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G = C4218D6order 192 = 26·3

16th semidirect product of C42 and D6 acting via D6/C3=C22

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C4218D6, C6.182+ 1+4, C4⋊C449D6, (C4×D4)⋊18S3, (C22×C4)⋊8D6, (D4×C12)⋊20C2, D6⋊Q88C2, C22⋊C448D6, (C4×C12)⋊32C22, (C2×D4).217D6, C232D6.5C2, C23.9D67C2, C4⋊Dic39C22, C423S316C2, C422S332C2, D6.17(C4○D4), (C2×C6).100C24, D6⋊C4.85C22, (C2×Dic6)⋊6C22, C12.48D411C2, C2.19(D46D6), (C2×C12).699C23, Dic3⋊C442C22, (C22×C12)⋊37C22, C23.11D67C2, (C4×Dic3)⋊52C22, (C6×D4).307C22, C6.D49C22, C33(C22.45C24), C22.12(C4○D12), C23.16D629C2, C23.23D618C2, C23.28D616C2, (C22×S3).35C23, (S3×C23).41C22, C22.125(S3×C23), C23.184(C22×S3), (C22×C6).170C23, (C2×Dic3).207C23, (C22×Dic3).98C22, C4⋊C4⋊S37C2, (C2×D6⋊C4)⋊22C2, (C4×C3⋊D4)⋊43C2, C2.23(S3×C4○D4), (S3×C22⋊C4)⋊29C2, (C3×C4⋊C4)⋊61C22, C2.49(C2×C4○D12), C6.140(C2×C4○D4), (C2×C6).16(C4○D4), (S3×C2×C4).201C22, (C3×C22⋊C4)⋊57C22, (C2×C4).284(C22×S3), (C2×C3⋊D4).16C22, SmallGroup(192,1115)

Series: Derived Chief Lower central Upper central

C1C2×C6 — C4218D6
C1C3C6C2×C6C22×S3S3×C23S3×C22⋊C4 — C4218D6
C3C2×C6 — C4218D6
C1C22C4×D4

Generators and relations for C4218D6
 G = < a,b,c,d | a4=b4=c6=d2=1, ab=ba, cac-1=dad=a-1b2, bc=cb, dbd=a2b, dcd=c-1 >

Subgroups: 664 in 248 conjugacy classes, 97 normal (91 characteristic)
C1, C2 [×3], C2 [×6], C3, C4 [×11], C22, C22 [×2], C22 [×16], S3 [×3], C6 [×3], C6 [×3], C2×C4 [×5], C2×C4 [×13], D4 [×5], Q8, C23 [×2], C23 [×7], Dic3 [×6], C12 [×5], D6 [×2], D6 [×9], C2×C6, C2×C6 [×2], C2×C6 [×5], C42, C42 [×2], C22⋊C4 [×2], C22⋊C4 [×12], C4⋊C4, C4⋊C4 [×7], C22×C4 [×2], C22×C4 [×3], C2×D4, C2×D4 [×2], C2×Q8, C24, Dic6, C4×S3 [×3], C2×Dic3 [×6], C2×Dic3 [×2], C3⋊D4 [×3], C2×C12 [×5], C2×C12 [×2], C3×D4 [×2], C22×S3 [×2], C22×S3 [×5], C22×C6 [×2], C2×C22⋊C4 [×2], C42⋊C2 [×2], C4×D4, C4×D4, C22≀C2, C22⋊Q8 [×2], C22.D4 [×3], C4.4D4, C422C2 [×2], C4×Dic3 [×2], Dic3⋊C4 [×6], C4⋊Dic3, D6⋊C4 [×8], C6.D4 [×4], C4×C12, C3×C22⋊C4 [×2], C3×C4⋊C4, C2×Dic6, S3×C2×C4 [×2], C22×Dic3, C2×C3⋊D4 [×2], C22×C12 [×2], C6×D4, S3×C23, C22.45C24, C422S3, C423S3, C23.16D6, S3×C22⋊C4, C23.9D6, C23.11D6, D6⋊Q8, C4⋊C4⋊S3, C12.48D4, C2×D6⋊C4, C4×C3⋊D4, C23.28D6, C23.23D6, C232D6, D4×C12, C4218D6
Quotients: C1, C2 [×15], C22 [×35], S3, C23 [×15], D6 [×7], C4○D4 [×4], C24, C22×S3 [×7], C2×C4○D4 [×2], 2+ 1+4, C4○D12 [×2], S3×C23, C22.45C24, C2×C4○D12, D46D6, S3×C4○D4, C4218D6

Smallest permutation representation of C4218D6
On 48 points
Generators in S48
(1 41 10 34)(2 39 11 32)(3 37 12 36)(4 31 7 38)(5 35 8 42)(6 33 9 40)(13 28 22 43)(14 26 23 47)(15 30 24 45)(16 46 19 25)(17 44 20 29)(18 48 21 27)
(1 16 4 13)(2 17 5 14)(3 18 6 15)(7 22 10 19)(8 23 11 20)(9 24 12 21)(25 38 43 34)(26 39 44 35)(27 40 45 36)(28 41 46 31)(29 42 47 32)(30 37 48 33)
(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 3)(4 6)(7 9)(10 12)(13 24)(14 23)(15 22)(16 21)(17 20)(18 19)(25 45)(26 44)(27 43)(28 48)(29 47)(30 46)(31 36)(32 35)(33 34)(37 38)(39 42)(40 41)

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

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

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

45 conjugacy classes

class 1 2A2B2C2D2E2F2G2H2I 3 4A···4F4G4H4I4J4K···4O6A6B6C6D6E6F6G12A12B12C12D12E···12L
order122222222234···444444···466666661212121212···12
size1111224661222···2446612···12222444422224···4

45 irreducible representations

dim1111111111111111222222222444
type+++++++++++++++++++++++
imageC1C2C2C2C2C2C2C2C2C2C2C2C2C2C2C2S3D6D6D6D6D6C4○D4C4○D4C4○D122+ 1+4D46D6S3×C4○D4
kernelC4218D6C422S3C423S3C23.16D6S3×C22⋊C4C23.9D6C23.11D6D6⋊Q8C4⋊C4⋊S3C12.48D4C2×D6⋊C4C4×C3⋊D4C23.28D6C23.23D6C232D6D4×C12C4×D4C42C22⋊C4C4⋊C4C22×C4C2×D4D6C2×C6C22C6C2C2
# reps1111111111111111112121448122

Matrix representation of C4218D6 in GL6(𝔽13)

100000
010000
0001200
0012000
000080
000008
,
100000
010000
005000
000500
000052
000008
,
0120000
1120000
001000
0001200
000010
000001
,
1120000
0120000
0012000
000100
000010
0000812

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

C4218D6 in GAP, Magma, Sage, TeX

C_4^2\rtimes_{18}D_6
% in TeX

G:=Group("C4^2:18D6");
// GroupNames label

G:=SmallGroup(192,1115);
// by ID

G=gap.SmallGroup(192,1115);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,219,184,1571,136,6278]);
// Polycyclic

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

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