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G = C32⋊D85C2order 288 = 25·32

The semidirect product of C32⋊D8 and C2 acting through Inn(C32⋊D8)

non-abelian, soluble, monomial

Aliases: C32⋊D85C2, C4.20S3≀C2, C32⋊Q165C2, (C3×C12).20D4, C321(C4○D8), D6.D61C2, C322SD167C2, C3⋊Dic3.1C23, D6⋊S3.5C22, C322C8.9C22, C322Q8.6C22, C2.7(C2×S3≀C2), C3⋊S33C87C2, (C3×C6).4(C2×D4), (C2×C3⋊S3).28D4, (C4×C3⋊S3).60C22, SmallGroup(288,871)

Series: Derived Chief Lower central Upper central

C1C32C3⋊Dic3 — C32⋊D85C2
C1C32C3×C6C3⋊Dic3D6⋊S3C32⋊D8 — C32⋊D85C2
C32C3×C6C3⋊Dic3 — C32⋊D85C2
C1C4

Generators and relations for C32⋊D85C2
 G = < a,b,c,d,e | a3=b3=c8=d2=e2=1, ab=ba, cac-1=b, dad=eae=cbc-1=a-1, bd=db, ebe=b-1, dcd=c-1, ce=ec, ede=c4d >

Subgroups: 528 in 102 conjugacy classes, 23 normal (15 characteristic)
C1, C2, C2 [×3], C3 [×2], C4, C4 [×3], C22 [×3], S3 [×4], C6 [×4], C8 [×2], C2×C4 [×3], D4 [×4], Q8 [×2], C32, Dic3 [×4], C12 [×4], D6 [×4], C2×C6 [×2], C2×C8, D8, SD16 [×2], Q16, C4○D4 [×2], C3×S3 [×2], C3⋊S3, C3×C6, Dic6 [×2], C4×S3 [×4], D12 [×2], C3⋊D4 [×4], C2×C12 [×2], C4○D8, C3×Dic3 [×2], C3⋊Dic3, C3×C12, S3×C6 [×2], C2×C3⋊S3, C4○D12 [×2], C322C8 [×2], D6⋊S3 [×2], C3⋊D12 [×2], C322Q8 [×2], S3×C12 [×2], C4×C3⋊S3, C32⋊D8, C322SD16 [×2], C32⋊Q16, C3⋊S33C8, D6.D6 [×2], C32⋊D85C2
Quotients: C1, C2 [×7], C22 [×7], D4 [×2], C23, C2×D4, C4○D8, S3≀C2, C2×S3≀C2, C32⋊D85C2

Character table of C32⋊D85C2

 class 12A2B2C2D3A3B4A4B4C4D4E6A6B6C6D6E6F8A8B8C8D12A12B12C12D12E12F12G12H
 size 111212184411121218441212121218181818444412121212
ρ1111111111111111111111111111111    trivial
ρ21111-111-1-1-1-11111111-111-1-1-1-1-1-1-1-1-1    linear of order 2
ρ311-1-1-111-1-111111-1-1-1-1-111-1-1-1-1-11111    linear of order 2
ρ411-1-111111-1-1111-1-1-1-111111111-1-1-1-1    linear of order 2
ρ5111-1-111-1-1-11111-111-11-1-11-1-1-1-1-11-11    linear of order 2
ρ6111-1111111-1111-111-1-1-1-1-111111-11-1    linear of order 2
ρ711-1111111-111111-1-11-1-1-1-11111-11-11    linear of order 2
ρ811-11-111-1-11-11111-1-111-1-11-1-1-1-11-11-1    linear of order 2
ρ92200-2222200-2220000000022220000    orthogonal lifted from D4
ρ102200222-2-200-22200000000-2-2-2-20000    orthogonal lifted from D4
ρ112-200022-2i2i000-2-20000--2-22-2-2i-2i2i2i0000    complex lifted from C4○D8
ρ122-200022-2i2i000-2-20000-22-2--2-2i-2i2i2i0000    complex lifted from C4○D8
ρ132-2000222i-2i000-2-20000-2-22--22i2i-2i-2i0000    complex lifted from C4○D8
ρ142-2000222i-2i000-2-20000--22-2-22i2i-2i-2i0000    complex lifted from C4○D8
ρ1544-200-2144-200-2101100000-211-21010    orthogonal lifted from S3≀C2
ρ1644200-2144200-210-1-100000-211-2-10-10    orthogonal lifted from S3≀C2
ρ17440-201-2-4-40201-210010000-122-10-10-1    orthogonal lifted from C2×S3≀C2
ρ18440201-2-4-40-201-2-100-10000-122-10101    orthogonal lifted from C2×S3≀C2
ρ1944-200-21-4-4200-21011000002-1-12-10-10    orthogonal lifted from C2×S3≀C2
ρ20440201-2440201-2-100-100001-2-210-10-1    orthogonal lifted from S3≀C2
ρ2144200-21-4-4-200-210-1-1000002-1-121010    orthogonal lifted from C2×S3≀C2
ρ22440-201-2440-201-2100100001-2-210101    orthogonal lifted from S3≀C2
ρ234-4000-21-4i4i0002-10-3--3000002i-ii-2i30-30    complex faithful
ρ244-40001-24i-4i000-12-300--30000i-2i2i-i0-303    complex faithful
ρ254-40001-24i-4i000-12--300-30000i-2i2i-i030-3    complex faithful
ρ264-40001-2-4i4i000-12--300-30000-i2i-2ii0-303    complex faithful
ρ274-40001-2-4i4i000-12-300--30000-i2i-2ii030-3    complex faithful
ρ284-4000-21-4i4i0002-10--3-3000002i-ii-2i-3030    complex faithful
ρ294-4000-214i-4i0002-10--3-300000-2ii-i2i30-30    complex faithful
ρ304-4000-214i-4i0002-10-3--300000-2ii-i2i-3030    complex faithful

Smallest permutation representation of C32⋊D85C2
On 48 points
Generators in S48
(2 25 47)(4 41 27)(6 29 43)(8 45 31)(9 35 23)(11 17 37)(13 39 19)(15 21 33)
(1 32 46)(3 48 26)(5 28 42)(7 44 30)(10 24 36)(12 38 18)(14 20 40)(16 34 22)
(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 18)(2 17)(3 24)(4 23)(5 22)(6 21)(7 20)(8 19)(9 27)(10 26)(11 25)(12 32)(13 31)(14 30)(15 29)(16 28)(33 43)(34 42)(35 41)(36 48)(37 47)(38 46)(39 45)(40 44)
(9 39)(10 40)(11 33)(12 34)(13 35)(14 36)(15 37)(16 38)(17 21)(18 22)(19 23)(20 24)(25 47)(26 48)(27 41)(28 42)(29 43)(30 44)(31 45)(32 46)

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

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

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

Matrix representation of C32⋊D85C2 in GL4(𝔽73) generated by

1000
0100
117272
0010
,
07200
17200
07210
07201
,
77052
14145252
37506659
7376659
,
436000
133000
3623714
4330766
,
0100
1000
2525720
242411
G:=sub<GL(4,GF(73))| [1,0,1,0,0,1,1,0,0,0,72,1,0,0,72,0],[0,1,0,0,72,72,72,72,0,0,1,0,0,0,0,1],[7,14,37,7,7,14,50,37,0,52,66,66,52,52,59,59],[43,13,36,43,60,30,23,30,0,0,7,7,0,0,14,66],[0,1,25,24,1,0,25,24,0,0,72,1,0,0,0,1] >;

C32⋊D85C2 in GAP, Magma, Sage, TeX

C_3^2\rtimes D_8\rtimes_5C_2
% in TeX

G:=Group("C3^2:D8:5C2");
// GroupNames label

G:=SmallGroup(288,871);
// by ID

G=gap.SmallGroup(288,871);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,3,141,100,675,346,80,2693,2028,362,797,1203]);
// Polycyclic

G:=Group<a,b,c,d,e|a^3=b^3=c^8=d^2=e^2=1,a*b=b*a,c*a*c^-1=b,d*a*d=e*a*e=c*b*c^-1=a^-1,b*d=d*b,e*b*e=b^-1,d*c*d=c^-1,c*e=e*c,e*d*e=c^4*d>;
// generators/relations

Export

Character table of C32⋊D85C2 in TeX

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