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G = C32⋊C4≀C2order 288 = 25·32

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

non-abelian, soluble, monomial

Aliases: C321C4≀C2, C4.17S3≀C2, D6⋊S33C4, (C3×C12).17D4, C322Q83C4, C12.31D68C2, 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

C1C32C3⋊Dic3 — C32⋊C4≀C2
C1C32C3×C6C3⋊Dic3C4×C3⋊S3D6.D6 — C32⋊C4≀C2
C32C3×C6C3⋊Dic3 — C32⋊C4≀C2
C1C4

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 [×2], C3 [×2], C4, C4 [×4], C22 [×2], S3 [×3], C6 [×3], C8, C2×C4 [×3], D4 [×2], Q8, C32, Dic3 [×3], C12 [×3], D6 [×3], C2×C6, C42, M4(2), C4○D4, C3×S3, C3⋊S3, C3×C6, C3⋊C8, C24, Dic6, C4×S3 [×3], D12, C3⋊D4 [×2], C2×C12, C4≀C2, C3×Dic3, C3⋊Dic3, C3×C12, C32⋊C4 [×2], S3×C6, C2×C3⋊S3, C8⋊S3, C4○D12, C3×C3⋊C8, D6⋊S3, C3⋊D12, C322Q8, S3×C12, C4×C3⋊S3, C2×C32⋊C4, C12.31D6, C4×C32⋊C4, D6.D6, C32⋊C4≀C2
Quotients: C1, C2 [×3], C4 [×2], C22, C2×C4, D4 [×2], 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-330000    complex faithful
ρ244-400-214i-4i000000-12--3-300-2i-i2ii-330000    complex faithful
ρ254-400-214i-4i000000-12-3--300-2i-i2ii3-30000    complex faithful
ρ264-400-21-4i4i000000-12--3-3002ii-2i-i3-30000    complex faithful
ρ274-4001-24i-4i0000002-10000i2i-i-2i0087ζ38785ζ38583ζ3838ζ38    complex faithful
ρ284-4001-2-4i4i0000002-10000-i-2ii2i008ζ3883ζ38385ζ38587ζ387    complex faithful
ρ294-4001-24i-4i0000002-10000i2i-i-2i0083ζ3838ζ3887ζ38785ζ385    complex faithful
ρ304-4001-2-4i4i0000002-10000-i-2ii2i0085ζ38587ζ3878ζ3883ζ383    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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