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G = C3×C327D4order 216 = 23·33

Direct product of C3 and C327D4

direct product, metabelian, supersoluble, monomial

Aliases: C3×C327D4, C3314D4, C6211S3, C6212C6, C6.28(S3×C6), C3⋊Dic36C6, (C3×C62)⋊3C2, (C3×C6).61D6, C3210(C3×D4), C3212(C3⋊D4), (C32×C6).25C22, (C6×C3⋊S3)⋊6C2, (C2×C3⋊S3)⋊6C6, (C2×C6)⋊6(C3×S3), C2.5(C6×C3⋊S3), C33(C3×C3⋊D4), (C2×C6)⋊3(C3⋊S3), C6.26(C2×C3⋊S3), C223(C3×C3⋊S3), (C3×C6).33(C2×C6), (C3×C3⋊Dic3)⋊8C2, SmallGroup(216,144)

Series: Derived Chief Lower central Upper central

C1C3×C6 — C3×C327D4
C1C3C32C3×C6C32×C6C6×C3⋊S3 — C3×C327D4
C32C3×C6 — C3×C327D4
C1C6C2×C6

Generators and relations for C3×C327D4
 G = < a,b,c,d,e | a3=b3=c3=d4=e2=1, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, dbd-1=ebe=b-1, dcd-1=ece=c-1, ede=d-1 >

Subgroups: 360 in 136 conjugacy classes, 42 normal (18 characteristic)
C1, C2, C2 [×2], C3, C3 [×4], C3 [×4], C4, C22, C22, S3 [×4], C6, C6 [×4], C6 [×18], D4, C32, C32 [×4], C32 [×4], Dic3 [×4], C12, D6 [×4], C2×C6, C2×C6 [×4], C2×C6 [×5], C3×S3 [×4], C3⋊S3, C3×C6, C3×C6 [×4], C3×C6 [×17], C3⋊D4 [×4], C3×D4, C33, C3×Dic3 [×4], C3⋊Dic3, S3×C6 [×4], C2×C3⋊S3, C62, C62 [×4], C62 [×4], C3×C3⋊S3, C32×C6, C32×C6, C3×C3⋊D4 [×4], C327D4, C3×C3⋊Dic3, C6×C3⋊S3, C3×C62, C3×C327D4
Quotients: C1, C2 [×3], C3, C22, S3 [×4], C6 [×3], D4, D6 [×4], C2×C6, C3×S3 [×4], C3⋊S3, C3⋊D4 [×4], C3×D4, S3×C6 [×4], C2×C3⋊S3, C3×C3⋊S3, C3×C3⋊D4 [×4], C327D4, C6×C3⋊S3, C3×C327D4

Smallest permutation representation of C3×C327D4
On 36 points
Generators in S36
(1 27 9)(2 28 10)(3 25 11)(4 26 12)(5 23 20)(6 24 17)(7 21 18)(8 22 19)(13 34 30)(14 35 31)(15 36 32)(16 33 29)
(1 16 7)(2 8 13)(3 14 5)(4 6 15)(9 29 18)(10 19 30)(11 31 20)(12 17 32)(21 27 33)(22 34 28)(23 25 35)(24 36 26)
(1 21 29)(2 30 22)(3 23 31)(4 32 24)(5 35 11)(6 12 36)(7 33 9)(8 10 34)(13 19 28)(14 25 20)(15 17 26)(16 27 18)
(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 4)(2 3)(5 8)(6 7)(9 12)(10 11)(13 14)(15 16)(17 18)(19 20)(21 24)(22 23)(25 28)(26 27)(29 32)(30 31)(33 36)(34 35)

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

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

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

C3×C327D4 is a maximal subgroup of   C3×S3×C3⋊D4  C62.91D6  C62.93D6  C6223D6  C62.96D6  C6224D6  C3×D4×C3⋊S3

63 conjugacy classes

class 1 2A2B2C3A3B3C···3N 4 6A6B6C···6AN6AO6AP12A12B
order1222333···34666···6661212
size11218112···218112···218181818

63 irreducible representations

dim1111111122222222
type+++++++
imageC1C2C2C2C3C6C6C6S3D4D6C3×S3C3⋊D4C3×D4S3×C6C3×C3⋊D4
kernelC3×C327D4C3×C3⋊Dic3C6×C3⋊S3C3×C62C327D4C3⋊Dic3C2×C3⋊S3C62C62C33C3×C6C2×C6C32C32C6C3
# reps11112222414882816

Matrix representation of C3×C327D4 in GL4(𝔽13) generated by

9000
0900
0010
0001
,
9000
0300
0030
00129
,
1000
0100
0090
0013
,
0100
12000
0084
0075
,
0100
1000
0084
0075
G:=sub<GL(4,GF(13))| [9,0,0,0,0,9,0,0,0,0,1,0,0,0,0,1],[9,0,0,0,0,3,0,0,0,0,3,12,0,0,0,9],[1,0,0,0,0,1,0,0,0,0,9,1,0,0,0,3],[0,12,0,0,1,0,0,0,0,0,8,7,0,0,4,5],[0,1,0,0,1,0,0,0,0,0,8,7,0,0,4,5] >;

C3×C327D4 in GAP, Magma, Sage, TeX

C_3\times C_3^2\rtimes_7D_4
% in TeX

G:=Group("C3xC3^2:7D4");
// GroupNames label

G:=SmallGroup(216,144);
// by ID

G=gap.SmallGroup(216,144);
# by ID

G:=PCGroup([6,-2,-2,-3,-2,-3,-3,169,1444,5189]);
// Polycyclic

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

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