p-group, metabelian, nilpotent (class 3), monomial
Aliases: C42.92D4, C42.20Q8, C42.630C23, C8⋊2C8⋊9C2, C4⋊C8.15C4, C8⋊1C8⋊13C2, C4.46(C8○D4), C22⋊C8.16C4, (C22×C4).31Q8, C4⋊C8.218C22, C23.54(C4⋊C4), (C4×C8).101C22, C42.126(C2×C4), (C22×C4).668D4, C4.139(C8⋊C22), C4.133(C8.C22), C42.6C4.31C2, C22.8(C8.C4), (C2×C42).229C22, C2.5(M4(2)⋊C4), C42.12C4.37C2, C2.8(C42.6C22), (C2×C4⋊C8).21C2, (C2×C8).29(C2×C4), (C2×C4).37(C4⋊C4), C2.9(C2×C8.C4), C22.87(C2×C4⋊C4), (C2×C4).157(C2×Q8), (C2×C4).1466(C2×D4), (C2×C4).512(C22×C4), (C22×C4).251(C2×C4), SmallGroup(128,305)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C42.92D4
G = < a,b,c,d | a4=b4=1, c4=a2, d2=a2b, ab=ba, cac-1=a-1b2, dad-1=ab2, bc=cb, bd=db, dcd-1=c3 >
Subgroups: 124 in 80 conjugacy classes, 48 normal (30 characteristic)
C1, C2 [×3], C2 [×2], C4 [×2], C4 [×2], C4 [×4], C22, C22 [×2], C22 [×2], C8 [×8], C2×C4 [×6], C2×C4 [×6], C23, C42 [×4], C2×C8 [×4], C2×C8 [×6], C22×C4 [×3], C4×C8 [×2], C8⋊C4, C22⋊C8 [×2], C22⋊C8, C4⋊C8 [×2], C4⋊C8 [×4], C4⋊C8, C2×C42, C22×C8, C8⋊2C8 [×2], C8⋊1C8 [×2], C2×C4⋊C8, C42.12C4, C42.6C4, C42.92D4
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C2×C4 [×6], D4 [×2], Q8 [×2], C23, C4⋊C4 [×4], C22×C4, C2×D4, C2×Q8, C8.C4 [×2], C2×C4⋊C4, C8○D4 [×2], C8⋊C22, C8.C22, C42.6C22, M4(2)⋊C4, C2×C8.C4, C42.92D4
►On 64 points
(1 3 5 7)(2 41 6 45)(4 43 8 47)(9 23 13 19)(10 12 14 16)(11 17 15 21)(18 20 22 24)(25 62 29 58)(26 28 30 32)(27 64 31 60)(33 35 37 39)(34 56 38 52)(36 50 40 54)(42 44 46 48)(49 51 53 55)(57 59 61 63)
(1 39 42 55)(2 40 43 56)(3 33 44 49)(4 34 45 50)(5 35 46 51)(6 36 47 52)(7 37 48 53)(8 38 41 54)(9 62 17 31)(10 63 18 32)(11 64 19 25)(12 57 20 26)(13 58 21 27)(14 59 22 28)(15 60 23 29)(16 61 24 30)
(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 27 35 9 42 58 51 17)(2 30 36 12 43 61 52 20)(3 25 37 15 44 64 53 23)(4 28 38 10 45 59 54 18)(5 31 39 13 46 62 55 21)(6 26 40 16 47 57 56 24)(7 29 33 11 48 60 49 19)(8 32 34 14 41 63 50 22)
G:=sub<Sym(64)| (1,3,5,7)(2,41,6,45)(4,43,8,47)(9,23,13,19)(10,12,14,16)(11,17,15,21)(18,20,22,24)(25,62,29,58)(26,28,30,32)(27,64,31,60)(33,35,37,39)(34,56,38,52)(36,50,40,54)(42,44,46,48)(49,51,53,55)(57,59,61,63), (1,39,42,55)(2,40,43,56)(3,33,44,49)(4,34,45,50)(5,35,46,51)(6,36,47,52)(7,37,48,53)(8,38,41,54)(9,62,17,31)(10,63,18,32)(11,64,19,25)(12,57,20,26)(13,58,21,27)(14,59,22,28)(15,60,23,29)(16,61,24,30), (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), (1,27,35,9,42,58,51,17)(2,30,36,12,43,61,52,20)(3,25,37,15,44,64,53,23)(4,28,38,10,45,59,54,18)(5,31,39,13,46,62,55,21)(6,26,40,16,47,57,56,24)(7,29,33,11,48,60,49,19)(8,32,34,14,41,63,50,22)>;
G:=Group( (1,3,5,7)(2,41,6,45)(4,43,8,47)(9,23,13,19)(10,12,14,16)(11,17,15,21)(18,20,22,24)(25,62,29,58)(26,28,30,32)(27,64,31,60)(33,35,37,39)(34,56,38,52)(36,50,40,54)(42,44,46,48)(49,51,53,55)(57,59,61,63), (1,39,42,55)(2,40,43,56)(3,33,44,49)(4,34,45,50)(5,35,46,51)(6,36,47,52)(7,37,48,53)(8,38,41,54)(9,62,17,31)(10,63,18,32)(11,64,19,25)(12,57,20,26)(13,58,21,27)(14,59,22,28)(15,60,23,29)(16,61,24,30), (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), (1,27,35,9,42,58,51,17)(2,30,36,12,43,61,52,20)(3,25,37,15,44,64,53,23)(4,28,38,10,45,59,54,18)(5,31,39,13,46,62,55,21)(6,26,40,16,47,57,56,24)(7,29,33,11,48,60,49,19)(8,32,34,14,41,63,50,22) );
G=PermutationGroup([(1,3,5,7),(2,41,6,45),(4,43,8,47),(9,23,13,19),(10,12,14,16),(11,17,15,21),(18,20,22,24),(25,62,29,58),(26,28,30,32),(27,64,31,60),(33,35,37,39),(34,56,38,52),(36,50,40,54),(42,44,46,48),(49,51,53,55),(57,59,61,63)], [(1,39,42,55),(2,40,43,56),(3,33,44,49),(4,34,45,50),(5,35,46,51),(6,36,47,52),(7,37,48,53),(8,38,41,54),(9,62,17,31),(10,63,18,32),(11,64,19,25),(12,57,20,26),(13,58,21,27),(14,59,22,28),(15,60,23,29),(16,61,24,30)], [(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)], [(1,27,35,9,42,58,51,17),(2,30,36,12,43,61,52,20),(3,25,37,15,44,64,53,23),(4,28,38,10,45,59,54,18),(5,31,39,13,46,62,55,21),(6,26,40,16,47,57,56,24),(7,29,33,11,48,60,49,19),(8,32,34,14,41,63,50,22)])
38 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 4A | 4B | 4C | 4D | 4E | ··· | 4J | 4K | 4L | 8A | ··· | 8P | 8Q | 8R | 8S | 8T |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 4 | 4 | 8 | ··· | 8 | 8 | 8 | 8 | 8 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 1 | 1 | 1 | 1 | 2 | ··· | 2 | 4 | 4 | 4 | ··· | 4 | 8 | 8 | 8 | 8 |
38 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | - | + | - | + | - | ||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C4 | C4 | D4 | Q8 | D4 | Q8 | C8○D4 | C8.C4 | C8⋊C22 | C8.C22 |
| kernel | C42.92D4 | C8⋊2C8 | C8⋊1C8 | C2×C4⋊C8 | C42.12C4 | C42.6C4 | C22⋊C8 | C4⋊C8 | C42 | C42 | C22×C4 | C22×C4 | C4 | C22 | C4 | C4 |
| # reps | 1 | 2 | 2 | 1 | 1 | 1 | 4 | 4 | 1 | 1 | 1 | 1 | 8 | 8 | 1 | 1 |
Matrix representation of C42.92D4 ►in GL4(𝔽17) generated by
| 16 | 0 | 0 | 0 |
| 1 | 1 | 0 | 0 |
| 0 | 0 | 16 | 15 |
| 0 | 0 | 1 | 1 |
| 4 | 0 | 0 | 0 |
| 0 | 4 | 0 | 0 |
| 0 | 0 | 4 | 0 |
| 0 | 0 | 0 | 4 |
| 16 | 15 | 0 | 0 |
| 1 | 1 | 0 | 0 |
| 0 | 0 | 7 | 7 |
| 0 | 0 | 5 | 0 |
| 2 | 4 | 0 | 0 |
| 0 | 15 | 0 | 0 |
| 0 | 0 | 0 | 3 |
| 0 | 0 | 10 | 0 |
G:=sub<GL(4,GF(17))| [16,1,0,0,0,1,0,0,0,0,16,1,0,0,15,1],[4,0,0,0,0,4,0,0,0,0,4,0,0,0,0,4],[16,1,0,0,15,1,0,0,0,0,7,5,0,0,7,0],[2,0,0,0,4,15,0,0,0,0,0,10,0,0,3,0] >;
C42.92D4 in GAP, Magma, Sage, TeX
C_4^2._{92}D_4 % in TeX
G:=Group("C4^2.92D4"); // GroupNames label
G:=SmallGroup(128,305);
// by ID
G=gap.SmallGroup(128,305);
# by ID
G:=PCGroup([7,-2,2,2,-2,2,-2,2,112,141,64,1059,1123,136,172]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^4=1,c^4=a^2,d^2=a^2*b,a*b=b*a,c*a*c^-1=a^-1*b^2,d*a*d^-1=a*b^2,b*c=c*b,b*d=d*b,d*c*d^-1=c^3>;
// generators/relations