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G = Dic3⋊Dic3order 144 = 24·32

The semidirect product of Dic3 and Dic3 acting via Dic3/C6=C2

metabelian, supersoluble, monomial

Aliases: Dic3⋊Dic3, C6.18D12, C6.1Dic6, C62.4C22, C22.6S32, (C3×C6).2Q8, C6.20(C4×S3), C324(C4⋊C4), (C3×C6).15D4, (C2×C6).11D6, C31(C4⋊Dic3), (C3×Dic3)⋊1C4, C6.6(C3⋊D4), C6.5(C2×Dic3), C2.5(S3×Dic3), C31(Dic3⋊C4), (C6×Dic3).3C2, (C2×Dic3).1S3, C2.3(C3⋊D12), C2.1(C322Q8), (C3×C6).15(C2×C4), (C2×C3⋊Dic3).2C2, SmallGroup(144,66)

Series: Derived Chief Lower central Upper central

C1C3×C6 — Dic3⋊Dic3
C1C3C32C3×C6C62C6×Dic3 — Dic3⋊Dic3
C32C3×C6 — Dic3⋊Dic3
C1C22

Generators and relations for Dic3⋊Dic3
 G = < a,b,c,d | a6=c6=1, b2=a3, d2=c3, bab-1=a-1, ac=ca, ad=da, bc=cb, dbd-1=a3b, dcd-1=c-1 >

Subgroups: 160 in 60 conjugacy classes, 30 normal (26 characteristic)
C1, C2 [×3], C3 [×2], C3, C4 [×4], C22, C6 [×6], C6 [×3], C2×C4 [×3], C32, Dic3 [×2], Dic3 [×5], C12 [×3], C2×C6 [×2], C2×C6, C4⋊C4, C3×C6 [×3], C2×Dic3 [×2], C2×Dic3 [×3], C2×C12 [×2], C3×Dic3 [×2], C3×Dic3, C3⋊Dic3, C62, Dic3⋊C4, C4⋊Dic3, C6×Dic3 [×2], C2×C3⋊Dic3, Dic3⋊Dic3
Quotients: C1, C2 [×3], C4 [×2], C22, S3 [×2], C2×C4, D4, Q8, Dic3 [×2], D6 [×2], C4⋊C4, Dic6 [×2], C4×S3, D12, C2×Dic3, C3⋊D4, S32, Dic3⋊C4, C4⋊Dic3, S3×Dic3, C3⋊D12, C322Q8, Dic3⋊Dic3

Character table of Dic3⋊Dic3

 class 12A2B2C3A3B3C4A4B4C4D4E4F6A6B6C6D6E6F6G6H6I12A12B12C12D12E12F12G12H
 size 11112246666181822222244466666666
ρ1111111111111111111111111111111    trivial
ρ211111111-11-1-1-1111111111-111-111-1-1    linear of order 2
ρ31111111-11-11-1-11111111111-1-11-1-111    linear of order 2
ρ41111111-1-1-1-111111111111-1-1-1-1-1-1-1-1    linear of order 2
ρ51-11-1111-1i1-ii-i-1-11-1-11-11-1-i11-i-1-1ii    linear of order 4
ρ61-11-1111-1-i1i-ii-1-11-1-11-11-1i11i-1-1-i-i    linear of order 4
ρ71-11-11111i-1-i-ii-1-11-1-11-11-1-i-1-1-i11ii    linear of order 4
ρ81-11-11111-i-1ii-i-1-11-1-11-11-1i-1-1i11-i-i    linear of order 4
ρ92222-12-1020200-12-1-122-1-1-1-100-100-1-1    orthogonal lifted from S3
ρ1022-2-2222000000-2-2-222-2-2-2200000000    orthogonal lifted from D4
ρ1122222-1-1-20-20002-122-1-1-1-1-101101100    orthogonal lifted from D6
ρ1222222-1-12020002-122-1-1-1-1-10-1-10-1-100    orthogonal lifted from S3
ρ1322-2-22-1-1000000-21-22-1111-103-303-300    orthogonal lifted from D12
ρ142222-12-10-20-200-12-1-122-1-1-110010011    orthogonal lifted from D6
ρ1522-2-22-1-1000000-21-22-1111-10-330-3300    orthogonal lifted from D12
ρ162-22-22-1-1-202000-212-21-11-110-1-101100    symplectic lifted from Dic3, Schur index 2
ρ172-22-22-1-120-2000-212-21-11-110110-1-100    symplectic lifted from Dic3, Schur index 2
ρ182-2-2222200000022-2-2-2-22-2-200000000    symplectic lifted from Q8, Schur index 2
ρ192-2-22-12-1000000-1211-2-2-111-3003003-3    symplectic lifted from Dic6, Schur index 2
ρ202-2-222-1-10000002-1-2-211-11103-30-3300    symplectic lifted from Dic6, Schur index 2
ρ212-2-22-12-1000000-1211-2-2-111300-300-33    symplectic lifted from Dic6, Schur index 2
ρ222-2-222-1-10000002-1-2-211-1110-3303-300    symplectic lifted from Dic6, Schur index 2
ρ232-22-2-12-10-2i02i001-2-11-221-11-i00-i00ii    complex lifted from C4×S3
ρ242-22-2-12-102i0-2i001-2-11-221-11i00i00-i-i    complex lifted from C4×S3
ρ2522-2-2-12-10000001-21-12-211-1--300-300--3-3    complex lifted from C3⋊D4
ρ2622-2-2-12-10000001-21-12-211-1-300--300-3--3    complex lifted from C3⋊D4
ρ2744-4-4-2-21000000222-2-22-1-1100000000    orthogonal lifted from C3⋊D12
ρ284444-2-21000000-2-2-2-2-2-211100000000    orthogonal lifted from S32
ρ294-4-44-2-21000000-2-222221-1-100000000    symplectic lifted from C322Q8, Schur index 2
ρ304-44-4-2-2100000022-222-2-11-100000000    symplectic lifted from S3×Dic3, Schur index 2

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

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

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

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

Dic3⋊Dic3 is a maximal subgroup of
C62.6C23  Dic35Dic6  C62.9C23  C62.10C23  Dic3×Dic6  C62.13C23  Dic3.Dic6  C62.16C23  C62.17C23  C62.18C23  D6⋊Dic6  C62.23C23  C62.25C23  D67Dic6  C62.28C23  C62.29C23  C62.32C23  Dic3⋊Dic6  C62.37C23  C62.38C23  C62.39C23  C62.40C23  C12.30D12  C62.42C23  S3×Dic3⋊C4  C62.48C23  Dic34D12  C62.53C23  D61Dic6  C62.58C23  S3×C4⋊Dic3  D6.D12  D6.9D12  D62Dic6  C62.65C23  D63Dic6  C4×C3⋊D12  C62.75C23  D6⋊D12  C4×C322Q8  C123Dic6  C62.97C23  C62.98C23  C62.100C23  C62.57D4  C623Q8  C62.60D4  C62.111C23  C62.112C23  Dic3×C3⋊D4  C62.115C23  C62.117C23  C626D4  C624Q8  Dic9⋊Dic3  Dic3⋊Dic9  C62.D6  C62.3D6  C62.80D6  C62.82D6  C62.85D6
Dic3⋊Dic3 is a maximal quotient of
C12.81D12  C12.Dic6  C6.18D24  C12.82D12  C62.6Q8  Dic9⋊Dic3  Dic3⋊Dic9  C62.D6  C62.80D6  C62.82D6  C62.85D6

Matrix representation of Dic3⋊Dic3 in GL6(𝔽13)

1200000
0120000
0001200
0011200
000010
000001
,
010000
1200000
0001200
0012000
000010
000001
,
100000
010000
0012000
0001200
00001212
000010
,
010000
100000
008000
000800
000010
00001212

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

Dic3⋊Dic3 in GAP, Magma, Sage, TeX

{\rm Dic}_3\rtimes {\rm Dic}_3
% in TeX

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

G:=SmallGroup(144,66);
// by ID

G=gap.SmallGroup(144,66);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-3,-3,48,121,31,490,3461]);
// Polycyclic

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

Export

Character table of Dic3⋊Dic3 in TeX

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