p-group, metabelian, nilpotent (class 3), monomial
Aliases: C82⋊3C2, C42.658C23, C4.6(C4○D8), (C2×C8).225D4, C4.SD16⋊8C2, C4.4D8.4C2, C4⋊Q8.83C22, (C4×C8).371C22, C4⋊1D4.44C22, C2.11(C8.12D4), C22.59(C4⋊1D4), (C2×C4).715(C2×D4), SmallGroup(128,443)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C82⋊3C2
G = < a,b,c | a8=b8=c2=1, ab=ba, cac=a-1b4, cbc=a4b3 >
Subgroups: 224 in 89 conjugacy classes, 36 normal (6 characteristic)
C1, C2, C2, C4, C4, C22, C22, C8, C2×C4, C2×C4, D4, Q8, C23, C42, C4⋊C4, C2×C8, C2×D4, C2×Q8, C4×C8, D4⋊C4, Q8⋊C4, C4⋊1D4, C4⋊Q8, C82, C4.4D8, C4.SD16, C82⋊3C2
Quotients: C1, C2, C22, D4, C23, C2×D4, C4⋊1D4, C4○D8, C8.12D4, C82⋊3C2
►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 11 57 25 22 41 34 50)(2 12 58 26 23 42 35 51)(3 13 59 27 24 43 36 52)(4 14 60 28 17 44 37 53)(5 15 61 29 18 45 38 54)(6 16 62 30 19 46 39 55)(7 9 63 31 20 47 40 56)(8 10 64 32 21 48 33 49)
(2 21)(3 7)(4 19)(6 17)(8 23)(9 31)(10 55)(11 29)(12 53)(13 27)(14 51)(15 25)(16 49)(20 24)(26 44)(28 42)(30 48)(32 46)(33 35)(34 57)(36 63)(37 39)(38 61)(40 59)(41 54)(43 52)(45 50)(47 56)(58 64)(60 62)
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), (1,11,57,25,22,41,34,50)(2,12,58,26,23,42,35,51)(3,13,59,27,24,43,36,52)(4,14,60,28,17,44,37,53)(5,15,61,29,18,45,38,54)(6,16,62,30,19,46,39,55)(7,9,63,31,20,47,40,56)(8,10,64,32,21,48,33,49), (2,21)(3,7)(4,19)(6,17)(8,23)(9,31)(10,55)(11,29)(12,53)(13,27)(14,51)(15,25)(16,49)(20,24)(26,44)(28,42)(30,48)(32,46)(33,35)(34,57)(36,63)(37,39)(38,61)(40,59)(41,54)(43,52)(45,50)(47,56)(58,64)(60,62)>;
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), (1,11,57,25,22,41,34,50)(2,12,58,26,23,42,35,51)(3,13,59,27,24,43,36,52)(4,14,60,28,17,44,37,53)(5,15,61,29,18,45,38,54)(6,16,62,30,19,46,39,55)(7,9,63,31,20,47,40,56)(8,10,64,32,21,48,33,49), (2,21)(3,7)(4,19)(6,17)(8,23)(9,31)(10,55)(11,29)(12,53)(13,27)(14,51)(15,25)(16,49)(20,24)(26,44)(28,42)(30,48)(32,46)(33,35)(34,57)(36,63)(37,39)(38,61)(40,59)(41,54)(43,52)(45,50)(47,56)(58,64)(60,62) );
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)], [(1,11,57,25,22,41,34,50),(2,12,58,26,23,42,35,51),(3,13,59,27,24,43,36,52),(4,14,60,28,17,44,37,53),(5,15,61,29,18,45,38,54),(6,16,62,30,19,46,39,55),(7,9,63,31,20,47,40,56),(8,10,64,32,21,48,33,49)], [(2,21),(3,7),(4,19),(6,17),(8,23),(9,31),(10,55),(11,29),(12,53),(13,27),(14,51),(15,25),(16,49),(20,24),(26,44),(28,42),(30,48),(32,46),(33,35),(34,57),(36,63),(37,39),(38,61),(40,59),(41,54),(43,52),(45,50),(47,56),(58,64),(60,62)]])
38 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 4A | ··· | 4F | 4G | 4H | 4I | 8A | ··· | 8X |
| order | 1 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 4 | 4 | 4 | 8 | ··· | 8 |
| size | 1 | 1 | 1 | 1 | 16 | 2 | ··· | 2 | 16 | 16 | 16 | 2 | ··· | 2 |
38 irreducible representations
| dim | 1 | 1 | 1 | 1 | 2 | 2 |
| type | + | + | + | + | + | |
| image | C1 | C2 | C2 | C2 | D4 | C4○D8 |
| kernel | C82⋊3C2 | C82 | C4.4D8 | C4.SD16 | C2×C8 | C4 |
| # reps | 1 | 1 | 3 | 3 | 6 | 24 |
Matrix representation of C82⋊3C2 ►in GL4(𝔽17) generated by
| 0 | 11 | 0 | 0 |
| 3 | 11 | 0 | 0 |
| 0 | 0 | 13 | 0 |
| 0 | 0 | 0 | 13 |
| 13 | 0 | 0 | 0 |
| 0 | 13 | 0 | 0 |
| 0 | 0 | 0 | 10 |
| 0 | 0 | 12 | 10 |
| 1 | 0 | 0 | 0 |
| 1 | 16 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 1 | 16 |
G:=sub<GL(4,GF(17))| [0,3,0,0,11,11,0,0,0,0,13,0,0,0,0,13],[13,0,0,0,0,13,0,0,0,0,0,12,0,0,10,10],[1,1,0,0,0,16,0,0,0,0,1,1,0,0,0,16] >;
C82⋊3C2 in GAP, Magma, Sage, TeX
C_8^2\rtimes_3C_2
% in TeX
G:=Group("C8^2:3C2"); // GroupNames label
G:=SmallGroup(128,443);
// by ID
G=gap.SmallGroup(128,443);
# by ID
G:=PCGroup([7,-2,2,2,-2,2,-2,2,141,736,422,268,1123,136,2804,172]);
// Polycyclic
G:=Group<a,b,c|a^8=b^8=c^2=1,a*b=b*a,c*a*c=a^-1*b^4,c*b*c=a^4*b^3>;
// generators/relations