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G = C32⋊D6⋊C4order 432 = 24·33

The semidirect product of C32⋊D6 and C4 acting via C4/C2=C2

non-abelian, soluble

Aliases: C32⋊D6⋊C4, C6.5S3≀C2, He3⋊(C22⋊C4), C2.2(He3⋊D4), (C2×He3).5D4, He3⋊C2.3D4, C3.(S32⋊C4), (C2×He3⋊C4)⋊1C2, He3⋊(C2×C4)⋊4C2, (C2×C32⋊D6).C2, He3⋊C2.4(C2×C4), (C2×He3⋊C2).2C22, SmallGroup(432,238)

Series: Derived Chief Lower central Upper central

C1C3He3He3⋊C2 — C32⋊D6⋊C4
C1C3He3He3⋊C2C2×He3⋊C2C2×C32⋊D6 — C32⋊D6⋊C4
He3He3⋊C2 — C32⋊D6⋊C4
C1C2

Generators and relations for C32⋊D6⋊C4
 G = < a,b,c,d,e | a3=b3=c6=d2=e4=1, ab=ba, cac-1=dad=a-1b-1, eae-1=c4, cbc-1=ebe-1=b-1, bd=db, dcd=c-1, ece-1=ac-1d, ede-1=ac2d >

Subgroups: 907 in 99 conjugacy classes, 15 normal (13 characteristic)
C1, C2, C2 [×4], C3, C3 [×2], C4 [×2], C22 [×5], S3 [×8], C6, C6 [×6], C2×C4 [×2], C23, C32 [×2], Dic3 [×2], C12 [×2], D6 [×11], C2×C6 [×2], C22⋊C4, C3×S3 [×6], C3⋊S3 [×2], C3×C6 [×2], C4×S3, C2×Dic3, C2×C12, C22×S3 [×2], He3, C3×Dic3, C3⋊Dic3, S32 [×4], S3×C6 [×3], C2×C3⋊S3, D6⋊C4, C32⋊C6 [×2], He3⋊C2 [×2], C2×He3, S3×Dic3, C2×S32, C32⋊C12, He3⋊C4, C32⋊D6 [×2], C32⋊D6, C2×C32⋊C6, C2×He3⋊C2, He3⋊(C2×C4), C2×He3⋊C4, C2×C32⋊D6, C32⋊D6⋊C4
Quotients: C1, C2 [×3], C4 [×2], C22, C2×C4, D4 [×2], C22⋊C4, S3≀C2, S32⋊C4, He3⋊D4, C32⋊D6⋊C4

Character table of C32⋊D6⋊C4

 class 12A2B2C2D2E3A3B3C4A4B4C4D6A6B6C6D6E6F6G12A12B12C12D12E12F
 size 1199181821212181818182121218183636181818183636
ρ111111111111111111111111111    trivial
ρ21111-1-1111-11-1111111-1-1-1-1-1-111    linear of order 2
ρ31111-1-11111-11-111111-1-11111-1-1    linear of order 2
ρ4111111111-1-1-1-11111111-1-1-1-1-1-1    linear of order 2
ρ51-1-111-1111-i-iii-1-1-11-1-11-ii-iii-i    linear of order 4
ρ61-1-11-11111i-i-ii-1-1-11-11-1i-ii-ii-i    linear of order 4
ρ71-1-11-11111-iii-i-1-1-11-11-1-ii-ii-ii    linear of order 4
ρ81-1-111-1111ii-i-i-1-1-11-1-11i-ii-i-ii    linear of order 4
ρ92-22-2002220000-2-2-2-2200000000    orthogonal lifted from D4
ρ1022-2-2002220000222-2-200000000    orthogonal lifted from D4
ρ114400004-21020241-200000000-1-1    orthogonal lifted from S3≀C2
ρ124-400-2241-20000-42-100-11000000    orthogonal lifted from S32⋊C4
ρ1344002241-200004-2100-1-1000000    orthogonal lifted from S3≀C2
ρ144-4002-241-20000-42-1001-1000000    orthogonal lifted from S32⋊C4
ρ154400004-210-20-241-20000000011    orthogonal lifted from S3≀C2
ρ164400-2-241-200004-210011000000    orthogonal lifted from S3≀C2
ρ174-400004-2102i0-2i-4-1200000000i-i    complex lifted from S32⋊C4
ρ184-400004-210-2i02i-4-1200000000-ii    complex lifted from S32⋊C4
ρ1966-2-200-3002020-3001100-1-1-1-100    orthogonal lifted from He3⋊D4
ρ2066-2-200-300-20-20-3001100111100    orthogonal lifted from He3⋊D4
ρ21662200-3000000-300-1-1003-3-3300    orthogonal lifted from He3⋊D4
ρ22662200-3000000-300-1-100-333-300    orthogonal lifted from He3⋊D4
ρ236-62-200-300-2i02i03001-100i-ii-i00    complex faithful
ρ246-62-200-3002i0-2i03001-100-ii-ii00    complex faithful
ρ256-6-2200-3000000300-1100-3-3--3--300    complex faithful
ρ266-6-2200-3000000300-1100--3--3-3-300    complex faithful

Smallest permutation representation of C32⋊D6⋊C4
On 36 points
Generators in S36
(1 36 33)(2 13 16)(3 32 31)(4 34 35)(5 17 18)(6 15 14)(7 25 28)(8 26 29)(10 23 20)(11 24 21)
(1 3 4)(2 6 5)(7 25 28)(8 29 26)(9 27 30)(10 23 20)(11 21 24)(12 19 22)(13 15 17)(14 18 16)(31 35 33)(32 34 36)
(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)
(8 9)(11 12)(13 18)(14 17)(15 16)(19 21)(22 24)(26 30)(27 29)(31 34)(32 33)(35 36)
(1 12 2 9)(3 22 6 30)(4 19 5 27)(7 36 10 13)(8 33 11 16)(14 26 31 24)(15 28 32 20)(17 25 34 23)(18 29 35 21)

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

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

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

Matrix representation of C32⋊D6⋊C4 in GL6(𝔽13)

0000121
11001112
0000120
1000120
0010120
0001120
,
0120000
1120000
10121200
0121000
10001212
0120010
,
010000
100000
11001212
000001
000100
001000
,
100000
010000
11001212
000010
000100
11121200
,
46911911
20112112
22911112
0211200
22112911
0200112

G:=sub<GL(6,GF(13))| [0,1,0,1,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,12,11,12,12,12,12,1,12,0,0,0,0],[0,1,1,0,1,0,12,12,0,12,0,12,0,0,12,1,0,0,0,0,12,0,0,0,0,0,0,0,12,1,0,0,0,0,12,0],[0,1,1,0,0,0,1,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,12,0,0,0,0,0,12,1,0,0],[1,0,1,0,0,1,0,1,1,0,0,1,0,0,0,0,0,12,0,0,0,0,1,12,0,0,12,1,0,0,0,0,12,0,0,0],[4,2,2,0,2,0,6,0,2,2,2,2,9,11,9,11,11,0,11,2,11,2,2,0,9,11,11,0,9,11,11,2,2,0,11,2] >;

C32⋊D6⋊C4 in GAP, Magma, Sage, TeX

C_3^2\rtimes D_6\rtimes C_4
% in TeX

G:=Group("C3^2:D6:C4");
// GroupNames label

G:=SmallGroup(432,238);
// by ID

G=gap.SmallGroup(432,238);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-3,3,-3,85,64,1124,851,298,348,1027,537,14118,7069]);
// Polycyclic

G:=Group<a,b,c,d,e|a^3=b^3=c^6=d^2=e^4=1,a*b=b*a,c*a*c^-1=d*a*d=a^-1*b^-1,e*a*e^-1=c^4,c*b*c^-1=e*b*e^-1=b^-1,b*d=d*b,d*c*d=c^-1,e*c*e^-1=a*c^-1*d,e*d*e^-1=a*c^2*d>;
// generators/relations

Export

Character table of C32⋊D6⋊C4 in TeX

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