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G = D6.Dic3order 144 = 24·32

The non-split extension by D6 of Dic3 acting via Dic3/C6=C2

metabelian, supersoluble, monomial

Aliases: D6.Dic3, C12.30D6, Dic3.Dic3, C322M4(2), C3⋊C84S3, C4.15S32, (C4×S3).2S3, (S3×C6).2C4, C6.17(C4×S3), C33(C8⋊S3), (S3×C12).1C2, C324C87C2, C2.3(S3×Dic3), C6.2(C2×Dic3), C31(C4.Dic3), (C3×Dic3).1C4, (C3×C12).29C22, (C3×C3⋊C8)⋊7C2, (C3×C6).10(C2×C4), SmallGroup(144,54)

Series: Derived Chief Lower central Upper central

C1C3×C6 — D6.Dic3
C1C3C32C3×C6C3×C12S3×C12 — D6.Dic3
C32C3×C6 — D6.Dic3
C1C4

Generators and relations for D6.Dic3
 G = < a,b,c,d | a6=b2=1, c6=a3, d2=a3c3, bab=a-1, ac=ca, ad=da, bc=cb, dbd-1=a3b, dcd-1=c5 >

6C2
2C3
3C22
3C4
2S3
2C6
6C6
3C8
3C2×C4
9C8
2C12
3C12
3C2×C6
2C3×S3
9M4(2)
3C24
3C3⋊C8
3C2×C12
3C3⋊C8
6C3⋊C8
3C8⋊S3
3C4.Dic3

Character table of D6.Dic3

 class 12A2B3A3B3C4A4B4C6A6B6C6D6E8A8B8C8D12A12B12C12D12E12F12G12H24A24B24C24D
 size 11622411622466661818222244666666
ρ1111111111111111111111111111111    trivial
ρ211111111111111-1-1-1-111111111-1-1-1-1    linear of order 2
ρ311-111111-1111-1-1-1-111111111-1-1-1-1-1-1    linear of order 2
ρ411-111111-1111-1-111-1-1111111-1-11111    linear of order 2
ρ511-1111-1-11111-1-1-iii-i-1-1-1-1-1-111i-i-ii    linear of order 4
ρ611-1111-1-11111-1-1i-i-ii-1-1-1-1-1-111-iii-i    linear of order 4
ρ7111111-1-1-111111i-ii-i-1-1-1-1-1-1-1-1-iii-i    linear of order 4
ρ8111111-1-1-111111-ii-ii-1-1-1-1-1-1-1-1i-i-ii    linear of order 4
ρ922-22-1-122-22-1-1110000-122-1-1-1110000    orthogonal lifted from D6
ρ10220-12-1220-12-10022002-1-12-1-100-1-1-1-1    orthogonal lifted from S3
ρ11220-12-1220-12-100-2-2002-1-12-1-1001111    orthogonal lifted from D6
ρ122222-1-12222-1-1-1-10000-122-1-1-1-1-10000    orthogonal lifted from S3
ρ1322-22-1-1-2-222-1-11100001-2-2111-1-10000    symplectic lifted from Dic3, Schur index 2
ρ142222-1-1-2-2-22-1-1-1-100001-2-2111110000    symplectic lifted from Dic3, Schur index 2
ρ15220-12-1-2-20-12-100-2i2i00-211-21100-iii-i    complex lifted from C4×S3
ρ16220-12-1-2-20-12-1002i-2i00-211-21100i-i-ii    complex lifted from C4×S3
ρ172-202222i-2i0-2-2-20000002i2i-2i-2i2i-2i000000    complex lifted from M4(2)
ρ182-20222-2i2i0-2-2-2000000-2i-2i2i2i-2i2i000000    complex lifted from M4(2)
ρ192-202-1-12i-2i0-211-3--30000-i2i-2ii-ii-330000    complex lifted from C4.Dic3
ρ202-202-1-1-2i2i0-211--3-30000i-2i2i-ii-i-330000    complex lifted from C4.Dic3
ρ212-202-1-1-2i2i0-211-3--30000i-2i2i-ii-i3-30000    complex lifted from C4.Dic3
ρ222-202-1-12i-2i0-211--3-30000-i2i-2ii-ii3-30000    complex lifted from C4.Dic3
ρ232-20-12-12i-2i01-210000002i-ii-2i-ii008ζ3883ζ38387ζ38785ζ385    complex lifted from C8⋊S3
ρ242-20-12-1-2i2i01-21000000-2ii-i2ii-i0083ζ3838ζ3885ζ38587ζ387    complex lifted from C8⋊S3
ρ252-20-12-12i-2i01-210000002i-ii-2i-ii0085ζ38587ζ38783ζ3838ζ38    complex lifted from C8⋊S3
ρ262-20-12-1-2i2i01-21000000-2ii-i2ii-i0087ζ38785ζ3858ζ3883ζ383    complex lifted from C8⋊S3
ρ27440-2-21440-2-21000000-2-2-2-211000000    orthogonal lifted from S32
ρ28440-2-21-4-40-2-210000002222-1-1000000    symplectic lifted from S3×Dic3, Schur index 2
ρ294-40-2-214i-4i022-1000000-2i-2i2i2ii-i000000    complex faithful
ρ304-40-2-21-4i4i022-10000002i2i-2i-2i-ii000000    complex faithful

Smallest permutation representation of D6.Dic3
On 48 points
Generators in S48
(1 11 9 7 5 3)(2 12 10 8 6 4)(13 15 17 19 21 23)(14 16 18 20 22 24)(25 27 29 31 33 35)(26 28 30 32 34 36)(37 47 45 43 41 39)(38 48 46 44 42 40)
(1 32)(2 33)(3 34)(4 35)(5 36)(6 25)(7 26)(8 27)(9 28)(10 29)(11 30)(12 31)(13 39)(14 40)(15 41)(16 42)(17 43)(18 44)(19 45)(20 46)(21 47)(22 48)(23 37)(24 38)
(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 22 10 19 7 16 4 13)(2 15 11 24 8 21 5 18)(3 20 12 17 9 14 6 23)(25 43 34 40 31 37 28 46)(26 48 35 45 32 42 29 39)(27 41 36 38 33 47 30 44)

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

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

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

D6.Dic3 is a maximal subgroup of
S3×C8⋊S3  C24⋊D6  C24.63D6  C24.64D6  S3×C4.Dic3  D12.2Dic3  D12.Dic3  Dic63D6  Dic6.19D6  D129D6  D12.7D6  D126D6  D12.11D6  D12.24D6  Dic6.22D6  C36.39D6  D6.Dic9  He3⋊M4(2)  He33M4(2)  C337M4(2)  C338M4(2)  C3310M4(2)
D6.Dic3 is a maximal quotient of
C3⋊C8⋊Dic3  C12.77D12  C12.81D12  C36.39D6  D6.Dic9  He3⋊M4(2)  C337M4(2)  C338M4(2)  C3310M4(2)

Matrix representation of D6.Dic3 in GL4(𝔽5) generated by

4020
0402
1020
0102
,
0104
4010
0204
3010
,
2010
0104
3010
0202
,
0101
0040
0300
2010
G:=sub<GL(4,GF(5))| [4,0,1,0,0,4,0,1,2,0,2,0,0,2,0,2],[0,4,0,3,1,0,2,0,0,1,0,1,4,0,4,0],[2,0,3,0,0,1,0,2,1,0,1,0,0,4,0,2],[0,0,0,2,1,0,3,0,0,4,0,1,1,0,0,0] >;

D6.Dic3 in GAP, Magma, Sage, TeX

D_6.{\rm Dic}_3
% in TeX

G:=Group("D6.Dic3");
// GroupNames label

G:=SmallGroup(144,54);
// by ID

G=gap.SmallGroup(144,54);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-3,-3,121,31,50,490,3461]);
// Polycyclic

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

Export

Subgroup lattice of D6.Dic3 in TeX
Character table of D6.Dic3 in TeX

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