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G = C3×Q8.14D6order 288 = 25·32

Direct product of C3 and Q8.14D6

direct product, metabelian, supersoluble, monomial

Aliases: C3×Q8.14D6, C62.68D4, D4.S36C6, C3⋊Q166C6, D4.9(S3×C6), C6.61(C6×D4), (C3×D4).49D6, C12.59(C3×D4), Q8.19(S3×C6), (C3×Q8).74D6, (C2×Dic6)⋊11C6, (C6×Dic6)⋊16C2, (C2×C12).248D6, (C3×C12).161D4, C4.Dic310C6, (C3×C12).90C23, C12.19(C22×C6), Dic6.12(C2×C6), C12.142(C3⋊D4), C12.170(C22×S3), (C6×C12).133C22, C3224(C8.C22), (C3×Dic6).42C22, (D4×C32).25C22, (Q8×C32).26C22, C3⋊C8.4(C2×C6), C4.19(S3×C2×C6), (C2×C4).21(S3×C6), C4○D4.6(C3×S3), (C3×D4).9(C2×C6), (C2×C6).11(C3×D4), C4.25(C3×C3⋊D4), C2.25(C6×C3⋊D4), C35(C3×C8.C22), (C2×C12).44(C2×C6), (C3×D4.S3)⋊12C2, (C3×C4○D4).21S3, (C3×C4○D4).11C6, (C3×C3⋊Q16)⋊14C2, (C3×C6).269(C2×D4), C6.162(C2×C3⋊D4), (C3×C3⋊C8).23C22, (C3×C4.Dic3)⋊8C2, (C3×Q8).21(C2×C6), C22.6(C3×C3⋊D4), (C2×C6).47(C3⋊D4), (C32×C4○D4).4C2, SmallGroup(288,722)

Series: Derived Chief Lower central Upper central

C1C12 — C3×Q8.14D6
C1C3C6C12C3×C12C3×Dic6C6×Dic6 — C3×Q8.14D6
C3C6C12 — C3×Q8.14D6
C1C6C2×C12C3×C4○D4

Generators and relations for C3×Q8.14D6
 G = < a,b,c,d,e | a3=b4=d6=1, c2=e2=b2, ab=ba, ac=ca, ad=da, ae=ea, cbc-1=ebe-1=b-1, bd=db, dcd-1=b2c, ece-1=bc, ede-1=d-1 >

Subgroups: 290 in 140 conjugacy classes, 58 normal (42 characteristic)
C1, C2, C2 [×2], C3 [×2], C3, C4 [×2], C4 [×3], C22, C22, C6 [×2], C6 [×8], C8 [×2], C2×C4, C2×C4 [×2], D4, D4, Q8, Q8 [×3], C32, Dic3 [×2], C12 [×4], C12 [×8], C2×C6 [×2], C2×C6 [×5], M4(2), SD16 [×2], Q16 [×2], C2×Q8, C4○D4, C3×C6, C3×C6 [×2], C3⋊C8 [×2], C24 [×2], Dic6 [×2], Dic6, C2×Dic3, C2×C12 [×2], C2×C12 [×6], C3×D4 [×2], C3×D4 [×5], C3×Q8 [×2], C3×Q8 [×4], C8.C22, C3×Dic3 [×2], C3×C12 [×2], C3×C12, C62, C62, C4.Dic3, D4.S3 [×2], C3⋊Q16 [×2], C3×M4(2), C3×SD16 [×2], C3×Q16 [×2], C2×Dic6, C6×Q8, C3×C4○D4 [×2], C3×C4○D4, C3×C3⋊C8 [×2], C3×Dic6 [×2], C3×Dic6, C6×Dic3, C6×C12, C6×C12, D4×C32, D4×C32, Q8×C32, Q8.14D6, C3×C8.C22, C3×C4.Dic3, C3×D4.S3 [×2], C3×C3⋊Q16 [×2], C6×Dic6, C32×C4○D4, C3×Q8.14D6
Quotients: C1, C2 [×7], C3, C22 [×7], S3, C6 [×7], D4 [×2], C23, D6 [×3], C2×C6 [×7], C2×D4, C3×S3, C3⋊D4 [×2], C3×D4 [×2], C22×S3, C22×C6, C8.C22, S3×C6 [×3], C2×C3⋊D4, C6×D4, C3×C3⋊D4 [×2], S3×C2×C6, Q8.14D6, C3×C8.C22, C6×C3⋊D4, C3×Q8.14D6

Smallest permutation representation of C3×Q8.14D6
On 48 points
Generators in S48
(1 3 2)(4 5 6)(7 9 8)(10 12 11)(13 15 14)(16 17 18)(19 20 21)(22 23 24)(25 27 29)(26 28 30)(31 33 35)(32 34 36)(37 41 39)(38 42 40)(43 47 45)(44 48 46)
(1 11 15 8)(2 12 13 9)(3 10 14 7)(4 22 21 16)(5 23 19 17)(6 24 20 18)(25 35 28 32)(26 36 29 33)(27 31 30 34)(37 46 40 43)(38 47 41 44)(39 48 42 45)
(1 36 15 33)(2 34 13 31)(3 32 14 35)(4 43 21 46)(5 47 19 44)(6 45 20 48)(7 25 10 28)(8 29 11 26)(9 27 12 30)(16 37 22 40)(17 41 23 38)(18 39 24 42)
(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 20 15 6)(2 19 13 5)(3 21 14 4)(7 16 10 22)(8 18 11 24)(9 17 12 23)(25 43 28 46)(26 48 29 45)(27 47 30 44)(31 38 34 41)(32 37 35 40)(33 42 36 39)

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

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

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

63 conjugacy classes

class 1 2A2B2C3A3B3C3D3E4A4B4C4D4E6A6B6C···6G6H···6R8A8B12A···12J12K···12U12V12W12X12Y24A24B24C24D
order12223333344444666···66···68812···1212···121212121224242424
size1124112222241212112···24···412122···24···41212121212121212

63 irreducible representations

dim11111111111122222222222222224444
type++++++++++++--
imageC1C2C2C2C2C2C3C6C6C6C6C6S3D4D4D6D6D6C3×S3C3⋊D4C3×D4C3⋊D4C3×D4S3×C6S3×C6S3×C6C3×C3⋊D4C3×C3⋊D4C8.C22Q8.14D6C3×C8.C22C3×Q8.14D6
kernelC3×Q8.14D6C3×C4.Dic3C3×D4.S3C3×C3⋊Q16C6×Dic6C32×C4○D4Q8.14D6C4.Dic3D4.S3C3⋊Q16C2×Dic6C3×C4○D4C3×C4○D4C3×C12C62C2×C12C3×D4C3×Q8C4○D4C12C12C2×C6C2×C6C2×C4D4Q8C4C22C32C3C3C1
# reps11221122442211111122222222441224

Matrix representation of C3×Q8.14D6 in GL6(𝔽73)

800000
080000
001000
000100
000010
000001
,
100000
010000
0014800
0037200
00060722
005660721
,
100000
010000
0000720
00056070
001000
00048017
,
900000
64650000
001000
000100
0000720
00013072
,
65660000
980000
000400
0018000
000686112
0044686712

G:=sub<GL(6,GF(73))| [8,0,0,0,0,0,0,8,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,3,0,56,0,0,48,72,60,60,0,0,0,0,72,72,0,0,0,0,2,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,56,0,48,0,0,72,0,0,0,0,0,0,70,0,17],[9,64,0,0,0,0,0,65,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,13,0,0,0,0,72,0,0,0,0,0,0,72],[65,9,0,0,0,0,66,8,0,0,0,0,0,0,0,18,0,44,0,0,4,0,68,68,0,0,0,0,61,67,0,0,0,0,12,12] >;

C3×Q8.14D6 in GAP, Magma, Sage, TeX

C_3\times Q_8._{14}D_6
% in TeX

G:=Group("C3xQ8.14D6");
// GroupNames label

G:=SmallGroup(288,722);
// by ID

G=gap.SmallGroup(288,722);
# by ID

G:=PCGroup([7,-2,-2,-2,-3,-2,-2,-3,336,590,555,2524,648,102,9414]);
// Polycyclic

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

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