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G = Dic6.9D6order 288 = 25·32

9th non-split extension by Dic6 of D6 acting via D6/C3=C22

metabelian, supersoluble, monomial

Aliases: Dic6.9D6, C3⋊S33Q16, C3⋊C8.16D6, C33(S3×Q16), Q8.14S32, C3⋊Q162S3, C6.63(S3×D4), C328(C2×Q16), (C3×Q8).31D6, C3⋊Dic3.23D4, (C3×C12).21C23, C12.21(C22×S3), C323Q1610C2, C2.23(Dic3⋊D6), Dic3.D6.4C2, C12.29D6.1C2, (Q8×C32).3C22, (C3×Dic6).17C22, C324Q8.12C22, C4.21(C2×S32), (Q8×C3⋊S3).2C2, (C2×C3⋊S3).60D4, (C3×C3⋊Q16)⋊2C2, (C3×C3⋊C8).7C22, (C3×C6).136(C2×D4), (C4×C3⋊S3).19C22, SmallGroup(288,592)

Series: Derived Chief Lower central Upper central

C1C3×C12 — Dic6.9D6
C1C3C32C3×C6C3×C12C3×Dic6Dic3.D6 — Dic6.9D6
C32C3×C6C3×C12 — Dic6.9D6
C1C2C4Q8

Generators and relations for Dic6.9D6
 G = < a,b,c,d | a12=1, b2=c6=a6, d2=a3, bab-1=a-1, cac-1=a7, ad=da, cbc-1=a9b, dbd-1=a3b, dcd-1=a3c5 >

Subgroups: 546 in 139 conjugacy classes, 40 normal (16 characteristic)
C1, C2, C2, C3, C3, C4, C4, C22, S3, C6, C6, C8, C2×C4, Q8, Q8, C32, Dic3, C12, C12, D6, C2×C8, Q16, C2×Q8, C3⋊S3, C3×C6, C3⋊C8, C24, Dic6, Dic6, C4×S3, C3×Q8, C3×Q8, C2×Q16, C3×Dic3, C3⋊Dic3, C3⋊Dic3, C3×C12, C3×C12, C2×C3⋊S3, S3×C8, Dic12, C3⋊Q16, C3⋊Q16, C3×Q16, S3×Q8, C3×C3⋊C8, C6.D6, C322Q8, C3×Dic6, C324Q8, C324Q8, C4×C3⋊S3, C4×C3⋊S3, Q8×C32, S3×Q16, C12.29D6, C323Q16, C3×C3⋊Q16, Dic3.D6, Q8×C3⋊S3, Dic6.9D6
Quotients: C1, C2, C22, S3, D4, C23, D6, Q16, C2×D4, C22×S3, C2×Q16, S32, S3×D4, C2×S32, S3×Q16, Dic3⋊D6, Dic6.9D6

Smallest permutation representation of Dic6.9D6
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 31 7 25)(2 30 8 36)(3 29 9 35)(4 28 10 34)(5 27 11 33)(6 26 12 32)(13 40 19 46)(14 39 20 45)(15 38 21 44)(16 37 22 43)(17 48 23 42)(18 47 24 41)
(1 17 3 19 5 21 7 23 9 13 11 15)(2 24 4 14 6 16 8 18 10 20 12 22)(25 45 35 43 33 41 31 39 29 37 27 47)(26 40 36 38 34 48 32 46 30 44 28 42)
(1 41 4 44 7 47 10 38)(2 42 5 45 8 48 11 39)(3 43 6 46 9 37 12 40)(13 26 16 29 19 32 22 35)(14 27 17 30 20 33 23 36)(15 28 18 31 21 34 24 25)

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

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

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

33 conjugacy classes

class 1 2A2B2C3A3B3C4A4B4C4D4E4F6A6B6C8A8B8C8D12A12B12C···12G12H12I24A24B24C24D
order12223334444446668888121212···12121224242424
size119922424121218362246666448···8242412121212

33 irreducible representations

dim1111112222222444448
type++++++++++++-+++-+-
imageC1C2C2C2C2C2S3D4D4D6D6D6Q16S32S3×D4C2×S32S3×Q16Dic3⋊D6Dic6.9D6
kernelDic6.9D6C12.29D6C323Q16C3×C3⋊Q16Dic3.D6Q8×C3⋊S3C3⋊Q16C3⋊Dic3C2×C3⋊S3C3⋊C8Dic6C3×Q8C3⋊S3Q8C6C4C3C2C1
# reps1122112112224121421

Matrix representation of Dic6.9D6 in GL6(𝔽73)

100000
010000
000100
0072000
0000072
0000172
,
100000
010000
00306200
00624300
000001
000010
,
7210000
7200000
0016100
00617200
000010
000001
,
7200000
7210000
00165700
00161600
000010
000001

G:=sub<GL(6,GF(73))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,72,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,72,72],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,30,62,0,0,0,0,62,43,0,0,0,0,0,0,0,1,0,0,0,0,1,0],[72,72,0,0,0,0,1,0,0,0,0,0,0,0,1,61,0,0,0,0,61,72,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[72,72,0,0,0,0,0,1,0,0,0,0,0,0,16,16,0,0,0,0,57,16,0,0,0,0,0,0,1,0,0,0,0,0,0,1] >;

Dic6.9D6 in GAP, Magma, Sage, TeX

{\rm Dic}_6._9D_6
% in TeX

G:=Group("Dic6.9D6");
// GroupNames label

G:=SmallGroup(288,592);
// by ID

G=gap.SmallGroup(288,592);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,112,120,254,135,100,675,346,185,80,1356,9414]);
// Polycyclic

G:=Group<a,b,c,d|a^12=1,b^2=c^6=a^6,d^2=a^3,b*a*b^-1=a^-1,c*a*c^-1=a^7,a*d=d*a,c*b*c^-1=a^9*b,d*b*d^-1=a^3*b,d*c*d^-1=a^3*c^5>;
// generators/relations

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