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G = C327D4order 72 = 23·32

2nd semidirect product of C32 and D4 acting via D4/C22=C2

metabelian, supersoluble, monomial

Aliases: C327D4, C623C2, C6.16D6, (C2×C6)⋊4S3, C33(C3⋊D4), C3⋊Dic33C2, C222(C3⋊S3), (C3×C6).15C22, (C2×C3⋊S3)⋊3C2, C2.5(C2×C3⋊S3), SmallGroup(72,35)

Series: Derived Chief Lower central Upper central

C1C3×C6 — C327D4
C1C3C32C3×C6C2×C3⋊S3 — C327D4
C32C3×C6 — C327D4
C1C2C22

Generators and relations for C327D4
 G = < a,b,c,d | a3=b3=c4=d2=1, ab=ba, cac-1=dad=a-1, cbc-1=dbd=b-1, dcd=c-1 >

2C2
18C2
9C4
9C22
2C6
2C6
2C6
2C6
6S3
6S3
6S3
6S3
9D4
3Dic3
3D6
3Dic3
3Dic3
3Dic3
3D6
3D6
3D6
2C3×C6
2C3⋊S3
3C3⋊D4
3C3⋊D4
3C3⋊D4
3C3⋊D4

Character table of C327D4

 class 12A2B2C3A3B3C3D46A6B6C6D6E6F6G6H6I6J6K6L
 size 11218222218222222222222
ρ1111111111111111111111    trivial
ρ211-1-11111111-1-1-1-1-1-1-1-111    linear of order 2
ρ311-111111-111-1-1-1-1-1-1-1-111    linear of order 2
ρ4111-11111-1111111111111    linear of order 2
ρ52220-1-1-120-1-1-1-1-12-1-1-12-12    orthogonal lifted from S3
ρ622-20-1-1-120-1-1111-2111-2-12    orthogonal lifted from D6
ρ722-20-12-1-10-12-2-2111111-1-1    orthogonal lifted from D6
ρ822-20-1-12-10-1-111-211-2112-1    orthogonal lifted from D6
ρ92220-12-1-10-1222-1-1-1-1-1-1-1-1    orthogonal lifted from S3
ρ102-20022220-2-200000000-2-2    orthogonal lifted from D4
ρ1122202-1-1-102-1-1-1-1-12-12-1-1-1    orthogonal lifted from S3
ρ1222-202-1-1-102-11111-21-21-1-1    orthogonal lifted from D6
ρ132220-1-12-10-1-1-1-12-1-12-1-12-1    orthogonal lifted from S3
ρ142-200-1-12-1011--3-30--3-30--3-3-21    complex lifted from C3⋊D4
ρ152-200-1-12-1011-3--30-3--30-3--3-21    complex lifted from C3⋊D4
ρ162-200-12-1-101-200-3-3-3--3--3--311    complex lifted from C3⋊D4
ρ172-200-12-1-101-200--3--3--3-3-3-311    complex lifted from C3⋊D4
ρ182-200-1-1-12011--3-3-30--3--3-301-2    complex lifted from C3⋊D4
ρ192-200-1-1-12011-3--3--30-3-3--301-2    complex lifted from C3⋊D4
ρ202-2002-1-1-10-21--3-3--3-30-30--311    complex lifted from C3⋊D4
ρ212-2002-1-1-10-21-3--3-3--30--30-311    complex lifted from C3⋊D4

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

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

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

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

C327D4 is a maximal subgroup of
D6.3D6  S3×C3⋊D4  C12.59D6  D4×C3⋊S3  C12.D6  He36D4  C6.D18  C32.3S4  C336D4  C337D4  C3315D4  C324S4  SL2(𝔽3).D6  (C2×C6)⋊4S4  C30.12D6  C327D20  C62⋊D5
C327D4 is a maximal quotient of
C6.Dic6  C6.11D12  C327D8  C329SD16  C3211SD16  C327Q16  C625C4  C6.D18  He37D4  C336D4  C337D4  C3315D4  (C2×C6)⋊4S4  C30.12D6  C327D20  C62⋊D5

Matrix representation of C327D4 in GL4(𝔽13) generated by

0100
121200
0010
0001
,
1000
0100
0001
001212
,
1000
121200
00112
0042
,
1000
121200
0011
00012
G:=sub<GL(4,GF(13))| [0,12,0,0,1,12,0,0,0,0,1,0,0,0,0,1],[1,0,0,0,0,1,0,0,0,0,0,12,0,0,1,12],[1,12,0,0,0,12,0,0,0,0,11,4,0,0,2,2],[1,12,0,0,0,12,0,0,0,0,1,0,0,0,1,12] >;

C327D4 in GAP, Magma, Sage, TeX

C_3^2\rtimes_7D_4
% in TeX

G:=Group("C3^2:7D4");
// GroupNames label

G:=SmallGroup(72,35);
// by ID

G=gap.SmallGroup(72,35);
# by ID

G:=PCGroup([5,-2,-2,-2,-3,-3,61,323,1204]);
// Polycyclic

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

Export

Subgroup lattice of C327D4 in TeX
Character table of C327D4 in TeX

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