metabelian, soluble, monomial, A-group
Aliases: C33:7(C2xC8), C33:4C8:10C2, C12.3(C32:C4), C3:Dic3.40D6, (C32xC12).5C4, C4.3(C33:C4), (C3xC12).13Dic3, C3:S3:4(C3:C8), (C3xC3:S3):7C8, C32:5(C2xC3:C8), (C4xC3:S3).9S3, C6.8(C2xC32:C4), (C6xC3:S3).12C4, C3:2(C3:S3:3C8), (C12xC3:S3).14C2, (C2xC3:S3).7Dic3, C2.1(C2xC33:C4), (C32xC6).15(C2xC4), (C3xC6).22(C2xDic3), (C3xC3:Dic3).48C22, SmallGroup(432,635)
Series: Derived ►Chief ►Lower central ►Upper central
| C33 — C33:7(C2xC8) |
Generators and relations for C33:7(C2xC8)
G = < a,b,c,d,e | a3=b3=c3=d2=e8=1, ab=ba, ac=ca, dad=a-1, eae-1=ab-1, bc=cb, dbd=b-1, ebe-1=a-1b-1, cd=dc, ece-1=c-1, de=ed >
Subgroups: 424 in 88 conjugacy classes, 25 normal (19 characteristic)
C1, C2, C2, C3, C3, C4, C4, C22, S3, C6, C6, C8, C2xC4, C32, C32, Dic3, C12, C12, D6, C2xC6, C2xC8, C3xS3, C3:S3, C3xC6, C3xC6, C3:C8, C4xS3, C2xC12, C33, C3xDic3, C3:Dic3, C3xC12, C3xC12, S3xC6, C2xC3:S3, C2xC3:C8, C3xC3:S3, C32xC6, C32:2C8, S3xC12, C4xC3:S3, C3xC3:Dic3, C32xC12, C6xC3:S3, C3:S3:3C8, C33:4C8, C12xC3:S3, C33:7(C2xC8)
Quotients: C1, C2, C4, C22, S3, C8, C2xC4, Dic3, D6, C2xC8, C3:C8, C2xDic3, C32:C4, C2xC3:C8, C2xC32:C4, C33:C4, C3:S3:3C8, C2xC33:C4, C33:7(C2xC8)
►On 48 points
(1 13 47)(2 14 48)(3 41 15)(4 42 16)(5 9 43)(6 10 44)(7 45 11)(8 46 12)(17 27 40)(18 33 28)(19 34 29)(20 30 35)(21 31 36)(22 37 32)(23 38 25)(24 26 39)
(2 48 14)(4 16 42)(6 44 10)(8 12 46)(17 40 27)(19 29 34)(21 36 31)(23 25 38)
(1 47 13)(2 14 48)(3 41 15)(4 16 42)(5 43 9)(6 10 44)(7 45 11)(8 12 46)(17 40 27)(18 28 33)(19 34 29)(20 30 35)(21 36 31)(22 32 37)(23 38 25)(24 26 39)
(1 30)(2 31)(3 32)(4 25)(5 26)(6 27)(7 28)(8 29)(9 24)(10 17)(11 18)(12 19)(13 20)(14 21)(15 22)(16 23)(33 45)(34 46)(35 47)(36 48)(37 41)(38 42)(39 43)(40 44)
(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 37 38 39 40)(41 42 43 44 45 46 47 48)
G:=sub<Sym(48)| (1,13,47)(2,14,48)(3,41,15)(4,42,16)(5,9,43)(6,10,44)(7,45,11)(8,46,12)(17,27,40)(18,33,28)(19,34,29)(20,30,35)(21,31,36)(22,37,32)(23,38,25)(24,26,39), (2,48,14)(4,16,42)(6,44,10)(8,12,46)(17,40,27)(19,29,34)(21,36,31)(23,25,38), (1,47,13)(2,14,48)(3,41,15)(4,16,42)(5,43,9)(6,10,44)(7,45,11)(8,12,46)(17,40,27)(18,28,33)(19,34,29)(20,30,35)(21,36,31)(22,32,37)(23,38,25)(24,26,39), (1,30)(2,31)(3,32)(4,25)(5,26)(6,27)(7,28)(8,29)(9,24)(10,17)(11,18)(12,19)(13,20)(14,21)(15,22)(16,23)(33,45)(34,46)(35,47)(36,48)(37,41)(38,42)(39,43)(40,44), (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)(33,34,35,36,37,38,39,40)(41,42,43,44,45,46,47,48)>;
G:=Group( (1,13,47)(2,14,48)(3,41,15)(4,42,16)(5,9,43)(6,10,44)(7,45,11)(8,46,12)(17,27,40)(18,33,28)(19,34,29)(20,30,35)(21,31,36)(22,37,32)(23,38,25)(24,26,39), (2,48,14)(4,16,42)(6,44,10)(8,12,46)(17,40,27)(19,29,34)(21,36,31)(23,25,38), (1,47,13)(2,14,48)(3,41,15)(4,16,42)(5,43,9)(6,10,44)(7,45,11)(8,12,46)(17,40,27)(18,28,33)(19,34,29)(20,30,35)(21,36,31)(22,32,37)(23,38,25)(24,26,39), (1,30)(2,31)(3,32)(4,25)(5,26)(6,27)(7,28)(8,29)(9,24)(10,17)(11,18)(12,19)(13,20)(14,21)(15,22)(16,23)(33,45)(34,46)(35,47)(36,48)(37,41)(38,42)(39,43)(40,44), (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)(33,34,35,36,37,38,39,40)(41,42,43,44,45,46,47,48) );
G=PermutationGroup([[(1,13,47),(2,14,48),(3,41,15),(4,42,16),(5,9,43),(6,10,44),(7,45,11),(8,46,12),(17,27,40),(18,33,28),(19,34,29),(20,30,35),(21,31,36),(22,37,32),(23,38,25),(24,26,39)], [(2,48,14),(4,16,42),(6,44,10),(8,12,46),(17,40,27),(19,29,34),(21,36,31),(23,25,38)], [(1,47,13),(2,14,48),(3,41,15),(4,16,42),(5,43,9),(6,10,44),(7,45,11),(8,12,46),(17,40,27),(18,28,33),(19,34,29),(20,30,35),(21,36,31),(22,32,37),(23,38,25),(24,26,39)], [(1,30),(2,31),(3,32),(4,25),(5,26),(6,27),(7,28),(8,29),(9,24),(10,17),(11,18),(12,19),(13,20),(14,21),(15,22),(16,23),(33,45),(34,46),(35,47),(36,48),(37,41),(38,42),(39,43),(40,44)], [(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),(33,34,35,36,37,38,39,40),(41,42,43,44,45,46,47,48)]])
48 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3A | 3B | ··· | 3G | 4A | 4B | 4C | 4D | 6A | 6B | ··· | 6G | 6H | 6I | 8A | ··· | 8H | 12A | 12B | 12C | ··· | 12N | 12O | 12P |
| order | 1 | 2 | 2 | 2 | 3 | 3 | ··· | 3 | 4 | 4 | 4 | 4 | 6 | 6 | ··· | 6 | 6 | 6 | 8 | ··· | 8 | 12 | 12 | 12 | ··· | 12 | 12 | 12 |
| size | 1 | 1 | 9 | 9 | 2 | 4 | ··· | 4 | 1 | 1 | 9 | 9 | 2 | 4 | ··· | 4 | 18 | 18 | 27 | ··· | 27 | 2 | 2 | 4 | ··· | 4 | 18 | 18 |
48 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | - | - | + | + | ||||||||
| image | C1 | C2 | C2 | C4 | C4 | C8 | S3 | D6 | Dic3 | Dic3 | C3:C8 | C32:C4 | C2xC32:C4 | C33:C4 | C3:S3:3C8 | C2xC33:C4 | C33:7(C2xC8) |
| kernel | C33:7(C2xC8) | C33:4C8 | C12xC3:S3 | C32xC12 | C6xC3:S3 | C3xC3:S3 | C4xC3:S3 | C3:Dic3 | C3xC12 | C2xC3:S3 | C3:S3 | C12 | C6 | C4 | C3 | C2 | C1 |
| # reps | 1 | 2 | 1 | 2 | 2 | 8 | 1 | 1 | 1 | 1 | 4 | 2 | 2 | 4 | 4 | 4 | 8 |
Matrix representation of C33:7(C2xC8) ►in GL6(F73)
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 64 | 0 | 54 | 0 |
| 0 | 0 | 0 | 8 | 0 | 19 |
| 0 | 0 | 0 | 0 | 8 | 0 |
| 0 | 0 | 0 | 0 | 0 | 64 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 67 | 48 |
| 0 | 0 | 0 | 1 | 6 | 48 |
| 0 | 0 | 0 | 0 | 64 | 0 |
| 0 | 0 | 0 | 0 | 0 | 8 |
| 0 | 72 | 0 | 0 | 0 | 0 |
| 1 | 72 | 0 | 0 | 0 | 0 |
| 0 | 0 | 8 | 0 | 19 | 19 |
| 0 | 0 | 0 | 8 | 54 | 19 |
| 0 | 0 | 0 | 0 | 64 | 0 |
| 0 | 0 | 0 | 0 | 0 | 64 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 45 | 0 |
| 0 | 0 | 1 | 0 | 0 | 28 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 10 | 0 | 0 | 0 | 0 |
| 10 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 67 | 67 | 51 | 51 |
| 0 | 0 | 67 | 6 | 51 | 51 |
| 0 | 0 | 0 | 63 | 67 | 6 |
| 0 | 0 | 63 | 0 | 6 | 6 |
G:=sub<GL(6,GF(73))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,64,0,0,0,0,0,0,8,0,0,0,0,54,0,8,0,0,0,0,19,0,64],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,67,6,64,0,0,0,48,48,0,8],[0,1,0,0,0,0,72,72,0,0,0,0,0,0,8,0,0,0,0,0,0,8,0,0,0,0,19,54,64,0,0,0,19,19,0,64],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,45,0,0,1,0,0,0,28,1,0],[0,10,0,0,0,0,10,0,0,0,0,0,0,0,67,67,0,63,0,0,67,6,63,0,0,0,51,51,67,6,0,0,51,51,6,6] >;
C33:7(C2xC8) in GAP, Magma, Sage, TeX
C_3^3\rtimes_7(C_2\times C_8)
% in TeX
G:=Group("C3^3:7(C2xC8)"); // GroupNames label
G:=SmallGroup(432,635);
// by ID
G=gap.SmallGroup(432,635);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-3,3,-3,28,64,58,2804,298,2693,1027,14118]);
// Polycyclic
G:=Group<a,b,c,d,e|a^3=b^3=c^3=d^2=e^8=1,a*b=b*a,a*c=c*a,d*a*d=a^-1,e*a*e^-1=a*b^-1,b*c=c*b,d*b*d=b^-1,e*b*e^-1=a^-1*b^-1,c*d=d*c,e*c*e^-1=c^-1,d*e=e*d>;
// generators/relations