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G = C32⋊C4≀C2  order 288 = 25·32

The semidirect product of C32 and C4≀C2 acting via C4≀C2/C4=D4

non-abelian, soluble, monomial

Aliases: C32⋊1C4≀C2, C4.17S3≀C2, D6⋊S3⋊3C4, (C3×C12).17D4, C32⋊2Q8⋊3C4, C12.31D6⋊8C2, D6.D6.1C2, C2.7(S32⋊C4), (C4×C32⋊C4)⋊6C2, (C2×C3⋊S3).7D4, C3⋊Dic3.7(C2×C4), (C4×C3⋊S3).51C22, (C3×C6).6(C22⋊C4), SmallGroup(288,379)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C32 — C3⋊Dic3 — C32⋊C4≀C2
Chief series
C1 — C32 — C3×C6 — C3⋊Dic3 — C4×C3⋊S3 — D6.D6 — C32⋊C4≀C2
Lower central
C32 — C3×C6 — C3⋊Dic3 — C32⋊C4≀C2
Upper central
C1 — C4

Generators and relations for C32⋊C4≀C2
 G = < a,b,c,d,e | a3=b3=c4=d2=e4=1, ab=ba, cac-1=a-1, ad=da, eae-1=cbc-1=dbd=b-1, ebe-1=a, dcd=c-1, ce=ec, ede-1=c-1d >

Subgroups: 400 in 74 conjugacy classes, 15 normal (all characteristic)
C1, C2, C2, C3, C4, C4, C22, S3, C6, C8, C2×C4, D4, Q8, C32, Dic3, C12, D6, C2×C6, C42, M4(2), C4○D4, C3×S3, C3⋊S3, C3×C6, C3⋊C8, C24, Dic6, C4×S3, D12, C3⋊D4, C2×C12, C4≀C2, C3×Dic3, C3⋊Dic3, C3×C12, C32⋊C4, S3×C6, C2×C3⋊S3, C8⋊S3, C4○D12, C3×C3⋊C8, D6⋊S3, C3⋊D12, C32⋊2Q8, S3×C12, C4×C3⋊S3, C2×C32⋊C4, C12.31D6, C4×C32⋊C4, D6.D6, C32⋊C4≀C2
Quotients: C1, C2, C4, C22, C2×C4, D4, C22⋊C4, C4≀C2, S3≀C2, S32⋊C4, C32⋊C4≀C2

Character table of C32⋊C4≀C2

 class 12A2B2C3A3B4A4B4C4D4E4F4G4H6A6B6C6D8A8B12A12B12C12D12E12F24A24B24C24D
 size 111218441112181818181844121212124444121212121212
ρ1111111111111111111111111111111    trivial
ρ211-111111-1-1-1-1-1111-1-1111111-1-11111    linear of order 2
ρ311-111111-11111111-1-1-1-11111-1-1-1-1-1-1    linear of order 2
ρ4111111111-1-1-1-111111-1-1111111-1-1-1-1    linear of order 2
ρ511-1-111-1-11-iii-i111-1-1-ii-1-1-1-111i-ii-i    linear of order 4
ρ6111-111-1-1-1i-i-ii11111-ii-1-1-1-1-1-1i-ii-i    linear of order 4
ρ7111-111-1-1-1-iii-i11111i-i-1-1-1-1-1-1-ii-ii    linear of order 4
ρ811-1-111-1-11i-i-ii111-1-1i-i-1-1-1-111-ii-ii    linear of order 4
ρ9220222-2-200000-2220000-2-2-2-2000000    orthogonal lifted from D4
ρ10220-2222200000-22200002222000000    orthogonal lifted from D4
ρ112-200222i-2i01-i1+i-1-i-1+i0-2-200002i-2i-2i2i000000    complex lifted from C4≀C2
ρ122-20022-2i2i01+i1-i-1+i-1-i0-2-20000-2i2i2i-2i000000    complex lifted from C4≀C2
ρ132-200222i-2i0-1+i-1-i1+i1-i0-2-200002i-2i-2i2i000000    complex lifted from C4≀C2
ρ142-20022-2i2i0-1-i-1+i1-i1+i0-2-20000-2i2i2i-2i000000    complex lifted from C4≀C2
ρ154420-21442000001-2-1-100-21-21-1-10000    orthogonal lifted from S3≀C2
ρ1644001-244000000-2100221-21-200-1-1-1-1    orthogonal lifted from S3≀C2
ρ174420-21-4-4-2000001-2-1-1002-12-1110000    orthogonal lifted from S32⋊C4
ρ1844001-244000000-2100-2-21-21-2001111    orthogonal lifted from S3≀C2
ρ1944-20-21-4-42000001-211002-12-1-1-10000    orthogonal lifted from S32⋊C4
ρ2044-20-2144-2000001-21100-21-21110000    orthogonal lifted from S3≀C2
ρ2144001-2-4-4000000-2100-2i2i-12-1200-ii-ii    complex lifted from S32⋊C4
ρ2244001-2-4-4000000-21002i-2i-12-1200i-ii-i    complex lifted from S32⋊C4
ρ234-400-21-4i4i000000-12√-3-√-3002ii-2i-i-√3√30000    complex faithful
ρ244-400-214i-4i000000-12-√-3√-300-2i-i2ii-√3√30000    complex faithful
ρ254-400-214i-4i000000-12√-3-√-300-2i-i2ii√3-√30000    complex faithful
ρ264-400-21-4i4i000000-12-√-3√-3002ii-2i-i√3-√30000    complex faithful
ρ274-4001-24i-4i0000002-10000i2i-i-2i002ζ87ζ3+ζ872ζ85ζ3+ζ852ζ83ζ3+ζ832ζ8ζ3+ζ8    complex faithful
ρ284-4001-2-4i4i0000002-10000-i-2ii2i002ζ8ζ3+ζ82ζ83ζ3+ζ832ζ85ζ3+ζ852ζ87ζ3+ζ87    complex faithful
ρ294-4001-24i-4i0000002-10000i2i-i-2i002ζ83ζ3+ζ832ζ8ζ3+ζ82ζ87ζ3+ζ872ζ85ζ3+ζ85    complex faithful
ρ304-4001-2-4i4i0000002-10000-i-2ii2i002ζ85ζ3+ζ852ζ87ζ3+ζ872ζ8ζ3+ζ82ζ83ζ3+ζ83    complex faithful

Smallest permutation representation of C32⋊C4≀C2
►On 48 points
Generators in S48
(5 10 47)(6 48 11)(7 12 45)(8 46 9)(21 26 31)(22 32 27)(23 28 29)(24 30 25)

(1 18 13)(2 14 19)(3 20 15)(4 16 17)(33 41 40)(34 37 42)(35 43 38)(36 39 44)

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

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

Matrix representation of C32⋊C4≀C2 ►in GL4(𝔽5) generated by

3004
3022
1242
3001
,
2100
3200
0343
2330
,
2010
0032
0030
0230
,
1130
2041
1313
4203
,
3400
0230
0203
0200
G:=sub<GL(4,GF(5))| [3,3,1,3,0,0,2,0,0,2,4,0,4,2,2,1],[2,3,0,2,1,2,3,3,0,0,4,3,0,0,3,0],[2,0,0,0,0,0,0,2,1,3,3,3,0,2,0,0],[1,2,1,4,1,0,3,2,3,4,1,0,0,1,3,3],[3,0,0,0,4,2,2,2,0,3,0,0,0,0,3,0] >;
 

C32⋊C4≀C2 in GAP, Magma, Sage, TeX 

C_3^2\rtimes C_4\wr C_2
 
% in TeX
 
G:=Group("C3^2:C4wrC2");
 
// GroupNames label
 
G:=SmallGroup(288,379);
 
// by ID
 
G=gap.SmallGroup(288,379);
 
# by ID
 
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,3,85,64,422,219,100,80,2693,2028,691,797,2372]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e|a^3=b^3=c^4=d^2=e^4=1,a*b=b*a,c*a*c^-1=a^-1,a*d=d*a,e*a*e^-1=c*b*c^-1=d*b*d=b^-1,e*b*e^-1=a,d*c*d=c^-1,c*e=e*c,e*d*e^-1=c^-1*d>;
 
// generators/relations
 

Export

Character table of C32⋊C4≀C2 in TeX

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