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G = C62.13D4order 288 = 25·32

13rd non-split extension by C62 of D4 acting faithfully

non-abelian, soluble, monomial

Aliases: C62.13D4, C32⋊D82C2, C32⋊Q163C2, C22.1S3≀C2, C322(C4○D8), D6.4D62C2, C3⋊Dic3.31D4, C322SD166C2, C3⋊Dic3.9C23, D6⋊S3.2C22, C322Q8.3C22, C322C8.10C22, C2.18(C2×S3≀C2), (C3×C6).18(C2×D4), (C2×C322C8)⋊5C2, (C2×C3⋊Dic3).96C22, SmallGroup(288,885)

Series: Derived Chief Lower central Upper central

C1C32C3⋊Dic3 — C62.13D4
C1C32C3×C6C3⋊Dic3D6⋊S3C32⋊D8 — C62.13D4
C32C3×C6C3⋊Dic3 — C62.13D4
C1C2C22

Generators and relations for C62.13D4
 G = < a,b,c,d | a6=b6=d2=1, c4=b3, ab=ba, cac-1=a3b4, dad=a-1b3, cbc-1=a2b3, bd=db, dcd=b3c3 >

Subgroups: 496 in 102 conjugacy classes, 23 normal (15 characteristic)
C1, C2, C2 [×3], C3 [×2], C4 [×4], C22, C22 [×2], S3 [×2], C6 [×6], C8 [×2], C2×C4 [×3], D4 [×4], Q8 [×2], C32, Dic3 [×6], C12 [×2], D6 [×2], C2×C6 [×4], C2×C8, D8, SD16 [×2], Q16, C4○D4 [×2], C3×S3 [×2], C3×C6, C3×C6, Dic6 [×2], C4×S3 [×2], C2×Dic3 [×4], C3⋊D4 [×4], C3×D4 [×2], C4○D8, C3×Dic3 [×2], C3⋊Dic3 [×2], S3×C6 [×2], C62, D42S3 [×2], C322C8 [×2], S3×Dic3 [×2], D6⋊S3 [×2], C322Q8 [×2], C3×C3⋊D4 [×2], C2×C3⋊Dic3, C32⋊D8, C322SD16 [×2], C32⋊Q16, C2×C322C8, D6.4D6 [×2], C62.13D4
Quotients: C1, C2 [×7], C22 [×7], D4 [×2], C23, C2×D4, C4○D8, S3≀C2, C2×S3≀C2, C62.13D4

Character table of C62.13D4

 class 12A2B2C2D3A3B4A4B4C4D4E6A6B6C6D6E6F8A8B8C8D12A12B
 size 1121212449912121844882424181818182424
ρ1111111111111111111111111    trivial
ρ211-11-111-1-11-1111-1-1-111-11-1-11    linear of order 2
ρ3111-1-11111-1-111111-1-11111-1-1    linear of order 2
ρ411-1-1111-1-1-11111-1-11-11-11-11-1    linear of order 2
ρ511-11111-1-1-1-1111-1-111-11-11-1-1    linear of order 2
ρ61111-11111-1111111-11-1-1-1-11-1    linear of order 2
ρ711-1-1-111-1-111111-1-1-1-1-11-1111    linear of order 2
ρ8111-1111111-1111111-1-1-1-1-1-11    linear of order 2
ρ922-200222200-222-2-200000000    orthogonal lifted from D4
ρ102220022-2-200-2222200000000    orthogonal lifted from D4
ρ112-2000222i-2i000-2-20000--2-2-2200    complex lifted from C4○D8
ρ122-200022-2i2i000-2-20000--22-2-200    complex lifted from C4○D8
ρ132-2000222i-2i000-2-20000-22--2-200    complex lifted from C4○D8
ρ142-200022-2i2i000-2-20000-2-2--2200    complex lifted from C4○D8
ρ154440-21-200-200-21-2110000001    orthogonal lifted from S3≀C2
ρ1644-40-21-200200-212-11000000-1    orthogonal lifted from C2×S3≀C2
ρ1744-4-20-21000201-2-12010000-10    orthogonal lifted from C2×S3≀C2
ρ1844420-21000201-21-20-10000-10    orthogonal lifted from S3≀C2
ρ1944-4021-200-200-212-1-10000001    orthogonal lifted from C2×S3≀C2
ρ20444021-200200-21-21-1000000-1    orthogonal lifted from S3≀C2
ρ2144-420-21000-201-2-120-1000010    orthogonal lifted from C2×S3≀C2
ρ22444-20-21000-201-21-201000010    orthogonal lifted from S3≀C2
ρ238-80002-4000004-20000000000    symplectic faithful, Schur index 2
ρ248-8000-4200000-240000000000    symplectic faithful, Schur index 2

Smallest permutation representation of C62.13D4
On 48 points
Generators in S48
(1 42 24 29 13 37)(2 38 14 30 17 43)(3 39 15 31 18 44)(4 45 19 32 16 40)(5 46 20 25 9 33)(6 34 10 26 21 47)(7 35 11 27 22 48)(8 41 23 28 12 36)
(1 20 13 5 24 9)(2 21 14 6 17 10)(3 11 18 7 15 22)(4 12 19 8 16 23)(25 37 46 29 33 42)(26 38 47 30 34 43)(27 44 35 31 48 39)(28 45 36 32 41 40)
(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 8)(2 7)(3 6)(4 5)(9 19)(10 18)(11 17)(12 24)(13 23)(14 22)(15 21)(16 20)(25 28)(26 27)(29 32)(30 31)(33 41)(34 48)(35 47)(36 46)(37 45)(38 44)(39 43)(40 42)

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

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

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

Matrix representation of C62.13D4 in GL6(𝔽73)

27540000
46460000
0017200
001000
000001
0000721
,
7200000
0720000
0007200
0017200
0000072
0000172
,
41410000
1600000
000001
000010
0072000
0007200
,
41410000
16320000
000010
000001
001000
000100

G:=sub<GL(6,GF(73))| [27,46,0,0,0,0,54,46,0,0,0,0,0,0,1,1,0,0,0,0,72,0,0,0,0,0,0,0,0,72,0,0,0,0,1,1],[72,0,0,0,0,0,0,72,0,0,0,0,0,0,0,1,0,0,0,0,72,72,0,0,0,0,0,0,0,1,0,0,0,0,72,72],[41,16,0,0,0,0,41,0,0,0,0,0,0,0,0,0,72,0,0,0,0,0,0,72,0,0,0,1,0,0,0,0,1,0,0,0],[41,16,0,0,0,0,41,32,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,0,1,0,0] >;

C62.13D4 in GAP, Magma, Sage, TeX

C_6^2._{13}D_4
% in TeX

G:=Group("C6^2.13D4");
// GroupNames label

G:=SmallGroup(288,885);
// by ID

G=gap.SmallGroup(288,885);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,3,141,422,219,675,346,80,2693,2028,362,797,1203]);
// Polycyclic

G:=Group<a,b,c,d|a^6=b^6=d^2=1,c^4=b^3,a*b=b*a,c*a*c^-1=a^3*b^4,d*a*d=a^-1*b^3,c*b*c^-1=a^2*b^3,b*d=d*b,d*c*d=b^3*c^3>;
// generators/relations

Export

Character table of C62.13D4 in TeX

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