p-group, metabelian, nilpotent (class 2), monomial
Aliases: C82:1C2, C23.25C42, C42.742C23, (C2xC8):9C8, C8o4(C4:C8), C4.4(C4xC8), C4:C8.26C4, (C4xC8).34C4, C8.23(C2xC8), C8o2(C8:C8), C8:C8:25C2, C8o3(C22:C8), C22.4(C4xC8), C4.52(C8oD4), C22:C8.24C4, C4.33(C22xC8), (C22xC8).43C4, (C2xC4).59C42, (C4xC8).359C22, C42.292(C2xC4), C22.22(C2xC42), C8o2(C42.12C4), C2.2(C8o2M4(2)), C42.12C4.50C2, (C2xC42).1029C22, C2.3(C2xC4xC8), (C4xC8)o(C4:C8), (C2xC4xC8).58C2, (C4xC8)o(C22:C8), (C2xC4).79(C2xC8), (C2xC8)o2(C8:C8), (C2xC8).263(C2xC4), (C2xC4).579(C22xC4), (C22xC4).374(C2xC4), (C2xC8)o2(C42.12C4), SmallGroup(128,182)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C82:C2
G = < a,b,c | a8=b8=c2=1, ab=ba, ac=ca, cbc=a4b >
Subgroups: 124 in 106 conjugacy classes, 88 normal (14 characteristic)
C1, C2, C2, C2, C4, C4, C22, C22, C22, C8, C8, C2xC4, C2xC4, C2xC4, C23, C42, C42, C2xC8, C22xC4, C22xC4, C4xC8, C4xC8, C22:C8, C4:C8, C2xC42, C22xC8, C82, C8:C8, C2xC4xC8, C42.12C4, C82:C2
Quotients: C1, C2, C4, C22, C8, C2xC4, C23, C42, C2xC8, C22xC4, C4xC8, C2xC42, C22xC8, C8oD4, C2xC4xC8, C8o2M4(2), C82:C2
►On 64 points
(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)(49 50 51 52 53 54 55 56)(57 58 59 60 61 62 63 64)
(1 42 40 55 20 10 57 25)(2 43 33 56 21 11 58 26)(3 44 34 49 22 12 59 27)(4 45 35 50 23 13 60 28)(5 46 36 51 24 14 61 29)(6 47 37 52 17 15 62 30)(7 48 38 53 18 16 63 31)(8 41 39 54 19 9 64 32)
(9 13)(10 14)(11 15)(12 16)(25 29)(26 30)(27 31)(28 32)(41 45)(42 46)(43 47)(44 48)(49 53)(50 54)(51 55)(52 56)
G:=sub<Sym(64)| (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)(49,50,51,52,53,54,55,56)(57,58,59,60,61,62,63,64), (1,42,40,55,20,10,57,25)(2,43,33,56,21,11,58,26)(3,44,34,49,22,12,59,27)(4,45,35,50,23,13,60,28)(5,46,36,51,24,14,61,29)(6,47,37,52,17,15,62,30)(7,48,38,53,18,16,63,31)(8,41,39,54,19,9,64,32), (9,13)(10,14)(11,15)(12,16)(25,29)(26,30)(27,31)(28,32)(41,45)(42,46)(43,47)(44,48)(49,53)(50,54)(51,55)(52,56)>;
G:=Group( (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)(49,50,51,52,53,54,55,56)(57,58,59,60,61,62,63,64), (1,42,40,55,20,10,57,25)(2,43,33,56,21,11,58,26)(3,44,34,49,22,12,59,27)(4,45,35,50,23,13,60,28)(5,46,36,51,24,14,61,29)(6,47,37,52,17,15,62,30)(7,48,38,53,18,16,63,31)(8,41,39,54,19,9,64,32), (9,13)(10,14)(11,15)(12,16)(25,29)(26,30)(27,31)(28,32)(41,45)(42,46)(43,47)(44,48)(49,53)(50,54)(51,55)(52,56) );
G=PermutationGroup([[(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),(49,50,51,52,53,54,55,56),(57,58,59,60,61,62,63,64)], [(1,42,40,55,20,10,57,25),(2,43,33,56,21,11,58,26),(3,44,34,49,22,12,59,27),(4,45,35,50,23,13,60,28),(5,46,36,51,24,14,61,29),(6,47,37,52,17,15,62,30),(7,48,38,53,18,16,63,31),(8,41,39,54,19,9,64,32)], [(9,13),(10,14),(11,15),(12,16),(25,29),(26,30),(27,31),(28,32),(41,45),(42,46),(43,47),(44,48),(49,53),(50,54),(51,55),(52,56)]])
80 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 4A | ··· | 4L | 4M | ··· | 4R | 8A | ··· | 8P | 8Q | ··· | 8BD |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 4 | ··· | 4 | 8 | ··· | 8 | 8 | ··· | 8 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 1 | ··· | 1 | 2 | ··· | 2 | 1 | ··· | 1 | 2 | ··· | 2 |
80 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 |
| type | + | + | + | + | + | ||||||
| image | C1 | C2 | C2 | C2 | C2 | C4 | C4 | C4 | C4 | C8 | C8oD4 |
| kernel | C82:C2 | C82 | C8:C8 | C2xC4xC8 | C42.12C4 | C4xC8 | C22:C8 | C4:C8 | C22xC8 | C2xC8 | C4 |
| # reps | 1 | 2 | 2 | 1 | 2 | 4 | 8 | 8 | 4 | 32 | 16 |
Matrix representation of C82:C2 ►in GL3(F17) generated by
| 16 | 0 | 0 |
| 0 | 15 | 0 |
| 0 | 0 | 15 |
| 8 | 0 | 0 |
| 0 | 0 | 1 |
| 0 | 16 | 0 |
| 16 | 0 | 0 |
| 0 | 1 | 0 |
| 0 | 0 | 16 |
G:=sub<GL(3,GF(17))| [16,0,0,0,15,0,0,0,15],[8,0,0,0,0,16,0,1,0],[16,0,0,0,1,0,0,0,16] >;
C82:C2 in GAP, Magma, Sage, TeX
C_8^2\rtimes C_2
% in TeX
G:=Group("C8^2:C2"); // GroupNames label
G:=SmallGroup(128,182);
// by ID
G=gap.SmallGroup(128,182);
# by ID
G:=PCGroup([7,-2,2,2,-2,2,-2,2,56,120,387,136,172]);
// Polycyclic
G:=Group<a,b,c|a^8=b^8=c^2=1,a*b=b*a,a*c=c*a,c*b*c=a^4*b>;
// generators/relations