metabelian, supersoluble, monomial
Aliases: C33:7D4, C32:10D12, C6.13S32, (S3xC6):4S3, D6:2(C3:S3), C3:Dic3:4S3, (C3xC6).32D6, C3:2(C3:D12), C32:7(C3:D4), C3:1(C32:7D4), (C32xC6).10C22, (S3xC3xC6):4C2, C6.5(C2xC3:S3), C2.5(S3xC3:S3), (C3xC3:Dic3):2C2, (C2xC33:C2):1C2, SmallGroup(216,128)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C33:7D4
G = < a,b,c,d,e | a3=b3=c3=d4=e2=1, ab=ba, ac=ca, dad-1=eae=a-1, bc=cb, dbd-1=ebe=b-1, cd=dc, ece=c-1, ede=d-1 >
Subgroups: 812 in 136 conjugacy classes, 34 normal (14 characteristic)
C1, C2, C2, C3, C3, C3, C4, C22, S3, C6, C6, C6, D4, C32, C32, C32, Dic3, C12, D6, D6, C2xC6, C3xS3, C3:S3, C3xC6, C3xC6, C3xC6, D12, C3:D4, C33, C3xDic3, C3:Dic3, S3xC6, C2xC3:S3, C62, S3xC32, C33:C2, C32xC6, C3:D12, C32:7D4, C3xC3:Dic3, S3xC3xC6, C2xC33:C2, C33:7D4
Quotients: C1, C2, C22, S3, D4, D6, C3:S3, D12, C3:D4, S32, C2xC3:S3, C3:D12, C32:7D4, S3xC3:S3, C33:7D4
(1 16 24)(2 21 13)(3 14 22)(4 23 15)(5 25 35)(6 36 26)(7 27 33)(8 34 28)(9 18 29)(10 30 19)(11 20 31)(12 32 17)
(1 29 7)(2 8 30)(3 31 5)(4 6 32)(9 27 16)(10 13 28)(11 25 14)(12 15 26)(17 23 36)(18 33 24)(19 21 34)(20 35 22)
(1 7 29)(2 8 30)(3 5 31)(4 6 32)(9 16 27)(10 13 28)(11 14 25)(12 15 26)(17 23 36)(18 24 33)(19 21 34)(20 22 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)
(1 3)(5 29)(6 32)(7 31)(8 30)(9 35)(10 34)(11 33)(12 36)(13 21)(14 24)(15 23)(16 22)(17 26)(18 25)(19 28)(20 27)
G:=sub<Sym(36)| (1,16,24)(2,21,13)(3,14,22)(4,23,15)(5,25,35)(6,36,26)(7,27,33)(8,34,28)(9,18,29)(10,30,19)(11,20,31)(12,32,17), (1,29,7)(2,8,30)(3,31,5)(4,6,32)(9,27,16)(10,13,28)(11,25,14)(12,15,26)(17,23,36)(18,33,24)(19,21,34)(20,35,22), (1,7,29)(2,8,30)(3,5,31)(4,6,32)(9,16,27)(10,13,28)(11,14,25)(12,15,26)(17,23,36)(18,24,33)(19,21,34)(20,22,35), (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), (1,3)(5,29)(6,32)(7,31)(8,30)(9,35)(10,34)(11,33)(12,36)(13,21)(14,24)(15,23)(16,22)(17,26)(18,25)(19,28)(20,27)>;
G:=Group( (1,16,24)(2,21,13)(3,14,22)(4,23,15)(5,25,35)(6,36,26)(7,27,33)(8,34,28)(9,18,29)(10,30,19)(11,20,31)(12,32,17), (1,29,7)(2,8,30)(3,31,5)(4,6,32)(9,27,16)(10,13,28)(11,25,14)(12,15,26)(17,23,36)(18,33,24)(19,21,34)(20,35,22), (1,7,29)(2,8,30)(3,5,31)(4,6,32)(9,16,27)(10,13,28)(11,14,25)(12,15,26)(17,23,36)(18,24,33)(19,21,34)(20,22,35), (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), (1,3)(5,29)(6,32)(7,31)(8,30)(9,35)(10,34)(11,33)(12,36)(13,21)(14,24)(15,23)(16,22)(17,26)(18,25)(19,28)(20,27) );
G=PermutationGroup([[(1,16,24),(2,21,13),(3,14,22),(4,23,15),(5,25,35),(6,36,26),(7,27,33),(8,34,28),(9,18,29),(10,30,19),(11,20,31),(12,32,17)], [(1,29,7),(2,8,30),(3,31,5),(4,6,32),(9,27,16),(10,13,28),(11,25,14),(12,15,26),(17,23,36),(18,33,24),(19,21,34),(20,35,22)], [(1,7,29),(2,8,30),(3,5,31),(4,6,32),(9,16,27),(10,13,28),(11,14,25),(12,15,26),(17,23,36),(18,24,33),(19,21,34),(20,22,35)], [(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)], [(1,3),(5,29),(6,32),(7,31),(8,30),(9,35),(10,34),(11,33),(12,36),(13,21),(14,24),(15,23),(16,22),(17,26),(18,25),(19,28),(20,27)]])
C33:7D4 is a maximal subgroup of
S3xC3:D12 D6:4S32 (S3xC6).D6 D6.3S32 D12:(C3:S3) C12.73S32 C12.58S32 C3:S3xD12 C62.90D6 S3xC32:7D4 C62:23D6
C33:7D4 is a maximal quotient of
C33:7D8 C33:14SD16 C33:15SD16 C33:7Q16 C62.77D6 C62.79D6 C62.82D6
33 conjugacy classes
class | 1 | 2A | 2B | 2C | 3A | ··· | 3E | 3F | 3G | 3H | 3I | 4 | 6A | ··· | 6E | 6F | 6G | 6H | 6I | 6J | ··· | 6Q | 12A | 12B |
order | 1 | 2 | 2 | 2 | 3 | ··· | 3 | 3 | 3 | 3 | 3 | 4 | 6 | ··· | 6 | 6 | 6 | 6 | 6 | 6 | ··· | 6 | 12 | 12 |
size | 1 | 1 | 6 | 54 | 2 | ··· | 2 | 4 | 4 | 4 | 4 | 18 | 2 | ··· | 2 | 4 | 4 | 4 | 4 | 6 | ··· | 6 | 18 | 18 |
33 irreducible representations
dim | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 |
type | + | + | + | + | + | + | + | + | + | + | + | |
image | C1 | C2 | C2 | C2 | S3 | S3 | D4 | D6 | D12 | C3:D4 | S32 | C3:D12 |
kernel | C33:7D4 | C3xC3:Dic3 | S3xC3xC6 | C2xC33:C2 | C3:Dic3 | S3xC6 | C33 | C3xC6 | C32 | C32 | C6 | C3 |
# reps | 1 | 1 | 1 | 1 | 1 | 4 | 1 | 5 | 2 | 8 | 4 | 4 |
Matrix representation of C33:7D4 ►in GL8(F13)
1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 |
0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 |
0 | 0 | 0 | 0 | 1 | 10 | 0 | 0 |
0 | 0 | 0 | 0 | 1 | 11 | 0 | 0 |
0 | 0 | 0 | 0 | 0 | 0 | 12 | 12 |
0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 |
1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 |
0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 |
0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 |
0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 |
0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 |
0 | 0 | 0 | 0 | 0 | 0 | 12 | 12 |
1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
0 | 0 | 0 | 12 | 0 | 0 | 0 | 0 |
0 | 0 | 1 | 12 | 0 | 0 | 0 | 0 |
0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 |
0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 |
0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 |
0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 |
7 | 3 | 0 | 0 | 0 | 0 | 0 | 0 |
5 | 6 | 0 | 0 | 0 | 0 | 0 | 0 |
0 | 0 | 12 | 0 | 0 | 0 | 0 | 0 |
0 | 0 | 0 | 12 | 0 | 0 | 0 | 0 |
0 | 0 | 0 | 0 | 1 | 10 | 0 | 0 |
0 | 0 | 0 | 0 | 0 | 12 | 0 | 0 |
0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 |
0 | 0 | 0 | 0 | 0 | 0 | 12 | 12 |
12 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
9 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 |
0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 |
0 | 0 | 0 | 0 | 12 | 3 | 0 | 0 |
0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 |
0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 |
0 | 0 | 0 | 0 | 0 | 0 | 12 | 12 |
G:=sub<GL(8,GF(13))| [1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,1,0,0,0,0,0,0,10,11,0,0,0,0,0,0,0,0,12,1,0,0,0,0,0,0,12,0],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,1,12],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,12,12,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[7,5,0,0,0,0,0,0,3,6,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,10,12,0,0,0,0,0,0,0,0,1,12,0,0,0,0,0,0,0,12],[12,9,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,3,1,0,0,0,0,0,0,0,0,1,12,0,0,0,0,0,0,0,12] >;
C33:7D4 in GAP, Magma, Sage, TeX
C_3^3\rtimes_7D_4
% in TeX
G:=Group("C3^3:7D4");
// GroupNames label
G:=SmallGroup(216,128);
// by ID
G=gap.SmallGroup(216,128);
# by ID
G:=PCGroup([6,-2,-2,-2,-3,-3,-3,73,31,201,730,5189]);
// 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,d*a*d^-1=e*a*e=a^-1,b*c=c*b,d*b*d^-1=e*b*e=b^-1,c*d=d*c,e*c*e=c^-1,e*d*e=d^-1>;
// generators/relations