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G = C32:7D4order 72 = 23·32

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

metabelian, supersoluble, monomial

Aliases: C32:7D4, C62:3C2, C6.16D6, (C2xC6):4S3, C3:3(C3:D4), C3:Dic3:3C2, C22:2(C3:S3), (C3xC6).15C22, (C2xC3:S3):3C2, C2.5(C2xC3:S3), SmallGroup(72,35)

Series: Derived Chief Lower central Upper central

C1C3xC6 — C32:7D4
C1C3C32C3xC6C2xC3:S3 — C32:7D4
C32C3xC6 — C32:7D4
C1C2C22

Generators and relations for C32:7D4
 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 >

Subgroups: 140 in 48 conjugacy classes, 21 normal (9 characteristic)
Quotients: C1, C2, C22, S3, D4, D6, C3:S3, C3:D4, C2xC3:S3, C32:7D4
2C2
18C2
9C4
9C22
2C6
2C6
2C6
2C6
6S3
6S3
6S3
6S3
9D4
3Dic3
3D6
3Dic3
3Dic3
3Dic3
3D6
3D6
3D6
2C3xC6
2C3:S3
3C3:D4
3C3:D4
3C3:D4
3C3:D4

Character table of C32:7D4

 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 C32:7D4
On 36 points
Generators in S36
(1 16 11)(2 12 13)(3 14 9)(4 10 15)(5 31 26)(6 27 32)(7 29 28)(8 25 30)(17 36 22)(18 23 33)(19 34 24)(20 21 35)
(1 8 21)(2 22 5)(3 6 23)(4 24 7)(9 32 18)(10 19 29)(11 30 20)(12 17 31)(13 36 26)(14 27 33)(15 34 28)(16 25 35)
(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 24)(6 23)(7 22)(8 21)(9 14)(10 13)(11 16)(12 15)(17 28)(18 27)(19 26)(20 25)(29 36)(30 35)(31 34)(32 33)

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

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

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

C32:7D4 is a maximal subgroup of
D6.3D6  S3xC3:D4  C12.59D6  D4xC3:S3  C12.D6  He3:6D4  C6.D18  C32.3S4  C33:6D4  C33:7D4  C33:15D4  C32:4S4  SL2(F3).D6  (C2xC6):4S4  C30.12D6  C32:7D20  C62:D5
C32:7D4 is a maximal quotient of
C6.Dic6  C6.11D12  C32:7D8  C32:9SD16  C32:11SD16  C32:7Q16  C62:5C4  C6.D18  He3:7D4  C33:6D4  C33:7D4  C33:15D4  (C2xC6):4S4  C30.12D6  C32:7D20  C62:D5

Matrix representation of C32:7D4 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] >;

C32:7D4 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 C32:7D4 in TeX
Character table of C32:7D4 in TeX

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