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G = C62.2Q8order 288 = 25·32

2nd non-split extension by C62 of Q8 acting faithfully

non-abelian, soluble, monomial

Aliases: C62.2Q8, C22.PSU3(𝔽2), C3⋊Dic3.8D4, C322C8.3C4, C322(C8.C4), C62.C4.1C2, C2.5(C2.PSU3(𝔽2)), (C3×C6).10(C4⋊C4), C3⋊Dic3.16(C2×C4), (C2×C322C8).7C2, (C2×C3⋊Dic3).6C22, SmallGroup(288,396)

Series: Derived Chief Lower central Upper central

C1C32C3⋊Dic3 — C62.2Q8
C1C32C3×C6C3⋊Dic3C2×C3⋊Dic3C2×C322C8 — C62.2Q8
C32C3×C6C3⋊Dic3 — C62.2Q8
C1C2C22

Generators and relations for C62.2Q8
 G = < a,b,c,d | a6=b6=1, c4=b3, d2=a3c2, ab=ba, cac-1=a3b-1, dad-1=a-1b4, cbc-1=a4b3, dbd-1=a4b, dcd-1=a3c3 >

2C2
4C3
9C4
9C4
4C6
4C6
4C6
9C8
9C8
9C2×C4
18C8
18C8
4C2×C6
12Dic3
12Dic3
2C3×C6
9C2×C8
9M4(2)
9M4(2)
12C2×Dic3
9C8.C4
2C322C8
2C322C8

Character table of C62.2Q8

 class 12A2B34A4B4C6A6B6C8A8B8C8D8E8F8G8H
 size 112899188881818181836363636
ρ1111111111111111111    trivial
ρ211111111111111-1-1-1-1    linear of order 2
ρ31111111111-1-1-1-1-1-111    linear of order 2
ρ41111111111-1-1-1-111-1-1    linear of order 2
ρ511-11-1-11-11-1-1-111i-i-ii    linear of order 4
ρ611-11-1-11-11-111-1-1i-ii-i    linear of order 4
ρ711-11-1-11-11-1-1-111-iii-i    linear of order 4
ρ811-11-1-11-11-111-1-1-ii-ii    linear of order 4
ρ922-2222-2-22-200000000    orthogonal lifted from D4
ρ102222-2-2-222200000000    symplectic lifted from Q8, Schur index 2
ρ112-202-2i2i00-20-22--2-20000    complex lifted from C8.C4
ρ122-202-2i2i00-202-2-2--20000    complex lifted from C8.C4
ρ132-2022i-2i00-202-2--2-20000    complex lifted from C8.C4
ρ142-2022i-2i00-20-22-2--20000    complex lifted from C8.C4
ρ1588-8-10001-1100000000    orthogonal lifted from C2.PSU3(𝔽2)
ρ16888-1000-1-1-100000000    orthogonal lifted from PSU3(𝔽2)
ρ178-80-100031-300000000    symplectic faithful, Schur index 2
ρ188-80-1000-31300000000    symplectic faithful, Schur index 2

Smallest permutation representation of C62.2Q8
On 48 points
Generators in S48
(1 39 29)(2 6)(3 31 33)(4 8)(5 35 25)(7 27 37)(9 17 48)(10 22 41 14 18 45)(11 42 19)(12 47 20 16 43 24)(13 21 44)(15 46 23)(26 30)(28 32)(34 38)(36 40)
(1 5)(2 36 30 6 40 26)(3 7)(4 28 34 8 32 38)(9 44 17 13 48 21)(10 22 41 14 18 45)(11 23 42 15 19 46)(12 47 20 16 43 24)(25 29)(27 31)(33 37)(35 39)
(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 13 3 15 5 9 7 11)(2 12 8 10 6 16 4 14)(17 27 19 29 21 31 23 25)(18 26 24 32 22 30 20 28)(33 46 35 48 37 42 39 44)(34 45 40 43 38 41 36 47)

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

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

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

Matrix representation of C62.2Q8 in GL10(𝔽73)

1000000000
07200000000
007272000000
0010000000
0000100000
0000010000
000000727200
0000001000
000000007272
0000000010
,
72000000000
07200000000
0010000000
0001000000
0000010000
000072720000
0000000100
000000727200
000000007272
0000000010
,
0100000000
46000000000
0000100000
0000010000
001311000000
007160000000
000000001311
000000007160
0000001000
0000000100
,
51000000000
06300000000
0000001000
0000000100
0000000010
0000000001
001311000000
007160000000
000013110000
000071600000

G:=sub<GL(10,GF(73))| [1,0,0,0,0,0,0,0,0,0,0,72,0,0,0,0,0,0,0,0,0,0,72,1,0,0,0,0,0,0,0,0,72,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,72,1,0,0,0,0,0,0,0,0,72,0,0,0,0,0,0,0,0,0,0,0,72,1,0,0,0,0,0,0,0,0,72,0],[72,0,0,0,0,0,0,0,0,0,0,72,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,72,0,0,0,0,0,0,0,0,1,72,0,0,0,0,0,0,0,0,0,0,0,72,0,0,0,0,0,0,0,0,1,72,0,0,0,0,0,0,0,0,0,0,72,1,0,0,0,0,0,0,0,0,72,0],[0,46,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,13,71,0,0,0,0,0,0,0,0,11,60,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,13,71,0,0,0,0,0,0,0,0,11,60,0,0],[51,0,0,0,0,0,0,0,0,0,0,63,0,0,0,0,0,0,0,0,0,0,0,0,0,0,13,71,0,0,0,0,0,0,0,0,11,60,0,0,0,0,0,0,0,0,0,0,13,71,0,0,0,0,0,0,0,0,11,60,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0] >;

C62.2Q8 in GAP, Magma, Sage, TeX

C_6^2._2Q_8
% in TeX

G:=Group("C6^2.2Q8");
// GroupNames label

G:=SmallGroup(288,396);
// by ID

G=gap.SmallGroup(288,396);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,3,28,85,92,219,100,346,9413,2028,691,12550,1581,2372]);
// Polycyclic

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

Export

Subgroup lattice of C62.2Q8 in TeX
Character table of C62.2Q8 in TeX

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