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G = C2×D6.3D6order 288 = 25·32

Direct product of C2 and D6.3D6

direct product, metabelian, supersoluble, monomial

Aliases: C2×D6.3D6, C62.142C23, C3⋊D413D6, C23.29S32, C64(C4○D12), C63(D42S3), (C2×Dic3)⋊21D6, (C3×C6).29C24, C6.29(S3×C23), (C22×C6).99D6, (S3×C6).16C23, D6.16(C22×S3), (C22×S3).56D6, C327D48C22, C3⋊D1210C22, (C22×Dic3)⋊11S3, (C6×Dic3)⋊29C22, (S3×Dic3)⋊16C22, C322Q815C22, C6.D611C22, C3⋊Dic3.27C23, (C2×C62).77C22, (C3×Dic3).23C23, Dic3.15(C22×S3), C35(C2×C4○D12), (C6×C3⋊D4)⋊4C2, C22.9(C2×S32), (C3×C6)⋊5(C4○D4), C34(C2×D42S3), (C2×C3⋊D4)⋊13S3, C2.30(C22×S32), (Dic3×C2×C6)⋊13C2, (C2×S3×Dic3)⋊24C2, C3211(C2×C4○D4), (C2×C6.D6)⋊5C2, (S3×C2×C6).66C22, (C2×C3⋊D12)⋊15C2, (C2×C322Q8)⋊17C2, (C2×C3⋊S3).30C23, (C2×C327D4)⋊15C2, (C3×C3⋊D4)⋊10C22, (C2×C6).157(C22×S3), (C22×C3⋊S3).59C22, (C2×C3⋊Dic3).105C22, SmallGroup(288,970)

Series: Derived Chief Lower central Upper central

C1C3×C6 — C2×D6.3D6
C1C3C32C3×C6S3×C6S3×Dic3C2×S3×Dic3 — C2×D6.3D6
C32C3×C6 — C2×D6.3D6
C1C22C23

Generators and relations for C2×D6.3D6
 G = < a,b,c,d,e | a2=b6=c2=d6=1, e2=b3, ab=ba, ac=ca, ad=da, ae=ea, cbc=b-1, bd=db, be=eb, dcd-1=ece-1=b3c, ede-1=d-1 >

Subgroups: 1218 in 355 conjugacy classes, 116 normal (36 characteristic)
C1, C2, C2 [×2], C2 [×6], C3 [×2], C3, C4 [×8], C22, C22 [×2], C22 [×10], S3 [×8], C6 [×2], C6 [×4], C6 [×11], C2×C4 [×16], D4 [×12], Q8 [×4], C23, C23 [×2], C32, Dic3 [×6], Dic3 [×6], C12 [×6], D6 [×2], D6 [×14], C2×C6 [×2], C2×C6 [×4], C2×C6 [×13], C22×C4 [×3], C2×D4 [×3], C2×Q8, C4○D4 [×8], C3×S3 [×2], C3⋊S3 [×2], C3×C6, C3×C6 [×2], C3×C6 [×2], Dic6 [×8], C4×S3 [×12], D12 [×4], C2×Dic3 [×3], C2×Dic3 [×4], C2×Dic3 [×7], C3⋊D4 [×4], C3⋊D4 [×16], C2×C12 [×7], C3×D4 [×4], C22×S3, C22×S3 [×3], C22×C6 [×2], C22×C6 [×2], C2×C4○D4, C3×Dic3 [×6], C3⋊Dic3 [×2], S3×C6 [×2], S3×C6 [×2], C2×C3⋊S3 [×2], C2×C3⋊S3 [×2], C62, C62 [×2], C62 [×2], C2×Dic6 [×2], S3×C2×C4 [×3], C2×D12, C4○D12 [×8], D42S3 [×8], C22×Dic3, C22×Dic3, C2×C3⋊D4, C2×C3⋊D4 [×4], C22×C12, C6×D4, S3×Dic3 [×4], C6.D6 [×4], C3⋊D12 [×4], C322Q8 [×4], C6×Dic3 [×3], C6×Dic3 [×4], C3×C3⋊D4 [×4], C2×C3⋊Dic3, C327D4 [×4], S3×C2×C6, C22×C3⋊S3, C2×C62, C2×C4○D12, C2×D42S3, C2×S3×Dic3, D6.3D6 [×8], C2×C6.D6, C2×C3⋊D12, C2×C322Q8, Dic3×C2×C6, C6×C3⋊D4, C2×C327D4, C2×D6.3D6
Quotients: C1, C2 [×15], C22 [×35], S3 [×2], C23 [×15], D6 [×14], C4○D4 [×2], C24, C22×S3 [×14], C2×C4○D4, S32, C4○D12 [×2], D42S3 [×2], S3×C23 [×2], C2×S32 [×3], C2×C4○D12, C2×D42S3, D6.3D6 [×2], C22×S32, C2×D6.3D6

Smallest permutation representation of C2×D6.3D6
On 48 points
Generators in S48
(1 13)(2 14)(3 15)(4 16)(5 17)(6 18)(7 19)(8 20)(9 21)(10 22)(11 23)(12 24)(25 37)(26 38)(27 39)(28 40)(29 41)(30 42)(31 43)(32 44)(33 45)(34 46)(35 47)(36 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 36)(2 35)(3 34)(4 33)(5 32)(6 31)(7 27)(8 26)(9 25)(10 30)(11 29)(12 28)(13 48)(14 47)(15 46)(16 45)(17 44)(18 43)(19 39)(20 38)(21 37)(22 42)(23 41)(24 40)
(1 9 5 7 3 11)(2 10 6 8 4 12)(13 21 17 19 15 23)(14 22 18 20 16 24)(25 35 27 31 29 33)(26 36 28 32 30 34)(37 47 39 43 41 45)(38 48 40 44 42 46)
(1 36 4 33)(2 31 5 34)(3 32 6 35)(7 30 10 27)(8 25 11 28)(9 26 12 29)(13 48 16 45)(14 43 17 46)(15 44 18 47)(19 42 22 39)(20 37 23 40)(21 38 24 41)

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

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

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

54 conjugacy classes

class 1 2A2B2C2D2E2F2G2H2I3A3B3C4A4B4C4D4E4F4G4H4I4J6A···6J6K···6S6T6U12A···12H12I12J
order122222222233344444444446···66···66612···121212
size1111226618182243333666618182···24···412126···61212

54 irreducible representations

dim111111111222222224444
type++++++++++++++++-+
imageC1C2C2C2C2C2C2C2C2S3S3D6D6D6D6C4○D4C4○D12S32D42S3C2×S32D6.3D6
kernelC2×D6.3D6C2×S3×Dic3D6.3D6C2×C6.D6C2×C3⋊D12C2×C322Q8Dic3×C2×C6C6×C3⋊D4C2×C327D4C22×Dic3C2×C3⋊D4C2×Dic3C3⋊D4C22×S3C22×C6C3×C6C6C23C6C22C2
# reps118111111117412481234

Matrix representation of C2×D6.3D6 in GL6(𝔽13)

100000
010000
001000
000100
0000120
0000012
,
1210000
1200000
0012000
0001200
000010
000001
,
100000
1120000
000500
008000
000010
000001
,
1200000
0120000
000100
001000
0000112
000010
,
1200000
0120000
000800
008000
0000120
0000121

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

C2×D6.3D6 in GAP, Magma, Sage, TeX

C_2\times D_6._3D_6
% in TeX

G:=Group("C2xD6.3D6");
// GroupNames label

G:=SmallGroup(288,970);
// by ID

G=gap.SmallGroup(288,970);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,120,675,1356,9414]);
// Polycyclic

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

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