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G = C3⋊S3.5D8order 288 = 25·32

The non-split extension by C3⋊S3 of D8 acting via D8/D4=C2

metabelian, soluble, monomial

Aliases: C3⋊S3.5D8, C12⋊S31C4, D41(C32⋊C4), (D4×C32)⋊1C4, C3⋊S3.6SD16, C3⋊Dic3.10D4, C327(D4⋊C4), C2.6(C62⋊C4), (D4×C3⋊S3).1C2, C3⋊S33C81C2, C4.1(C2×C32⋊C4), C4⋊(C32⋊C4)⋊2C2, (C3×C12).1(C2×C4), (C2×C3⋊S3).51D4, (C4×C3⋊S3).5C22, (C3×C6).16(C22⋊C4), SmallGroup(288,430)

Series: Derived Chief Lower central Upper central

C1C3×C12 — C3⋊S3.5D8
C1C32C3×C6C2×C3⋊S3C4×C3⋊S3C4⋊(C32⋊C4) — C3⋊S3.5D8
C32C3×C6C3×C12 — C3⋊S3.5D8
C1C2C4D4

Generators and relations for C3⋊S3.5D8
 G = < a,b,c,d,e | a3=b3=c2=d8=1, e2=c, ab=ba, cac=a-1, dad-1=eae-1=ab-1, cbc=b-1, dbd-1=ebe-1=a-1b-1, cd=dc, ce=ec, ede-1=cd-1 >

Subgroups: 720 in 104 conjugacy classes, 18 normal (all characteristic)
C1, C2, C2 [×4], C3 [×2], C4, C4 [×2], C22 [×5], S3 [×8], C6 [×6], C8, C2×C4 [×2], D4, D4 [×2], C23, C32, Dic3 [×2], C12 [×2], D6 [×14], C2×C6 [×4], C4⋊C4, C2×C8, C2×D4, C3⋊S3 [×2], C3⋊S3, C3×C6, C3×C6, C4×S3 [×2], D12 [×2], C3⋊D4 [×4], C3×D4 [×2], C22×S3 [×4], D4⋊C4, C3⋊Dic3, C3×C12, C32⋊C4, C2×C3⋊S3, C2×C3⋊S3 [×3], C62, S3×D4 [×2], C322C8, C4×C3⋊S3, C12⋊S3, C327D4, D4×C32, C2×C32⋊C4, C22×C3⋊S3, C3⋊S33C8, C4⋊(C32⋊C4), D4×C3⋊S3, C3⋊S3.5D8
Quotients: C1, C2 [×3], C4 [×2], C22, C2×C4, D4 [×2], C22⋊C4, D8, SD16, D4⋊C4, C32⋊C4, C2×C32⋊C4, C62⋊C4, C3⋊S3.5D8

Character table of C3⋊S3.5D8

 class 12A2B2C2D2E3A3B4A4B4C4D6A6B6C6D6E6F8A8B8C8D12A12B
 size 11499364421836364488881818181888
ρ1111111111111111111111111    trivial
ρ211-111-111111111-1-1-1-1-1-1-1-111    linear of order 2
ρ311-111-11111-1-111-1-1-1-1111111    linear of order 2
ρ41111111111-1-1111111-1-1-1-111    linear of order 2
ρ5111-1-1-1111-1i-i111111i-i-ii11    linear of order 4
ρ611-1-1-11111-1i-i11-1-1-1-1-iii-i11    linear of order 4
ρ711-1-1-11111-1-ii11-1-1-1-1i-i-ii11    linear of order 4
ρ8111-1-1-1111-1-ii111111-iii-i11    linear of order 4
ρ922022022-2-2002200000000-2-2    orthogonal lifted from D4
ρ10220-2-2022-22002200000000-2-2    orthogonal lifted from D4
ρ112-20-220220000-2-20000-22-2200    orthogonal lifted from D8
ρ122-20-220220000-2-200002-22-200    orthogonal lifted from D8
ρ132-202-20220000-2-20000-2-2--2--200    complex lifted from SD16
ρ142-202-20220000-2-20000--2--2-2-200    complex lifted from SD16
ρ1544-40001-24000-212-1-120000-21    orthogonal lifted from C2×C32⋊C4
ρ16440000-21-40001-2300-30000-12    orthogonal lifted from C62⋊C4
ρ17440000-21-40001-2-30030000-12    orthogonal lifted from C62⋊C4
ρ184440001-24000-21-211-20000-21    orthogonal lifted from C32⋊C4
ρ194400001-2-4000-2103-3000002-1    orthogonal lifted from C62⋊C4
ρ2044-4000-2140001-2-122-100001-2    orthogonal lifted from C2×C32⋊C4
ρ21444000-2140001-21-2-2100001-2    orthogonal lifted from C32⋊C4
ρ224400001-2-4000-210-33000002-1    orthogonal lifted from C62⋊C4
ρ238-800002-400004-20000000000    orthogonal faithful
ρ248-80000-420000-240000000000    orthogonal faithful

Permutation representations of C3⋊S3.5D8
On 24 points - transitive group 24T676
Generators in S24
(1 10 21)(3 23 12)(5 14 17)(7 19 16)
(1 10 21)(2 11 22)(3 23 12)(4 24 13)(5 14 17)(6 15 18)(7 19 16)(8 20 9)
(9 20)(10 21)(11 22)(12 23)(13 24)(14 17)(15 18)(16 19)
(1 2 3 4 5 6 7 8)(9 10 11 12 13 14 15 16)(17 18 19 20 21 22 23 24)
(1 8)(2 7)(3 6)(4 5)(9 10 20 21)(11 16 22 19)(12 18 23 15)(13 14 24 17)

G:=sub<Sym(24)| (1,10,21)(3,23,12)(5,14,17)(7,19,16), (1,10,21)(2,11,22)(3,23,12)(4,24,13)(5,14,17)(6,15,18)(7,19,16)(8,20,9), (9,20)(10,21)(11,22)(12,23)(13,24)(14,17)(15,18)(16,19), (1,2,3,4,5,6,7,8)(9,10,11,12,13,14,15,16)(17,18,19,20,21,22,23,24), (1,8)(2,7)(3,6)(4,5)(9,10,20,21)(11,16,22,19)(12,18,23,15)(13,14,24,17)>;

G:=Group( (1,10,21)(3,23,12)(5,14,17)(7,19,16), (1,10,21)(2,11,22)(3,23,12)(4,24,13)(5,14,17)(6,15,18)(7,19,16)(8,20,9), (9,20)(10,21)(11,22)(12,23)(13,24)(14,17)(15,18)(16,19), (1,2,3,4,5,6,7,8)(9,10,11,12,13,14,15,16)(17,18,19,20,21,22,23,24), (1,8)(2,7)(3,6)(4,5)(9,10,20,21)(11,16,22,19)(12,18,23,15)(13,14,24,17) );

G=PermutationGroup([(1,10,21),(3,23,12),(5,14,17),(7,19,16)], [(1,10,21),(2,11,22),(3,23,12),(4,24,13),(5,14,17),(6,15,18),(7,19,16),(8,20,9)], [(9,20),(10,21),(11,22),(12,23),(13,24),(14,17),(15,18),(16,19)], [(1,2,3,4,5,6,7,8),(9,10,11,12,13,14,15,16),(17,18,19,20,21,22,23,24)], [(1,8),(2,7),(3,6),(4,5),(9,10,20,21),(11,16,22,19),(12,18,23,15),(13,14,24,17)])

G:=TransitiveGroup(24,676);

On 24 points - transitive group 24T677
Generators in S24
(1 13 22)(2 14 23)(3 24 15)(4 17 16)(5 9 18)(6 10 19)(7 20 11)(8 21 12)
(2 23 14)(4 16 17)(6 19 10)(8 12 21)
(1 5)(2 6)(3 7)(4 8)(9 22)(10 23)(11 24)(12 17)(13 18)(14 19)(15 20)(16 21)
(1 2 3 4 5 6 7 8)(9 10 11 12 13 14 15 16)(17 18 19 20 21 22 23 24)
(1 4 5 8)(2 7 6 3)(9 21 22 16)(10 15 23 20)(11 19 24 14)(12 13 17 18)

G:=sub<Sym(24)| (1,13,22)(2,14,23)(3,24,15)(4,17,16)(5,9,18)(6,10,19)(7,20,11)(8,21,12), (2,23,14)(4,16,17)(6,19,10)(8,12,21), (1,5)(2,6)(3,7)(4,8)(9,22)(10,23)(11,24)(12,17)(13,18)(14,19)(15,20)(16,21), (1,2,3,4,5,6,7,8)(9,10,11,12,13,14,15,16)(17,18,19,20,21,22,23,24), (1,4,5,8)(2,7,6,3)(9,21,22,16)(10,15,23,20)(11,19,24,14)(12,13,17,18)>;

G:=Group( (1,13,22)(2,14,23)(3,24,15)(4,17,16)(5,9,18)(6,10,19)(7,20,11)(8,21,12), (2,23,14)(4,16,17)(6,19,10)(8,12,21), (1,5)(2,6)(3,7)(4,8)(9,22)(10,23)(11,24)(12,17)(13,18)(14,19)(15,20)(16,21), (1,2,3,4,5,6,7,8)(9,10,11,12,13,14,15,16)(17,18,19,20,21,22,23,24), (1,4,5,8)(2,7,6,3)(9,21,22,16)(10,15,23,20)(11,19,24,14)(12,13,17,18) );

G=PermutationGroup([(1,13,22),(2,14,23),(3,24,15),(4,17,16),(5,9,18),(6,10,19),(7,20,11),(8,21,12)], [(2,23,14),(4,16,17),(6,19,10),(8,12,21)], [(1,5),(2,6),(3,7),(4,8),(9,22),(10,23),(11,24),(12,17),(13,18),(14,19),(15,20),(16,21)], [(1,2,3,4,5,6,7,8),(9,10,11,12,13,14,15,16),(17,18,19,20,21,22,23,24)], [(1,4,5,8),(2,7,6,3),(9,21,22,16),(10,15,23,20),(11,19,24,14),(12,13,17,18)])

G:=TransitiveGroup(24,677);

Matrix representation of C3⋊S3.5D8 in GL6(𝔽73)

100000
010000
0072100
0072000
000001
00007272
,
100000
010000
0007200
0017200
000010
000001
,
7200000
0720000
000100
001000
000010
00007272
,
61670000
1200000
0000721
00007172
00494900
00494800
,
060000
1200000
0000172
000021
00494900
00494800

G:=sub<GL(6,GF(73))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,72,72,0,0,0,0,1,0,0,0,0,0,0,0,0,72,0,0,0,0,1,72],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,72,72,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[72,0,0,0,0,0,0,72,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,1,72,0,0,0,0,0,72],[61,12,0,0,0,0,67,0,0,0,0,0,0,0,0,0,49,49,0,0,0,0,49,48,0,0,72,71,0,0,0,0,1,72,0,0],[0,12,0,0,0,0,6,0,0,0,0,0,0,0,0,0,49,49,0,0,0,0,49,48,0,0,1,2,0,0,0,0,72,1,0,0] >;

C3⋊S3.5D8 in GAP, Magma, Sage, TeX

C_3\rtimes S_3._5D_8
% in TeX

G:=Group("C3:S3.5D8");
// GroupNames label

G:=SmallGroup(288,430);
// by ID

G=gap.SmallGroup(288,430);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,3,28,141,675,346,80,9413,691,12550,2372]);
// Polycyclic

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

Export

Character table of C3⋊S3.5D8 in TeX

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