p-group, metabelian, nilpotent (class 3), monomial
Aliases: C42.376D4, C42.147C23, C4.35C4≀C2, (C4×D4).3C4, (C4×Q8).3C4, C4⋊D4.10C4, C42.88(C2×C4), C22⋊Q8.10C4, (C4×M4(2))⋊19C2, (C22×C4).227D4, C8⋊C4.85C22, C42.6C4⋊34C2, C23.58(C22⋊C4), C42.2C22⋊7C2, (C2×C42).191C22, C42.C22⋊8C2, C42.C2.95C22, C4.4D4.114C22, C2.29(C42⋊C22), C23.36C23.10C2, C2.11(M4(2).8C22), C2.34(C2×C4≀C2), C4⋊C4.26(C2×C4), (C2×D4).21(C2×C4), (C2×Q8).21(C2×C4), (C2×C4).1175(C2×D4), (C22×C4).213(C2×C4), (C2×C4).141(C22×C4), (C2×C4).179(C22⋊C4), C22.205(C2×C22⋊C4), SmallGroup(128,261)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C42.376D4
G = < a,b,c,d | a4=b4=1, c4=a2, d2=b, ab=ba, cac-1=a-1b2, ad=da, cbc-1=a2b, bd=db, dcd-1=a2bc3 >
Subgroups: 212 in 108 conjugacy classes, 44 normal (34 characteristic)
C1, C2, C2, C4, C4, C22, C22, C8, C2×C4, C2×C4, D4, Q8, C23, C23, C42, C42, C22⋊C4, C4⋊C4, C4⋊C4, C2×C8, M4(2), C22×C4, C22×C4, C2×D4, C2×D4, C2×Q8, C4×C8, C8⋊C4, C22⋊C8, C4⋊C8, C2×C42, C42⋊C2, C4×D4, C4×D4, C4×Q8, C4⋊D4, C22⋊Q8, C22.D4, C4.4D4, C42.C2, C42⋊2C2, C2×M4(2), C42.C22, C42.2C22, C4×M4(2), C42.6C4, C23.36C23, C42.376D4
Quotients: C1, C2, C4, C22, C2×C4, D4, C23, C22⋊C4, C22×C4, C2×D4, C4≀C2, C2×C22⋊C4, M4(2).8C22, C2×C4≀C2, C42⋊C22, C42.376D4
►On 64 points
(1 14 5 10)(2 56 6 52)(3 16 7 12)(4 50 8 54)(9 64 13 60)(11 58 15 62)(17 29 21 25)(18 42 22 46)(19 31 23 27)(20 44 24 48)(26 35 30 39)(28 37 32 33)(34 41 38 45)(36 43 40 47)(49 63 53 59)(51 57 55 61)
(1 22 61 35)(2 19 62 40)(3 24 63 37)(4 21 64 34)(5 18 57 39)(6 23 58 36)(7 20 59 33)(8 17 60 38)(9 45 54 29)(10 42 55 26)(11 47 56 31)(12 44 49 28)(13 41 50 25)(14 46 51 30)(15 43 52 27)(16 48 53 32)
(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 17 22 60 61 38 35 8)(2 63 19 37 62 3 40 24)(4 5 21 18 64 57 34 39)(6 59 23 33 58 7 36 20)(9 51 45 30 54 14 29 46)(10 25 42 13 55 41 26 50)(11 16 47 48 56 53 31 32)(12 43 44 52 49 27 28 15)
G:=sub<Sym(64)| (1,14,5,10)(2,56,6,52)(3,16,7,12)(4,50,8,54)(9,64,13,60)(11,58,15,62)(17,29,21,25)(18,42,22,46)(19,31,23,27)(20,44,24,48)(26,35,30,39)(28,37,32,33)(34,41,38,45)(36,43,40,47)(49,63,53,59)(51,57,55,61), (1,22,61,35)(2,19,62,40)(3,24,63,37)(4,21,64,34)(5,18,57,39)(6,23,58,36)(7,20,59,33)(8,17,60,38)(9,45,54,29)(10,42,55,26)(11,47,56,31)(12,44,49,28)(13,41,50,25)(14,46,51,30)(15,43,52,27)(16,48,53,32), (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), (1,17,22,60,61,38,35,8)(2,63,19,37,62,3,40,24)(4,5,21,18,64,57,34,39)(6,59,23,33,58,7,36,20)(9,51,45,30,54,14,29,46)(10,25,42,13,55,41,26,50)(11,16,47,48,56,53,31,32)(12,43,44,52,49,27,28,15)>;
G:=Group( (1,14,5,10)(2,56,6,52)(3,16,7,12)(4,50,8,54)(9,64,13,60)(11,58,15,62)(17,29,21,25)(18,42,22,46)(19,31,23,27)(20,44,24,48)(26,35,30,39)(28,37,32,33)(34,41,38,45)(36,43,40,47)(49,63,53,59)(51,57,55,61), (1,22,61,35)(2,19,62,40)(3,24,63,37)(4,21,64,34)(5,18,57,39)(6,23,58,36)(7,20,59,33)(8,17,60,38)(9,45,54,29)(10,42,55,26)(11,47,56,31)(12,44,49,28)(13,41,50,25)(14,46,51,30)(15,43,52,27)(16,48,53,32), (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), (1,17,22,60,61,38,35,8)(2,63,19,37,62,3,40,24)(4,5,21,18,64,57,34,39)(6,59,23,33,58,7,36,20)(9,51,45,30,54,14,29,46)(10,25,42,13,55,41,26,50)(11,16,47,48,56,53,31,32)(12,43,44,52,49,27,28,15) );
G=PermutationGroup([[(1,14,5,10),(2,56,6,52),(3,16,7,12),(4,50,8,54),(9,64,13,60),(11,58,15,62),(17,29,21,25),(18,42,22,46),(19,31,23,27),(20,44,24,48),(26,35,30,39),(28,37,32,33),(34,41,38,45),(36,43,40,47),(49,63,53,59),(51,57,55,61)], [(1,22,61,35),(2,19,62,40),(3,24,63,37),(4,21,64,34),(5,18,57,39),(6,23,58,36),(7,20,59,33),(8,17,60,38),(9,45,54,29),(10,42,55,26),(11,47,56,31),(12,44,49,28),(13,41,50,25),(14,46,51,30),(15,43,52,27),(16,48,53,32)], [(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)], [(1,17,22,60,61,38,35,8),(2,63,19,37,62,3,40,24),(4,5,21,18,64,57,34,39),(6,59,23,33,58,7,36,20),(9,51,45,30,54,14,29,46),(10,25,42,13,55,41,26,50),(11,16,47,48,56,53,31,32),(12,43,44,52,49,27,28,15)]])
32 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 4A | ··· | 4J | 4K | 4L | 4M | 4N | 8A | ··· | 8H | 8I | 8J | 8K | 8L |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 4 | 4 | 4 | 4 | 8 | ··· | 8 | 8 | 8 | 8 | 8 |
| size | 1 | 1 | 1 | 1 | 4 | 8 | 2 | ··· | 2 | 4 | 8 | 8 | 8 | 4 | ··· | 4 | 8 | 8 | 8 | 8 |
32 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | |||||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C4 | C4 | C4 | C4 | D4 | D4 | C4≀C2 | M4(2).8C22 | C42⋊C22 |
| kernel | C42.376D4 | C42.C22 | C42.2C22 | C4×M4(2) | C42.6C4 | C23.36C23 | C4×D4 | C4×Q8 | C4⋊D4 | C22⋊Q8 | C42 | C22×C4 | C4 | C2 | C2 |
| # reps | 1 | 2 | 2 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 8 | 2 | 2 |
Matrix representation of C42.376D4 ►in GL6(𝔽17)
| 0 | 4 | 0 | 0 | 0 | 0 |
| 4 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 16 | 0 | 15 |
| 0 | 0 | 1 | 0 | 2 | 0 |
| 0 | 0 | 0 | 1 | 0 | 1 |
| 0 | 0 | 16 | 0 | 16 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 4 | 0 | 0 | 0 |
| 0 | 0 | 0 | 4 | 0 | 0 |
| 0 | 0 | 0 | 0 | 4 | 0 |
| 0 | 0 | 0 | 0 | 0 | 4 |
| 10 | 6 | 0 | 0 | 0 | 0 |
| 11 | 7 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 2 | 15 |
| 0 | 0 | 0 | 0 | 2 | 2 |
| 0 | 0 | 1 | 16 | 0 | 0 |
| 0 | 0 | 1 | 1 | 0 | 0 |
| 10 | 6 | 0 | 0 | 0 | 0 |
| 6 | 10 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 2 | 15 |
| 0 | 0 | 0 | 0 | 15 | 15 |
| 0 | 0 | 1 | 16 | 0 | 0 |
| 0 | 0 | 16 | 16 | 0 | 0 |
G:=sub<GL(6,GF(17))| [0,4,0,0,0,0,4,0,0,0,0,0,0,0,0,1,0,16,0,0,16,0,1,0,0,0,0,2,0,16,0,0,15,0,1,0],[0,1,0,0,0,0,1,0,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,4],[10,11,0,0,0,0,6,7,0,0,0,0,0,0,0,0,1,1,0,0,0,0,16,1,0,0,2,2,0,0,0,0,15,2,0,0],[10,6,0,0,0,0,6,10,0,0,0,0,0,0,0,0,1,16,0,0,0,0,16,16,0,0,2,15,0,0,0,0,15,15,0,0] >;
C42.376D4 in GAP, Magma, Sage, TeX
C_4^2._{376}D_4 % in TeX
G:=Group("C4^2.376D4"); // GroupNames label
G:=SmallGroup(128,261);
// by ID
G=gap.SmallGroup(128,261);
# by ID
G:=PCGroup([7,-2,2,2,-2,2,-2,2,112,141,758,184,1123,1018,248,1971,102]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^4=1,c^4=a^2,d^2=b,a*b=b*a,c*a*c^-1=a^-1*b^2,a*d=d*a,c*b*c^-1=a^2*b,b*d=d*b,d*c*d^-1=a^2*b*c^3>;
// generators/relations