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G = C623Q8order 288 = 25·32

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

metabelian, supersoluble, monomial

Aliases: C623Q8, C62.106C23, C23.22S32, (C2×C6)⋊7Dic6, C6.165(S3×D4), C6.24(C2×Dic6), (C22×C6).69D6, Dic3⋊Dic39C2, C6.64(C4○D12), C625C4.3C2, (C2×Dic3).42D6, (C3×Dic3).38D4, C62.C225C2, C6.D4.3S3, C3214(C22⋊Q8), C6.51(D42S3), C33(C12.48D4), C222(C322Q8), (C2×C62).25C22, (C22×Dic3).4S3, C37(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×C322Q8)⋊7C2, (C3×C6).152(C2×D4), C2.8(C2×C322Q8), (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

C1C62 — C623Q8
C1C3C32C3×C6C62C6×Dic3Dic3⋊Dic3 — C623Q8
C32C62 — C623Q8
C1C22C23

Generators and relations for C623Q8
 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 [×3], C2 [×2], C3 [×2], C3, C4 [×7], C22, C22 [×2], C22 [×2], C6 [×6], C6 [×9], C2×C4 [×8], Q8 [×2], C23, C32, Dic3 [×2], Dic3 [×9], C12 [×5], C2×C6 [×2], C2×C6 [×4], C2×C6 [×9], C22⋊C4 [×2], C4⋊C4 [×3], C22×C4, C2×Q8, C3×C6 [×3], C3×C6 [×2], Dic6 [×4], C2×Dic3 [×4], C2×Dic3 [×8], C2×C12 [×6], C22×C6 [×2], C22×C6, C22⋊Q8, C3×Dic3 [×2], C3×Dic3 [×3], C3⋊Dic3 [×2], C62, C62 [×2], C62 [×2], Dic3⋊C4 [×4], C4⋊Dic3 [×2], C6.D4, C6.D4 [×3], C3×C22⋊C4, C2×Dic6 [×2], C22×Dic3, C22×C12, C322Q8 [×2], C6×Dic3 [×4], C6×Dic3 [×2], C2×C3⋊Dic3 [×2], C2×C62, Dic3.D4, C12.48D4, Dic3⋊Dic3 [×2], C62.C22, C3×C6.D4, C625C4, C2×C322Q8, Dic3×C2×C6, C623Q8
Quotients: C1, C2 [×7], C22 [×7], S3 [×2], D4 [×2], Q8 [×2], C23, D6 [×6], C2×D4, C2×Q8, C4○D4, Dic6 [×4], C3⋊D4 [×2], C22×S3 [×2], C22⋊Q8, S32, C2×Dic6 [×2], C4○D12, S3×D4, D42S3, C2×C3⋊D4, C322Q8 [×2], C2×S32, Dic3.D4, C12.48D4, D6.3D6, C2×C322Q8, S3×C3⋊D4, C623Q8

Smallest permutation representation of C623Q8
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 10 2 11 3 12)(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 36 35 34 33 32)(37 38 39 40 41 42)(43 48 47 46 45 44)
(1 19 8 13)(2 21 9 15)(3 20 7 14)(4 23 10 17)(5 22 11 16)(6 24 12 18)(25 34 41 47)(26 33 42 46)(27 32 37 45)(28 31 38 44)(29 36 39 43)(30 35 40 48)
(1 39 8 29)(2 37 9 27)(3 41 7 25)(4 28 10 38)(5 26 11 42)(6 30 12 40)(13 43 19 36)(14 47 20 34)(15 45 21 32)(16 33 22 46)(17 31 23 44)(18 35 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), (1,10,2,11,3,12)(4,9,5,7,6,8)(13,23,15,22,14,24)(16,20,18,19,17,21)(25,26,27,28,29,30)(31,36,35,34,33,32)(37,38,39,40,41,42)(43,48,47,46,45,44), (1,19,8,13)(2,21,9,15)(3,20,7,14)(4,23,10,17)(5,22,11,16)(6,24,12,18)(25,34,41,47)(26,33,42,46)(27,32,37,45)(28,31,38,44)(29,36,39,43)(30,35,40,48), (1,39,8,29)(2,37,9,27)(3,41,7,25)(4,28,10,38)(5,26,11,42)(6,30,12,40)(13,43,19,36)(14,47,20,34)(15,45,21,32)(16,33,22,46)(17,31,23,44)(18,35,24,48)>;

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

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

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×D4D42S3C322Q8C2×S32D6.3D6S3×C3⋊D4
kernelC623Q8Dic3⋊Dic3C62.C22C3×C6.D4C625C4C2×C322Q8Dic3×C2×C6C6.D4C22×Dic3C3×Dic3C62C2×Dic3C22×C6C3×C6Dic3C2×C6C6C23C6C6C22C22C2C2
# reps121111111224224841112122

Matrix representation of C623Q8 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] >;

C623Q8 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

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