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G = D6⋊Dic3order 144 = 24·32

The semidirect product of D6 and Dic3 acting via Dic3/C6=C2

metabelian, supersoluble, monomial

Aliases: D6⋊Dic3, C6.16D12, C62.2C22, (S3×C6)⋊1C4, C22.4S32, (C2×C6).9D6, C33(D6⋊C4), C6.19(C4×S3), (C3×C6).13D4, (C22×S3).S3, (C2×Dic3)⋊1S3, (C6×Dic3)⋊1C2, C6.4(C2×Dic3), C2.4(S3×Dic3), C6.10(C3⋊D4), C323(C22⋊C4), C2.1(D6⋊S3), C2.1(C3⋊D12), C31(C6.D4), (S3×C2×C6).1C2, (C3×C6).13(C2×C4), (C2×C3⋊Dic3)⋊1C2, SmallGroup(144,64)

Series: Derived Chief Lower central Upper central

C1C3×C6 — D6⋊Dic3
C1C3C32C3×C6C62S3×C2×C6 — D6⋊Dic3
C32C3×C6 — D6⋊Dic3
C1C22

Generators and relations for D6⋊Dic3
 G = < a,b,c,d | a6=b2=c6=1, d2=c3, bab=a-1, ac=ca, ad=da, bc=cb, dbd-1=a3b, dcd-1=c-1 >

Subgroups: 224 in 76 conjugacy classes, 30 normal (26 characteristic)
C1, C2 [×3], C2 [×2], C3 [×2], C3, C4 [×2], C22, C22 [×4], S3 [×2], C6 [×6], C6 [×5], C2×C4 [×2], C23, C32, Dic3 [×5], C12, D6 [×2], D6 [×2], C2×C6 [×2], C2×C6 [×5], C22⋊C4, C3×S3 [×2], C3×C6 [×3], C2×Dic3, C2×Dic3 [×3], C2×C12, C22×S3, C22×C6, C3×Dic3, C3⋊Dic3, S3×C6 [×2], S3×C6 [×2], C62, D6⋊C4, C6.D4, C6×Dic3, C2×C3⋊Dic3, S3×C2×C6, D6⋊Dic3
Quotients: C1, C2 [×3], C4 [×2], C22, S3 [×2], C2×C4, D4 [×2], Dic3 [×2], D6 [×2], C22⋊C4, C4×S3, D12, C2×Dic3, C3⋊D4 [×3], S32, D6⋊C4, C6.D4, S3×Dic3, D6⋊S3, C3⋊D12, D6⋊Dic3

Character table of D6⋊Dic3

 class 12A2B2C2D2E3A3B3C4A4B4C4D6A6B6C6D6E6F6G6H6I6J6K6L6M12A12B12C12D
 size 11116622466181822222244466666666
ρ1111111111111111111111111111111    trivial
ρ2111111111-1-1-1-11111111111111-1-1-1-1    linear of order 2
ρ31111-1-111111-1-1111111111-1-1-1-11111    linear of order 2
ρ41111-1-1111-1-111111111111-1-1-1-1-1-1-1-1    linear of order 2
ρ51-11-1-11111i-ii-i-1-11-1-11-11-11-1-11ii-i-i    linear of order 4
ρ61-11-1-11111-ii-ii-1-11-1-11-11-11-1-11-i-iii    linear of order 4
ρ71-11-11-1111i-i-ii-1-11-1-11-11-1-111-1ii-i-i    linear of order 4
ρ81-11-11-1111-iii-i-1-11-1-11-11-1-111-1-i-iii    linear of order 4
ρ9222200-12-12200-12-1-122-1-1-10000-1-1-1-1    orthogonal lifted from S3
ρ102222-2-22-1-100002-122-1-1-1-1-111110000    orthogonal lifted from D6
ρ112222222-1-100002-122-1-1-1-1-1-1-1-1-10000    orthogonal lifted from S3
ρ122-2-2200222000022-2-2-2-2-2-2200000000    orthogonal lifted from D4
ρ1322-2-2002220000-2-2-222-22-2-200000000    orthogonal lifted from D4
ρ14222200-12-1-2-200-12-1-122-1-1-100001111    orthogonal lifted from D6
ρ152-2-2200-12-10000-1211-2-211-10000-33-33    orthogonal lifted from D12
ρ162-2-2200-12-10000-1211-2-211-100003-33-3    orthogonal lifted from D12
ρ172-22-2-222-1-10000-212-21-11-11-111-10000    symplectic lifted from Dic3, Schur index 2
ρ182-22-22-22-1-10000-212-21-11-111-1-110000    symplectic lifted from Dic3, Schur index 2
ρ192-2-22002-1-100002-1-2-21111-1-3-3--3--30000    complex lifted from C3⋊D4
ρ2022-2-2002-1-10000-21-22-11-111--3-3--3-30000    complex lifted from C3⋊D4
ρ2122-2-200-12-100001-21-12-2-1110000-3--3--3-3    complex lifted from C3⋊D4
ρ2222-2-2002-1-10000-21-22-11-111-3--3-3--30000    complex lifted from C3⋊D4
ρ232-22-200-12-1-2i2i001-2-11-221-110000ii-i-i    complex lifted from C4×S3
ρ242-22-200-12-12i-2i001-2-11-221-110000-i-iii    complex lifted from C4×S3
ρ252-2-22002-1-100002-1-2-21111-1--3--3-3-30000    complex lifted from C3⋊D4
ρ2622-2-200-12-100001-21-12-2-1110000--3-3-3--3    complex lifted from C3⋊D4
ρ274-4-4400-2-210000-2-22222-1-1100000000    orthogonal lifted from C3⋊D12
ρ28444400-2-210000-2-2-2-2-2-211100000000    orthogonal lifted from S32
ρ2944-4-400-2-210000222-2-221-1-100000000    symplectic lifted from D6⋊S3, Schur index 2
ρ304-44-400-2-21000022-222-2-11-100000000    symplectic lifted from S3×Dic3, Schur index 2

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

D6⋊Dic3 is a maximal subgroup of
C62.11C23  C62.20C23  D6⋊Dic6  Dic3.D12  C62.23C23  C62.24C23  C62.25C23  D66Dic6  D67Dic6  C62.28C23  C62.29C23  C12.27D12  C62.31C23  C62.32C23  C62.33C23  C62.47C23  C62.48C23  C62.49C23  C62.54C23  C62.55C23  Dic3⋊D12  D61Dic6  D6.9D12  Dic3×D12  D62Dic6  D63Dic6  D12⋊Dic3  D64Dic6  C4×D6⋊S3  C4×C3⋊D12  C62.74C23  C62.75C23  C62.77C23  D62D12  C62.83C23  C62.85C23  C122D12  S3×D6⋊C4  C62.91C23  D64D12  C62.100C23  C62.101C23  C62.56D4  C62.57D4  S3×C6.D4  C62.111C23  C62.112C23  Dic3×C3⋊D4  C62.115C23  C624D4  C625D4  C626D4  C627D4  C62.125C23  D18⋊Dic3  D6⋊Dic9  C62.4D6  C62.5D6  C62.77D6  C62.78D6  C62.84D6
D6⋊Dic3 is a maximal quotient of
C12.77D12  C12.D12  C12.14D12  D123Dic3  C6.16D24  Dic6⋊Dic3  C6.Dic12  D124Dic3  D122Dic3  C62.6Q8  C62.31D4  D18⋊Dic3  D6⋊Dic9  C62.4D6  C62.77D6  C62.78D6  C62.84D6

Matrix representation of D6⋊Dic3 in GL6(𝔽13)

1200000
0120000
0012100
0012000
000010
000001
,
010000
100000
0012000
0012100
000010
000001
,
1200000
0120000
0012000
0001200
0000012
0000112
,
010000
1200000
005000
000500
000001
000010

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

D6⋊Dic3 in GAP, Magma, Sage, TeX

D_6\rtimes {\rm Dic}_3
% in TeX

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

G:=SmallGroup(144,64);
// by ID

G=gap.SmallGroup(144,64);
# by ID

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

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

Export

Character table of D6⋊Dic3 in TeX

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