Copied to
clipboard

G = C62⋊3Q8  order 288 = 25·32

1st semidirect product of C62 and Q8 acting via Q8/C2=C22

metabelian, supersoluble, monomial

Aliases: C62⋊3Q8, C62.106C23, C23.22S32, (C2×C6)⋊7Dic6, C6.165(S3×D4), C6.24(C2×Dic6), (C22×C6).69D6, Dic3⋊Dic3⋊9C2, C6.64(C4○D12), C62⋊5C4.3C2, (C2×Dic3).42D6, (C3×Dic3).38D4, C62.C22⋊5C2, C6.D4.3S3, C32⋊14(C22⋊Q8), C6.51(D4⋊2S3), C3⋊3(C12.48D4), C22⋊2(C32⋊2Q8), (C2×C62).25C22, (C22×Dic3).4S3, C3⋊7(Dic3.D4), C2.25(D6.3D6), Dic3.19(C3⋊D4), (C6×Dic3).145C22, C2.38(S3×C3⋊D4), C6.61(C2×C3⋊D4), (C3×C6).40(C2×Q8), (Dic3×C2×C6).6C2, C22.135(C2×S32), (C2×C32⋊2Q8)⋊7C2, (C3×C6).152(C2×D4), C2.8(C2×C32⋊2Q8), (C3×C6).80(C4○D4), (C2×C6).125(C22×S3), (C3×C6.D4).2C2, (C2×C3⋊Dic3).65C22, SmallGroup(288,612)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C62 — C62⋊3Q8
Chief series
C1 — C3 — C32 — C3×C6 — C62 — C6×Dic3 — Dic3⋊Dic3 — C62⋊3Q8
Lower central
C32 — C62 — C62⋊3Q8
Upper central
C1 — C22 — C23

Generators and relations for C62⋊3Q8
 G = < a,b,c,d | a6=b6=c4=1, d2=c2, ab=ba, cac-1=a-1, dad-1=ab3, bc=cb, dbd-1=b-1, dcd-1=c-1 >

Subgroups: 538 in 165 conjugacy classes, 56 normal (44 characteristic)
C1, C2, C2, C3, C3, C4, C22, C22, C22, C6, C6, C2×C4, Q8, C23, C32, Dic3, Dic3, C12, C2×C6, C2×C6, C2×C6, C22⋊C4, C4⋊C4, C22×C4, C2×Q8, C3×C6, C3×C6, Dic6, C2×Dic3, C2×Dic3, C2×C12, C22×C6, C22×C6, C22⋊Q8, C3×Dic3, C3×Dic3, C3⋊Dic3, C62, C62, C62, Dic3⋊C4, C4⋊Dic3, C6.D4, C6.D4, C3×C22⋊C4, C2×Dic6, C22×Dic3, C22×C12, C32⋊2Q8, C6×Dic3, C6×Dic3, C2×C3⋊Dic3, C2×C62, Dic3.D4, C12.48D4, Dic3⋊Dic3, C62.C22, C3×C6.D4, C62⋊5C4, C2×C32⋊2Q8, Dic3×C2×C6, C62⋊3Q8
Quotients: C1, C2, C22, S3, D4, Q8, C23, D6, C2×D4, C2×Q8, C4○D4, Dic6, C3⋊D4, C22×S3, C22⋊Q8, S32, C2×Dic6, C4○D12, S3×D4, D4⋊2S3, C2×C3⋊D4, C32⋊2Q8, C2×S32, Dic3.D4, C12.48D4, D6.3D6, C2×C32⋊2Q8, S3×C3⋊D4, C62⋊3Q8

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

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

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

48 conjugacy classes

class 1 2A2B2C2D2E3A3B3C4A4B4C4D4E4F4G4H6A···6J6K···6S12A···12H12I12J12K12L
order122222333444444446···66···612···1212121212
size1111222246666121236362···24···46···612121212

48 irreducible representations

dim111111122222222224444444
type++++++++++-++-++--+
imageC1C2C2C2C2C2C2S3S3D4Q8D6D6C4○D4C3⋊D4Dic6C4○D12S32S3×D4D4⋊2S3C32⋊2Q8C2×S32D6.3D6S3×C3⋊D4
kernelC62⋊3Q8Dic3⋊Dic3C62.C22C3×C6.D4C62⋊5C4C2×C32⋊2Q8Dic3×C2×C6C6.D4C22×Dic3C3×Dic3C62C2×Dic3C22×C6C3×C6Dic3C2×C6C6C23C6C6C22C22C2C2
# reps121111111224224841112122

Matrix representation of C62⋊3Q8 ►in GL8(𝔽13)

120000000
012000000
00100000
000120000
00001000
00000100
000000112
00000010
,
01000000
1212000000
001200000
000120000
00001000
00000100
00000010
00000001
,
120000000
012000000
00100000
00010000
00005000
00000800
00000001
00000010
,
10000000
1212000000
00030000
00900000
00000100
000012000
00000010
00000001

G:=sub<GL(8,GF(13))| [12,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,1,0,0,0,0,0,0,12,0],[0,12,0,0,0,0,0,0,1,12,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[12,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,5,0,0,0,0,0,0,0,0,8,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0],[1,12,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,0,9,0,0,0,0,0,0,3,0,0,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1] >;
 

C62⋊3Q8 in GAP, Magma, Sage, TeX 

C_6^2\rtimes_3Q_8
 
% in TeX
 
G:=Group("C6^2:3Q8");
 
// GroupNames label
 
G:=SmallGroup(288,612);
 
// by ID
 
G=gap.SmallGroup(288,612);
 
# by ID
 
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,112,253,64,590,1356,9414]);
 
// Polycyclic
 
G:=Group<a,b,c,d|a^6=b^6=c^4=1,d^2=c^2,a*b=b*a,c*a*c^-1=a^-1,d*a*d^-1=a*b^3,b*c=c*b,d*b*d^-1=b^-1,d*c*d^-1=c^-1>;
 
// generators/relations
 

׿
×
⋊
ℤ
𝔽
○
≀
ℚ
◁