direct product, metabelian, soluble, monomial, A-group
Aliases: C6xC32:2C8, C62.7C12, (C3xC6):3C24, (C32xC6):3C8, C33:14(C2xC8), C32:7(C2xC24), (C3xC62).1C4, C3:Dic3.7C12, C2.3(C6xC32:C4), C6.23(C2xC32:C4), (C3xC6).14(C2xC12), (C2xC6).8(C32:C4), (C6xC3:Dic3).6C2, (C3xC3:Dic3).4C4, C22.2(C3xC32:C4), C3:Dic3.14(C2xC6), (C2xC3:Dic3).10C6, (C32xC6).12(C2xC4), (C3xC3:Dic3).39C22, SmallGroup(432,632)
Series: Derived ►Chief ►Lower central ►Upper central
| C1 — C32 — C3xC6 — C3:Dic3 — C3xC3:Dic3 — C3xC32:2C8 — C6xC32:2C8 |
| C32 — C6xC32:2C8 |
Generators and relations for C6xC32:2C8
G = < a,b,c,d | a6=b3=c3=d8=1, ab=ba, ac=ca, ad=da, dcd-1=bc=cb, dbd-1=b-1c >
Subgroups: 332 in 96 conjugacy classes, 32 normal (20 characteristic)
C1, C2, C2, C3, C3, C4, C22, C6, C6, C6, C8, C2xC4, C32, C32, Dic3, C12, C2xC6, C2xC6, C2xC8, C3xC6, C3xC6, C3xC6, C24, C2xDic3, C2xC12, C33, C3xDic3, C3:Dic3, C62, C62, C2xC24, C32xC6, C32xC6, C32:2C8, C6xDic3, C2xC3:Dic3, C3xC3:Dic3, C3xC62, C2xC32:2C8, C3xC32:2C8, C6xC3:Dic3, C6xC32:2C8
Quotients: C1, C2, C3, C4, C22, C6, C8, C2xC4, C12, C2xC6, C2xC8, C24, C2xC12, C32:C4, C2xC24, C32:2C8, C2xC32:C4, C3xC32:C4, C2xC32:2C8, C3xC32:2C8, C6xC32:C4, C6xC32:2C8
►On 48 points
(1 33 13 30 47 18)(2 34 14 31 48 19)(3 35 15 32 41 20)(4 36 16 25 42 21)(5 37 9 26 43 22)(6 38 10 27 44 23)(7 39 11 28 45 24)(8 40 12 29 46 17)
(1 13 47)(2 48 14)(3 41 15)(4 16 42)(5 9 43)(6 44 10)(7 45 11)(8 12 46)(17 40 29)(18 33 30)(19 31 34)(20 32 35)(21 36 25)(22 37 26)(23 27 38)(24 28 39)
(2 14 48)(4 42 16)(6 10 44)(8 46 12)(17 29 40)(19 34 31)(21 25 36)(23 38 27)
(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)| (1,33,13,30,47,18)(2,34,14,31,48,19)(3,35,15,32,41,20)(4,36,16,25,42,21)(5,37,9,26,43,22)(6,38,10,27,44,23)(7,39,11,28,45,24)(8,40,12,29,46,17), (1,13,47)(2,48,14)(3,41,15)(4,16,42)(5,9,43)(6,44,10)(7,45,11)(8,12,46)(17,40,29)(18,33,30)(19,31,34)(20,32,35)(21,36,25)(22,37,26)(23,27,38)(24,28,39), (2,14,48)(4,42,16)(6,10,44)(8,46,12)(17,29,40)(19,34,31)(21,25,36)(23,38,27), (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( (1,33,13,30,47,18)(2,34,14,31,48,19)(3,35,15,32,41,20)(4,36,16,25,42,21)(5,37,9,26,43,22)(6,38,10,27,44,23)(7,39,11,28,45,24)(8,40,12,29,46,17), (1,13,47)(2,48,14)(3,41,15)(4,16,42)(5,9,43)(6,44,10)(7,45,11)(8,12,46)(17,40,29)(18,33,30)(19,31,34)(20,32,35)(21,36,25)(22,37,26)(23,27,38)(24,28,39), (2,14,48)(4,42,16)(6,10,44)(8,46,12)(17,29,40)(19,34,31)(21,25,36)(23,38,27), (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([[(1,33,13,30,47,18),(2,34,14,31,48,19),(3,35,15,32,41,20),(4,36,16,25,42,21),(5,37,9,26,43,22),(6,38,10,27,44,23),(7,39,11,28,45,24),(8,40,12,29,46,17)], [(1,13,47),(2,48,14),(3,41,15),(4,16,42),(5,9,43),(6,44,10),(7,45,11),(8,12,46),(17,40,29),(18,33,30),(19,31,34),(20,32,35),(21,36,25),(22,37,26),(23,27,38),(24,28,39)], [(2,14,48),(4,42,16),(6,10,44),(8,46,12),(17,29,40),(19,34,31),(21,25,36),(23,38,27)], [(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)]])
72 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3A | 3B | 3C | ··· | 3H | 4A | 4B | 4C | 4D | 6A | ··· | 6F | 6G | ··· | 6X | 8A | ··· | 8H | 12A | ··· | 12H | 24A | ··· | 24P |
| order | 1 | 2 | 2 | 2 | 3 | 3 | 3 | ··· | 3 | 4 | 4 | 4 | 4 | 6 | ··· | 6 | 6 | ··· | 6 | 8 | ··· | 8 | 12 | ··· | 12 | 24 | ··· | 24 |
| size | 1 | 1 | 1 | 1 | 1 | 1 | 4 | ··· | 4 | 9 | 9 | 9 | 9 | 1 | ··· | 1 | 4 | ··· | 4 | 9 | ··· | 9 | 9 | ··· | 9 | 9 | ··· | 9 |
72 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 4 | 4 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | - | + | ||||||||||||
| image | C1 | C2 | C2 | C3 | C4 | C4 | C6 | C6 | C8 | C12 | C12 | C24 | C32:C4 | C32:2C8 | C2xC32:C4 | C3xC32:C4 | C3xC32:2C8 | C6xC32:C4 |
| kernel | C6xC32:2C8 | C3xC32:2C8 | C6xC3:Dic3 | C2xC32:2C8 | C3xC3:Dic3 | C3xC62 | C32:2C8 | C2xC3:Dic3 | C32xC6 | C3:Dic3 | C62 | C3xC6 | C2xC6 | C6 | C6 | C22 | C2 | C2 |
| # reps | 1 | 2 | 1 | 2 | 2 | 2 | 4 | 2 | 8 | 4 | 4 | 16 | 2 | 4 | 2 | 4 | 8 | 4 |
Matrix representation of C6xC32:2C8 ►in GL5(F73)
| 65 | 0 | 0 | 0 | 0 |
| 0 | 9 | 0 | 0 | 0 |
| 0 | 0 | 9 | 0 | 0 |
| 0 | 0 | 0 | 9 | 0 |
| 0 | 0 | 0 | 0 | 9 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 8 | 0 | 0 | 12 |
| 0 | 0 | 64 | 25 | 0 |
| 0 | 0 | 0 | 8 | 0 |
| 0 | 0 | 0 | 0 | 64 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 58 | 38 |
| 0 | 0 | 1 | 54 | 67 |
| 0 | 0 | 0 | 64 | 0 |
| 0 | 0 | 0 | 0 | 8 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 35 | 47 | 69 | 38 |
| 0 | 6 | 6 | 38 | 4 |
| 0 | 0 | 7 | 67 | 6 |
| 0 | 66 | 0 | 47 | 38 |
G:=sub<GL(5,GF(73))| [65,0,0,0,0,0,9,0,0,0,0,0,9,0,0,0,0,0,9,0,0,0,0,0,9],[1,0,0,0,0,0,8,0,0,0,0,0,64,0,0,0,0,25,8,0,0,12,0,0,64],[1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,58,54,64,0,0,38,67,0,8],[1,0,0,0,0,0,35,6,0,66,0,47,6,7,0,0,69,38,67,47,0,38,4,6,38] >;
C6xC32:2C8 in GAP, Magma, Sage, TeX
C_6\times C_3^2\rtimes_2C_8
% in TeX
G:=Group("C6xC3^2:2C8"); // GroupNames label
G:=SmallGroup(432,632);
// by ID
G=gap.SmallGroup(432,632);
# by ID
G:=PCGroup([7,-2,-2,-3,-2,-2,-3,3,84,80,14117,362,18822,1203]);
// Polycyclic
G:=Group<a,b,c,d|a^6=b^3=c^3=d^8=1,a*b=b*a,a*c=c*a,a*d=d*a,d*c*d^-1=b*c=c*b,d*b*d^-1=b^-1*c>;
// generators/relations