C2.7(C62:C4), C12.D6.1C2, C32:M4(2):2C2, (C4xC32:C4):1C2, C4.2(C2xC32:C4), (C3xC12).2(C2xC4), (C2xC3:S3).14D4, (C4xC3:S3).6C22, (C3xC6).17(C22:C4), SmallGroup(288,431)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C3xC12 — C32:6C4wrC2
Chief series
C1 — C32 — C3xC6 — C3:Dic3 — C4xC3:S3 — C32:M4(2) — C32:6C4wrC2
Lower central
C32 — C3xC6 — C3xC12 — C32:6C4wrC2
Upper central
C1 — C2 — C4 — D4

Generators and relations for C32:6C4wrC2
 G = < a,b,c,d,e | a3=b3=c4=d2=e4=1, ebe-1=ab=ba, ac=ca, ad=da, eae-1=a-1b, bc=cb, bd=db, dcd=c-1, ce=ec, ede-1=c-1d >

Subgroups: 456 in 84 conjugacy classes, 16 normal (all characteristic)
C1, C2, C2, C3, C4, C4, C22, S3, C6, C8, C2xC4, D4, D4, Q8, C32, Dic3, C12, D6, C2xC6, C42, M4(2), C4oD4, C3:S3, C3xC6, C3xC6, Dic6, C4xS3, C2xDic3, C3:D4, C3xD4, C4wrC2, C3:Dic3, C3:Dic3, C3xC12, C32:C4, C2xC3:S3, C62, D4:2S3, C32:2C8, C32:4Q8, C4xC3:S3, C2xC3:Dic3, C32:7D4, D4xC32, C2xC32:C4, C32:M4(2), C4xC32:C4, C12.D6, C32:6C4wrC2
Quotients: C1, C2, C4, C22, C2xC4, D4, C22:C4, C4wrC2, C32:C4, C2xC32:C4, C62:C4, C32:6C4wrC2

Character table of C32:6C4wrC2

 class 12A2B2C3A3B4A4B4C4D4E4F4G4H6A6B6C6D6E6F8A8B12A12B
 size 11418442991818181836448888363688
ρ1111111111111111111111111    trivial
ρ211-11111111111-111-1-1-1-1-1-111    linear of order 2
ρ3111111111-1-1-1-11111111-1-111    linear of order 2
ρ411-1111111-1-1-1-1-111-1-1-1-11111    linear of order 2
ρ5111-1111-1-1-ii-ii-1111111i-i11    linear of order 4
ρ611-1-1111-1-1-ii-ii111-1-1-1-1-ii11    linear of order 4
ρ7111-1111-1-1i-ii-i-1111111-ii11    linear of order 4
ρ811-1-1111-1-1i-ii-i111-1-1-1-1i-i11    linear of order 4
ρ9220222-2-2-20000022000000-2-2    orthogonal lifted from D4
ρ10220-222-2220000022000000-2-2    orthogonal lifted from D4
ρ112-2002202i-2i-1-i1-i1+i-1+i0-2-200000000    complex lifted from C4wrC2
ρ122-2002202i-2i1+i-1+i-1-i1-i0-2-200000000    complex lifted from C4wrC2
ρ132-200220-2i2i1-i-1-i-1+i1+i0-2-200000000    complex lifted from C4wrC2
ρ142-200220-2i2i-1+i1+i1-i-1-i0-2-200000000    complex lifted from C4wrC2
ρ1544001-2-400000001-230-3000-12    orthogonal lifted from C62:C4
ρ1644-401-2400000001-2-12-12001-2    orthogonal lifted from C2xC32:C4
ρ1744-40-2140000000-212-12-100-21    orthogonal lifted from C2xC32:C4
ρ184440-2140000000-21-21-2100-21    orthogonal lifted from C32:C4
ρ194400-21-40000000-210-303002-1    orthogonal lifted from C62:C4
ρ2044401-2400000001-21-21-2001-2    orthogonal lifted from C32:C4
ρ2144001-2-400000001-2-303000-12    orthogonal lifted from C62:C4
ρ224400-21-40000000-21030-3002-1    orthogonal lifted from C62:C4
ρ238-8002-400000000-2400000000    symplectic faithful, Schur index 2
ρ248-800-42000000004-200000000    symplectic faithful, Schur index 2

Smallest permutation representation of C32:6C4wrC2
►On 48 points
Generators in S48
(5 10 47)(6 11 48)(7 12 45)(8 9 46)(33 37 42)(34 38 43)(35 39 44)(36 40 41)

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

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

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

Matrix representation of C32:6C4wrC2 ►in GL6(F73)

100000
010000
001000
000100
00007272
000010
,
100000
010000
000100
00727200
00007272
000010
,
2700000
46460000
001000
000100
000010
000001
,
27540000
46460000
001000
000100
000010
000001
,
4600000
1410000
000010
000001
001000
00727200

G:=sub<GL(6,GF(73))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,72,1,0,0,0,0,72,0],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,72,0,0,0,0,1,72,0,0,0,0,0,0,72,1,0,0,0,0,72,0],[27,46,0,0,0,0,0,46,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[27,46,0,0,0,0,54,46,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[46,14,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,72,0,0,0,0,0,72,0,0,1,0,0,0,0,0,0,1,0,0] >;
 

C32:6C4wrC2 in GAP, Magma, Sage, TeX 

C_3^2\rtimes_6C_4\wr C_2
 
% in TeX
 
G:=Group("C3^2:6C4wrC2");
 
// GroupNames label
 
G:=SmallGroup(288,431);
 
// by ID
 
G=gap.SmallGroup(288,431);
 
# by ID
 
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,3,28,141,100,675,346,80,9413,691,12550,2372]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e|a^3=b^3=c^4=d^2=e^4=1,e*b*e^-1=a*b=b*a,a*c=c*a,a*d=d*a,e*a*e^-1=a^-1*b,b*c=c*b,b*d=d*b,d*c*d=c^-1,c*e=e*c,e*d*e^-1=c^-1*d>;
 
// generators/relations
 

Export

Character table of C32:6C4wrC2 in TeX

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