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G = C2×D6.6D6order 288 = 25·32

Direct product of C2 and D6.6D6

direct product, metabelian, supersoluble, monomial

Aliases: C2×D6.6D6, Dic624D6, C62.133C23, (C4×S3)⋊13D6, C61(C4○D12), C6.8(S3×C23), (C3×C6).8C24, C61(Q83S3), (C6×Dic6)⋊21C2, (C2×Dic6)⋊15S3, (C2×C12).285D6, (S3×C12)⋊17C22, (S3×C6).22C23, D6.20(C22×S3), (C22×S3).72D6, C6.D65C22, C12⋊S320C22, C3⋊D1212C22, C12.132(C22×S3), (C6×C12).162C22, (C3×C12).116C23, (C2×Dic3).117D6, (C3×Dic6)⋊27C22, (C3×Dic3).6C23, Dic3.26(C22×S3), (C6×Dic3).47C22, (S3×C2×C4)⋊4S3, (S3×C2×C12)⋊8C2, C4.63(C2×S32), (C2×C4).87S32, C31(C2×C4○D12), C324(C2×C4○D4), (C3×C6)⋊4(C4○D4), C2.11(C22×S32), C22.63(C2×S32), C31(C2×Q83S3), (C2×C6.D6)⋊3C2, (C2×C12⋊S3)⋊17C2, (C2×C3⋊D12)⋊19C2, (C2×C3⋊S3).18C23, (S3×C2×C6).105C22, (C2×C6).150(C22×S3), (C22×C3⋊S3).57C22, SmallGroup(288,949)

Series: Derived Chief Lower central Upper central

C1C3×C6 — C2×D6.6D6
C1C3C32C3×C6S3×C6C3⋊D12C2×C3⋊D12 — C2×D6.6D6
C32C3×C6 — C2×D6.6D6
C1C22C2×C4

Generators and relations for C2×D6.6D6
 G = < a,b,c,d,e | a2=b6=c2=1, d6=e2=b3, ab=ba, ac=ca, ad=da, ae=ea, cbc=b-1, bd=db, be=eb, cd=dc, ece-1=b3c, ede-1=b3d5 >

Subgroups: 1346 in 355 conjugacy classes, 116 normal (26 characteristic)
C1, C2, C2 [×2], C2 [×6], C3 [×2], C3, C4 [×2], C4 [×6], C22, C22 [×12], S3 [×14], C6 [×2], C6 [×4], C6 [×5], C2×C4, C2×C4 [×15], D4 [×12], Q8 [×4], C23 [×3], C32, Dic3 [×6], C12 [×4], C12 [×8], D6 [×2], D6 [×26], C2×C6 [×2], C2×C6 [×5], C22×C4 [×3], C2×D4 [×3], C2×Q8, C4○D4 [×8], C3×S3 [×2], C3⋊S3 [×4], C3×C6, C3×C6 [×2], Dic6 [×4], C4×S3 [×4], C4×S3 [×16], D12 [×20], C2×Dic3, C2×Dic3 [×2], C3⋊D4 [×8], C2×C12 [×2], C2×C12 [×8], C3×Q8 [×4], C22×S3, C22×S3 [×6], C22×C6, C2×C4○D4, C3×Dic3 [×6], C3×C12 [×2], S3×C6 [×2], S3×C6 [×2], C2×C3⋊S3 [×4], C2×C3⋊S3 [×4], C62, C2×Dic6, S3×C2×C4, S3×C2×C4 [×4], C2×D12 [×5], C4○D12 [×8], Q83S3 [×8], C2×C3⋊D4 [×2], C22×C12, C6×Q8, C6.D6 [×8], C3⋊D12 [×8], C3×Dic6 [×4], S3×C12 [×4], C6×Dic3, C6×Dic3 [×2], C12⋊S3 [×4], C6×C12, S3×C2×C6, C22×C3⋊S3 [×2], C2×C4○D12, C2×Q83S3, D6.6D6 [×8], C2×C6.D6 [×2], C2×C3⋊D12 [×2], C6×Dic6, S3×C2×C12, C2×C12⋊S3, C2×D6.6D6
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], Q83S3 [×2], S3×C23 [×2], C2×S32 [×3], C2×C4○D12, C2×Q83S3, D6.6D6 [×2], C22×S32, C2×D6.6D6

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

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

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

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

54 conjugacy classes

class 1 2A2B2C2D2E2F2G2H2I3A3B3C4A4B4C4D4E4F4G4H4I4J6A···6F6G6H6I6J6K6L6M12A12B12C12D12E···12J12K12L12M12N12O12P12Q12R
order122222222233344444444446···666666661212121212···121212121212121212
size1111661818181822422333366662···2444666622224···4666612121212

54 irreducible representations

dim111111122222222244444
type+++++++++++++++++++
imageC1C2C2C2C2C2C2S3S3D6D6D6D6D6C4○D4C4○D12S32Q83S3C2×S32C2×S32D6.6D6
kernelC2×D6.6D6D6.6D6C2×C6.D6C2×C3⋊D12C6×Dic6S3×C2×C12C2×C12⋊S3C2×Dic6S3×C2×C4Dic6C4×S3C2×Dic3C2×C12C22×S3C3×C6C6C2×C4C6C4C22C2
# reps182211111443214812214

Matrix representation of C2×D6.6D6 in GL6(𝔽13)

1200000
0120000
001000
000100
000010
000001
,
1210000
1200000
0012000
0001200
000010
000001
,
1200000
1210000
0081200
0011500
0000120
0000012
,
100000
010000
001800
0031200
0000012
0000112
,
100000
010000
005000
002800
0000121
000001

G:=sub<GL(6,GF(13))| [12,0,0,0,0,0,0,12,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[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],[12,12,0,0,0,0,0,1,0,0,0,0,0,0,8,11,0,0,0,0,12,5,0,0,0,0,0,0,12,0,0,0,0,0,0,12],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,3,0,0,0,0,8,12,0,0,0,0,0,0,0,1,0,0,0,0,12,12],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,5,2,0,0,0,0,0,8,0,0,0,0,0,0,12,0,0,0,0,0,1,1] >;

C2×D6.6D6 in GAP, Magma, Sage, TeX

C_2\times D_6._6D_6
% in TeX

G:=Group("C2xD6.6D6");
// GroupNames label

G:=SmallGroup(288,949);
// by ID

G=gap.SmallGroup(288,949);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,253,120,346,80,1356,9414]);
// Polycyclic

G:=Group<a,b,c,d,e|a^2=b^6=c^2=1,d^6=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,c*d=d*c,e*c*e^-1=b^3*c,e*d*e^-1=b^3*d^5>;
// generators/relations

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