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G = C62.Q8order 288 = 25·32

1st non-split extension by C62 of Q8 acting faithfully

non-abelian, soluble, monomial

Aliases: C62.1Q8, C22.2PSU3(𝔽2), C3⋊S3.3C42, C322(C2.C42), C2.2(C2.PSU3(𝔽2)), (C2×C32⋊C4)⋊2C4, (C3×C6).9(C4⋊C4), (C2×C3⋊S3).12D4, C3⋊S3.5(C22⋊C4), (C22×C32⋊C4).2C2, (C22×C3⋊S3).3C22, (C2×C3⋊S3).12(C2×C4), SmallGroup(288,395)

Series: Derived Chief Lower central Upper central

C1C32C3⋊S3 — C62.Q8
C1C32C3⋊S3C2×C3⋊S3C22×C3⋊S3C22×C32⋊C4 — C62.Q8
C32C3⋊S3 — C62.Q8
C1C22

Generators and relations for C62.Q8
 G = < a,b,c,d | a6=b6=c4=1, d2=a3c2, ab=ba, cac-1=a3b2, dad-1=a-1b4, cbc-1=a4b3, dbd-1=a4b, dcd-1=a3b3c-1 >

Subgroups: 612 in 92 conjugacy classes, 31 normal (7 characteristic)
C1, C2 [×3], C2 [×4], C3, C4 [×6], C22, C22 [×6], S3 [×4], C6 [×3], C2×C4 [×12], C23, C32, D6 [×6], C2×C6, C22×C4 [×3], C3⋊S3, C3⋊S3 [×3], C3×C6 [×3], C22×S3, C2.C42, C32⋊C4 [×6], C2×C3⋊S3 [×6], C62, C2×C32⋊C4 [×6], C2×C32⋊C4 [×6], C22×C3⋊S3, C22×C32⋊C4 [×3], C62.Q8
Quotients: C1, C2 [×3], C4 [×6], C22, C2×C4 [×3], D4 [×3], Q8, C42, C22⋊C4 [×3], C4⋊C4 [×3], C2.C42, PSU3(𝔽2), C2.PSU3(𝔽2) [×3], C62.Q8

Character table of C62.Q8

 class 12A2B2C2D2E2F2G34A4B4C4D4E4F4G4H4I4J4K4L6A6B6C
 size 111199998181818181818181818181818888
ρ1111111111111111111111111    trivial
ρ2111111111-1-1-1-11111-1-1-1-1111    linear of order 2
ρ31111111111-1-1-1-1-1-1-1111-1111    linear of order 2
ρ4111111111-1111-1-1-1-1-1-1-11111    linear of order 2
ρ51-1-1111-1-11ii-i-i-1-111-i-iii1-1-1    linear of order 4
ρ61-1-1111-1-11-i-iii-1-111ii-i-i1-1-1    linear of order 4
ρ711-1-1-111-11-i11-1i-i-ii-iii-1-11-1    linear of order 4
ρ811-1-1-111-11i-1-11i-i-iii-i-i1-11-1    linear of order 4
ρ911-1-1-111-11-i-1-11-iii-i-iii1-11-1    linear of order 4
ρ101-1-1111-1-11i-iii11-1-1-i-ii-i1-1-1    linear of order 4
ρ111-1-1111-1-11-ii-i-i11-1-1ii-ii1-1-1    linear of order 4
ρ1211-1-1-111-11i11-1-iii-ii-i-i-1-11-1    linear of order 4
ρ131-11-1-11-1111i-ii-ii-ii-11-1-i-1-11    linear of order 4
ρ141-11-1-11-111-1-ii-i-ii-ii1-11i-1-11    linear of order 4
ρ151-11-1-11-1111-ii-ii-ii-i-11-1i-1-11    linear of order 4
ρ161-11-1-11-111-1i-iii-ii-i1-11-i-1-11    linear of order 4
ρ172-22-22-22-22000000000000-2-22    orthogonal lifted from D4
ρ1822-2-22-2-222000000000000-22-2    orthogonal lifted from D4
ρ192-2-22-2-22220000000000002-2-2    orthogonal lifted from D4
ρ202222-2-2-2-22000000000000222    symplectic lifted from Q8, Schur index 2
ρ2188-8-80000-10000000000001-11    orthogonal lifted from C2.PSU3(𝔽2)
ρ228-88-80000-100000000000011-1    orthogonal lifted from C2.PSU3(𝔽2)
ρ238-8-880000-1000000000000-111    orthogonal lifted from C2.PSU3(𝔽2)
ρ2488880000-1000000000000-1-1-1    orthogonal lifted from PSU3(𝔽2)

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

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

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

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

Matrix representation of C62.Q8 in GL10(𝔽13)

12000000000
01200000000
0010000000
0001000000
00000120000
00001120000
00000012100
00000012000
006610051212
00000128010
,
1000000000
0100000000
0001000000
00121000000
00001200000
00000120000
00000011200
0000001000
00060050012
007012120811
,
0100000000
1000000000
0000100000
0000010000
00012000000
00120000000
006611551112
00000000121
0011880080
00118812080
,
8000000000
0500000000
0000001000
0000000100
00000000121
006611551112
00012000000
00120000000
00995588120
00995488120

G:=sub<GL(10,GF(13))| [12,0,0,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,6,0,0,0,0,1,0,0,0,0,6,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,12,12,0,0,0,12,0,0,0,0,0,0,12,12,0,8,0,0,0,0,0,0,1,0,5,0,0,0,0,0,0,0,0,0,12,1,0,0,0,0,0,0,0,0,12,0],[1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,12,0,0,0,0,0,7,0,0,1,1,0,0,0,0,6,0,0,0,0,0,12,0,0,0,0,12,0,0,0,0,0,12,0,0,0,12,0,0,0,0,0,0,1,1,5,0,0,0,0,0,0,0,12,0,0,8,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,12,1],[0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,12,6,0,1,1,0,0,0,0,12,0,6,0,1,1,0,0,1,0,0,0,1,0,8,8,0,0,0,1,0,0,1,0,8,8,0,0,0,0,0,0,5,0,0,12,0,0,0,0,0,0,5,0,0,0,0,0,0,0,0,0,11,12,8,8,0,0,0,0,0,0,12,1,0,0],[8,0,0,0,0,0,0,0,0,0,0,5,0,0,0,0,0,0,0,0,0,0,0,0,0,6,0,12,9,9,0,0,0,0,0,6,12,0,9,9,0,0,0,0,0,1,0,0,5,5,0,0,0,0,0,1,0,0,5,4,0,0,1,0,0,5,0,0,8,8,0,0,0,1,0,5,0,0,8,8,0,0,0,0,12,11,0,0,12,12,0,0,0,0,1,12,0,0,0,0] >;

C62.Q8 in GAP, Magma, Sage, TeX

C_6^2.Q_8
% in TeX

G:=Group("C6^2.Q8");
// GroupNames label

G:=SmallGroup(288,395);
// by ID

G=gap.SmallGroup(288,395);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,3,28,309,92,9413,2028,691,12550,1581,2372]);
// Polycyclic

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

Export

Character table of C62.Q8 in TeX

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