Copied to
clipboard

G = A4⋊D12order 288 = 25·32

The semidirect product of A4 and D12 acting via D12/D6=C2

non-abelian, soluble, monomial

Aliases: D62S4, A42D12, A4⋊C4⋊S3, (C3×A4)⋊3D4, C23.7S32, C2.15(S3×S4), C6.15(C2×S4), (C2×A4).7D6, (S3×C23)⋊2S3, C31(A4⋊D4), (C22×C6).7D6, (C6×A4).7C22, C222(C3⋊D12), (C2×S3×A4)⋊2C2, (C2×C3⋊S4)⋊2C2, (C3×A4⋊C4)⋊1C2, (C2×C6)⋊2(C3⋊D4), SmallGroup(288,858)

Series: Derived Chief Lower central Upper central

C1C22C6×A4 — A4⋊D12
C1C22C2×C6C3×A4C6×A4C2×S3×A4 — A4⋊D12
C3×A4C6×A4 — A4⋊D12
C1C2

Generators and relations for A4⋊D12
 G = < a,b,c,d,e | a2=b2=c3=d12=e2=1, cac-1=dad-1=eae=ab=ba, cbc-1=a, bd=db, be=eb, dcd-1=ece=c-1, ede=d-1 >

Subgroups: 942 in 134 conjugacy classes, 19 normal (all characteristic)
C1, C2, C2 [×5], C3, C3 [×2], C4 [×3], C22, C22 [×12], S3 [×6], C6, C6 [×5], C2×C4 [×3], D4 [×6], C23, C23 [×5], C32, Dic3 [×2], C12 [×2], A4, A4, D6, D6 [×11], C2×C6, C2×C6 [×3], C22⋊C4 [×3], C2×D4 [×3], C24, C3×S3, C3⋊S3, C3×C6, D12 [×4], C2×Dic3, C3⋊D4 [×3], C2×C12 [×2], S4 [×3], C2×A4, C2×A4 [×2], C22×S3 [×5], C22×C6, C22≀C2, C3×Dic3, C3×A4, S3×C6, C2×C3⋊S3, D6⋊C4 [×2], C3×C22⋊C4, A4⋊C4, C2×D12 [×2], C2×C3⋊D4, C2×S4 [×2], C22×A4, S3×C23, C3⋊D12, C3⋊S4, S3×A4, C6×A4, D6⋊D4, A4⋊D4, C3×A4⋊C4, C2×C3⋊S4, C2×S3×A4, A4⋊D12
Quotients: C1, C2 [×3], C22, S3 [×2], D4, D6 [×2], D12, C3⋊D4, S4, S32, C2×S4, C3⋊D12, A4⋊D4, S3×S4, A4⋊D12

Character table of A4⋊D12

 class 12A2B2C2D2E2F3A3B3C4A4B4C6A6B6C6D6E6F6G12A12B12C12D
 size 1133618362816121236266816242412121212
ρ1111111111111111111111111    trivial
ρ21111-1-1-111111-111111-1-11111    linear of order 2
ρ3111111-1111-1-1-11111111-1-1-1-1    linear of order 2
ρ41111-1-11111-1-1111111-1-1-1-1-1-1    linear of order 2
ρ52222000-12-1220-1-1-12-100-1-1-1-1    orthogonal lifted from S3
ρ62222000-12-1-2-20-1-1-12-1001111    orthogonal lifted from D6
ρ72-2-22000222000-2-22-2-2000000    orthogonal lifted from D4
ρ822222202-1-1000222-1-1-1-10000    orthogonal lifted from S3
ρ92222-2-202-1-1000222-1-1110000    orthogonal lifted from D6
ρ102-2-22000-12-100011-1-2100-3-333    orthogonal lifted from D12
ρ112-2-22000-12-100011-1-210033-3-3    orthogonal lifted from D12
ρ122-2-220002-1-1000-2-2211--3-30000    complex lifted from C3⋊D4
ρ132-2-220002-1-1000-2-2211-3--30000    complex lifted from C3⋊D4
ρ1433-1-1-31-13001-113-1-100001-11-1    orthogonal lifted from C2×S4
ρ1533-1-13-113001-1-13-1-100001-11-1    orthogonal lifted from S4
ρ1633-1-1-311300-11-13-1-10000-11-11    orthogonal lifted from C2×S4
ρ1733-1-13-1-1300-1113-1-10000-11-11    orthogonal lifted from S4
ρ184444000-2-21000-2-2-2-21000000    orthogonal lifted from S32
ρ194-4-44000-2-2100022-22-1000000    orthogonal lifted from C3⋊D12
ρ2066-2-2000-3002-20-3110000-11-11    orthogonal lifted from S3×S4
ρ2166-2-2000-300-220-31100001-11-1    orthogonal lifted from S3×S4
ρ226-62-2000600000-62-200000000    orthogonal lifted from A4⋊D4
ρ236-62-2000-3000003-110000-333-3    orthogonal faithful
ρ246-62-2000-3000003-1100003-3-33    orthogonal faithful

Smallest permutation representation of A4⋊D12
On 36 points
Generators in S36
(1 7)(2 8)(3 9)(4 10)(5 11)(6 12)(14 20)(16 22)(18 24)(26 32)(28 34)(30 36)
(13 19)(14 20)(15 21)(16 22)(17 23)(18 24)(25 31)(26 32)(27 33)(28 34)(29 35)(30 36)
(1 22 35)(2 36 23)(3 24 25)(4 26 13)(5 14 27)(6 28 15)(7 16 29)(8 30 17)(9 18 31)(10 32 19)(11 20 33)(12 34 21)
(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)
(1 3)(4 12)(5 11)(6 10)(7 9)(13 34)(14 33)(15 32)(16 31)(17 30)(18 29)(19 28)(20 27)(21 26)(22 25)(23 36)(24 35)

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

G:=Group( (1,7)(2,8)(3,9)(4,10)(5,11)(6,12)(14,20)(16,22)(18,24)(26,32)(28,34)(30,36), (13,19)(14,20)(15,21)(16,22)(17,23)(18,24)(25,31)(26,32)(27,33)(28,34)(29,35)(30,36), (1,22,35)(2,36,23)(3,24,25)(4,26,13)(5,14,27)(6,28,15)(7,16,29)(8,30,17)(9,18,31)(10,32,19)(11,20,33)(12,34,21), (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), (1,3)(4,12)(5,11)(6,10)(7,9)(13,34)(14,33)(15,32)(16,31)(17,30)(18,29)(19,28)(20,27)(21,26)(22,25)(23,36)(24,35) );

G=PermutationGroup([(1,7),(2,8),(3,9),(4,10),(5,11),(6,12),(14,20),(16,22),(18,24),(26,32),(28,34),(30,36)], [(13,19),(14,20),(15,21),(16,22),(17,23),(18,24),(25,31),(26,32),(27,33),(28,34),(29,35),(30,36)], [(1,22,35),(2,36,23),(3,24,25),(4,26,13),(5,14,27),(6,28,15),(7,16,29),(8,30,17),(9,18,31),(10,32,19),(11,20,33),(12,34,21)], [(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)], [(1,3),(4,12),(5,11),(6,10),(7,9),(13,34),(14,33),(15,32),(16,31),(17,30),(18,29),(19,28),(20,27),(21,26),(22,25),(23,36),(24,35)])

Matrix representation of A4⋊D12 in GL5(𝔽13)

10000
01000
000121
000120
001120
,
10000
01000
000112
001012
000012
,
10000
01000
00001
00100
00010
,
117000
611000
00010
00100
00001
,
711000
116000
000120
001200
000012

G:=sub<GL(5,GF(13))| [1,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,12,12,12,0,0,1,0,0],[1,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,1,0,0,0,0,12,12,12],[1,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,1,0,0],[11,6,0,0,0,7,11,0,0,0,0,0,0,1,0,0,0,1,0,0,0,0,0,0,1],[7,11,0,0,0,11,6,0,0,0,0,0,0,12,0,0,0,12,0,0,0,0,0,0,12] >;

A4⋊D12 in GAP, Magma, Sage, TeX

A_4\rtimes D_{12}
% in TeX

G:=Group("A4:D12");
// GroupNames label

G:=SmallGroup(288,858);
// by ID

G=gap.SmallGroup(288,858);
# by ID

G:=PCGroup([7,-2,-2,-2,-3,-3,-2,2,85,36,234,1684,3036,782,1777,1350]);
// Polycyclic

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

Export

Character table of A4⋊D12 in TeX

׿
×
𝔽