p-group, metabelian, nilpotent (class 3), monomial
Aliases: C42.399D4, C42.600C23, Q8⋊C8⋊2C2, (C2×Q8)⋊5C8, Q8.5(C2×C8), (C4×C8).2C22, C4.6(C22×C8), (C4×Q8).14C4, C42.53(C2×C4), C4.6(C2×M4(2)), C4⋊C8.246C22, C4.12(C22⋊C8), (C22×C4).656D4, (C2×C4).17M4(2), (C22×Q8).19C4, (C4×Q8).250C22, C4.124(C8.C22), C22.28(C22⋊C8), (C2×C42).156C22, C23.168(C22⋊C4), C42.12C4.15C2, C2.3(C42⋊C22), C2.1(C23.38D4), (C2×C4×Q8).4C2, (C2×C4⋊C4).39C4, (C2×C4).18(C2×C8), C4⋊C4.178(C2×C4), C2.15(C2×C22⋊C8), (C2×C4).1443(C2×D4), (C2×Q8).174(C2×C4), (C22×C4).178(C2×C4), (C2×C4).305(C22×C4), C22.99(C2×C22⋊C4), (C2×C4).238(C22⋊C4), SmallGroup(128,211)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C42.399D4
G = < a,b,c,d | a4=b4=1, c4=a2b2, d2=b, ab=ba, cac-1=a-1, ad=da, bc=cb, bd=db, dcd-1=b-1c3 >
Subgroups: 204 in 124 conjugacy classes, 62 normal (28 characteristic)
C1, C2, C2, C4, C4, C4, C22, C22, C22, C8, C2×C4, C2×C4, C2×C4, Q8, Q8, C23, C42, C42, C4⋊C4, C4⋊C4, C2×C8, C22×C4, C22×C4, C2×Q8, C2×Q8, C4×C8, C4×C8, C22⋊C8, C4⋊C8, C4⋊C8, C2×C42, C2×C42, C2×C4⋊C4, C2×C4⋊C4, C4×Q8, C4×Q8, C22×Q8, Q8⋊C8, C42.12C4, C2×C4×Q8, C42.399D4
Quotients: C1, C2, C4, C22, C8, C2×C4, D4, C23, C22⋊C4, C2×C8, M4(2), C22×C4, C2×D4, C22⋊C8, C2×C22⋊C4, C22×C8, C2×M4(2), C8.C22, C2×C22⋊C8, C23.38D4, C42⋊C22, C42.399D4
►On 64 points
(1 19 59 15)(2 16 60 20)(3 21 61 9)(4 10 62 22)(5 23 63 11)(6 12 64 24)(7 17 57 13)(8 14 58 18)(25 44 56 37)(26 38 49 45)(27 46 50 39)(28 40 51 47)(29 48 52 33)(30 34 53 41)(31 42 54 35)(32 36 55 43)
(1 21 63 13)(2 22 64 14)(3 23 57 15)(4 24 58 16)(5 17 59 9)(6 18 60 10)(7 19 61 11)(8 20 62 12)(25 39 52 42)(26 40 53 43)(27 33 54 44)(28 34 55 45)(29 35 56 46)(30 36 49 47)(31 37 50 48)(32 38 51 41)
(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 33 21 54 63 44 13 27)(2 30 22 36 64 49 14 47)(3 42 23 25 57 39 15 52)(4 55 24 45 58 28 16 34)(5 37 17 50 59 48 9 31)(6 26 18 40 60 53 10 43)(7 46 19 29 61 35 11 56)(8 51 20 41 62 32 12 38)
G:=sub<Sym(64)| (1,19,59,15)(2,16,60,20)(3,21,61,9)(4,10,62,22)(5,23,63,11)(6,12,64,24)(7,17,57,13)(8,14,58,18)(25,44,56,37)(26,38,49,45)(27,46,50,39)(28,40,51,47)(29,48,52,33)(30,34,53,41)(31,42,54,35)(32,36,55,43), (1,21,63,13)(2,22,64,14)(3,23,57,15)(4,24,58,16)(5,17,59,9)(6,18,60,10)(7,19,61,11)(8,20,62,12)(25,39,52,42)(26,40,53,43)(27,33,54,44)(28,34,55,45)(29,35,56,46)(30,36,49,47)(31,37,50,48)(32,38,51,41), (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), (1,33,21,54,63,44,13,27)(2,30,22,36,64,49,14,47)(3,42,23,25,57,39,15,52)(4,55,24,45,58,28,16,34)(5,37,17,50,59,48,9,31)(6,26,18,40,60,53,10,43)(7,46,19,29,61,35,11,56)(8,51,20,41,62,32,12,38)>;
G:=Group( (1,19,59,15)(2,16,60,20)(3,21,61,9)(4,10,62,22)(5,23,63,11)(6,12,64,24)(7,17,57,13)(8,14,58,18)(25,44,56,37)(26,38,49,45)(27,46,50,39)(28,40,51,47)(29,48,52,33)(30,34,53,41)(31,42,54,35)(32,36,55,43), (1,21,63,13)(2,22,64,14)(3,23,57,15)(4,24,58,16)(5,17,59,9)(6,18,60,10)(7,19,61,11)(8,20,62,12)(25,39,52,42)(26,40,53,43)(27,33,54,44)(28,34,55,45)(29,35,56,46)(30,36,49,47)(31,37,50,48)(32,38,51,41), (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), (1,33,21,54,63,44,13,27)(2,30,22,36,64,49,14,47)(3,42,23,25,57,39,15,52)(4,55,24,45,58,28,16,34)(5,37,17,50,59,48,9,31)(6,26,18,40,60,53,10,43)(7,46,19,29,61,35,11,56)(8,51,20,41,62,32,12,38) );
G=PermutationGroup([[(1,19,59,15),(2,16,60,20),(3,21,61,9),(4,10,62,22),(5,23,63,11),(6,12,64,24),(7,17,57,13),(8,14,58,18),(25,44,56,37),(26,38,49,45),(27,46,50,39),(28,40,51,47),(29,48,52,33),(30,34,53,41),(31,42,54,35),(32,36,55,43)], [(1,21,63,13),(2,22,64,14),(3,23,57,15),(4,24,58,16),(5,17,59,9),(6,18,60,10),(7,19,61,11),(8,20,62,12),(25,39,52,42),(26,40,53,43),(27,33,54,44),(28,34,55,45),(29,35,56,46),(30,36,49,47),(31,37,50,48),(32,38,51,41)], [(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)], [(1,33,21,54,63,44,13,27),(2,30,22,36,64,49,14,47),(3,42,23,25,57,39,15,52),(4,55,24,45,58,28,16,34),(5,37,17,50,59,48,9,31),(6,26,18,40,60,53,10,43),(7,46,19,29,61,35,11,56),(8,51,20,41,62,32,12,38)]])
44 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 4A | 4B | 4C | 4D | 4E | ··· | 4N | 4O | ··· | 4V | 8A | ··· | 8P |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 4 | ··· | 4 | 8 | ··· | 8 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 1 | 1 | 1 | 1 | 2 | ··· | 2 | 4 | ··· | 4 | 4 | ··· | 4 |
44 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | - | ||||||
| image | C1 | C2 | C2 | C2 | C4 | C4 | C4 | C8 | D4 | D4 | M4(2) | C8.C22 | C42⋊C22 |
| kernel | C42.399D4 | Q8⋊C8 | C42.12C4 | C2×C4×Q8 | C2×C4⋊C4 | C4×Q8 | C22×Q8 | C2×Q8 | C42 | C22×C4 | C2×C4 | C4 | C2 |
| # reps | 1 | 4 | 2 | 1 | 2 | 4 | 2 | 16 | 2 | 2 | 4 | 2 | 2 |
Matrix representation of C42.399D4 ►in GL6(𝔽17)
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 16 | 2 | 0 | 0 |
| 0 | 0 | 16 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 16 | 2 |
| 0 | 0 | 0 | 0 | 16 | 1 |
| 13 | 0 | 0 | 0 | 0 | 0 |
| 0 | 13 | 0 | 0 | 0 | 0 |
| 0 | 0 | 16 | 0 | 0 | 0 |
| 0 | 0 | 0 | 16 | 0 | 0 |
| 0 | 0 | 0 | 0 | 16 | 0 |
| 0 | 0 | 0 | 0 | 0 | 16 |
| 0 | 9 | 0 | 0 | 0 | 0 |
| 8 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 10 |
| 0 | 0 | 0 | 0 | 5 | 0 |
| 0 | 0 | 10 | 7 | 0 | 0 |
| 0 | 0 | 5 | 7 | 0 | 0 |
| 9 | 0 | 0 | 0 | 0 | 0 |
| 0 | 8 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 16 | 0 | 0 | 0 |
| 0 | 0 | 0 | 16 | 0 | 0 |
G:=sub<GL(6,GF(17))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,16,16,0,0,0,0,2,1,0,0,0,0,0,0,16,16,0,0,0,0,2,1],[13,0,0,0,0,0,0,13,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,16],[0,8,0,0,0,0,9,0,0,0,0,0,0,0,0,0,10,5,0,0,0,0,7,7,0,0,0,5,0,0,0,0,10,0,0,0],[9,0,0,0,0,0,0,8,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,1,0,0,0,0,0,0,1,0,0] >;
C42.399D4 in GAP, Magma, Sage, TeX
C_4^2._{399}D_4 % in TeX
G:=Group("C4^2.399D4"); // GroupNames label
G:=SmallGroup(128,211);
// by ID
G=gap.SmallGroup(128,211);
# by ID
G:=PCGroup([7,-2,2,2,-2,2,-2,2,112,141,232,387,184,1123,570,136,172]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^4=1,c^4=a^2*b^2,d^2=b,a*b=b*a,c*a*c^-1=a^-1,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d^-1=b^-1*c^3>;
// generators/relations