p-group, metabelian, nilpotent (class 3), monomial
Aliases: C42.374D4, Q8.3M4(2), D4.3M4(2), C42.609C23, D4⋊C8⋊27C2, Q8⋊C8⋊31C2, (C4×D4).16C4, (C4×C8).7C22, (C4×Q8).16C4, C42.62(C2×C4), C4.120(C4○D8), C4⋊C8.194C22, (C22×C4).204D4, C4.20(C2×M4(2)), C42⋊C2.17C4, C42.6C4⋊25C2, (C4×D4).265C22, (C4×Q8).252C22, C42.12C4⋊12C2, C23.46(C22⋊C4), (C2×C42).165C22, C2.14(C24.4C4), C2.6(C23.24D4), C2.11(C42⋊C22), (C4×C4○D4).3C2, C4⋊C4.181(C2×C4), (C2×C4○D4).15C4, (C2×D4).193(C2×C4), (C2×C4).1451(C2×D4), (C2×Q8).176(C2×C4), (C2×C4).314(C22×C4), (C22×C4).187(C2×C4), (C2×C4).315(C22⋊C4), C22.164(C2×C22⋊C4), SmallGroup(128,220)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C42.374D4
G = < a,b,c,d | a4=b4=1, c4=a2b2, d2=b, ab=ba, cac-1=dad-1=ab2, bc=cb, bd=db, dcd-1=a2b-1c3 >
Subgroups: 228 in 123 conjugacy classes, 50 normal (36 characteristic)
C1, C2, C2, C4, C4, C4, C22, C22, C8, C2×C4, C2×C4, D4, D4, Q8, Q8, C23, C23, C42, C42, C22⋊C4, C4⋊C4, C4⋊C4, C2×C8, C22×C4, C22×C4, C2×D4, C2×D4, C2×Q8, C4○D4, C4×C8, C8⋊C4, C22⋊C8, C4⋊C8, C4⋊C8, C2×C42, C2×C42, C42⋊C2, C42⋊C2, C4×D4, C4×D4, C4×Q8, C2×C4○D4, D4⋊C8, Q8⋊C8, C42.12C4, C42.6C4, C4×C4○D4, C42.374D4
Quotients: C1, C2, C4, C22, C2×C4, D4, C23, C22⋊C4, M4(2), C22×C4, C2×D4, C2×C22⋊C4, C2×M4(2), C4○D8, C24.4C4, C23.24D4, C42⋊C22, C42.374D4
►On 64 points
(1 42 38 26)(2 31 39 47)(3 44 40 28)(4 25 33 41)(5 46 34 30)(6 27 35 43)(7 48 36 32)(8 29 37 45)(9 59 50 18)(10 23 51 64)(11 61 52 20)(12 17 53 58)(13 63 54 22)(14 19 55 60)(15 57 56 24)(16 21 49 62)
(1 61 34 24)(2 62 35 17)(3 63 36 18)(4 64 37 19)(5 57 38 20)(6 58 39 21)(7 59 40 22)(8 60 33 23)(9 44 54 32)(10 45 55 25)(11 46 56 26)(12 47 49 27)(13 48 50 28)(14 41 51 29)(15 42 52 30)(16 43 53 31)
(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 23 61 8 34 60 24 33)(2 7 62 59 35 40 17 22)(3 58 63 39 36 21 18 6)(4 38 64 20 37 5 19 57)(9 47 44 49 54 27 32 12)(10 56 45 26 55 11 25 46)(13 43 48 53 50 31 28 16)(14 52 41 30 51 15 29 42)
G:=sub<Sym(64)| (1,42,38,26)(2,31,39,47)(3,44,40,28)(4,25,33,41)(5,46,34,30)(6,27,35,43)(7,48,36,32)(8,29,37,45)(9,59,50,18)(10,23,51,64)(11,61,52,20)(12,17,53,58)(13,63,54,22)(14,19,55,60)(15,57,56,24)(16,21,49,62), (1,61,34,24)(2,62,35,17)(3,63,36,18)(4,64,37,19)(5,57,38,20)(6,58,39,21)(7,59,40,22)(8,60,33,23)(9,44,54,32)(10,45,55,25)(11,46,56,26)(12,47,49,27)(13,48,50,28)(14,41,51,29)(15,42,52,30)(16,43,53,31), (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), (1,23,61,8,34,60,24,33)(2,7,62,59,35,40,17,22)(3,58,63,39,36,21,18,6)(4,38,64,20,37,5,19,57)(9,47,44,49,54,27,32,12)(10,56,45,26,55,11,25,46)(13,43,48,53,50,31,28,16)(14,52,41,30,51,15,29,42)>;
G:=Group( (1,42,38,26)(2,31,39,47)(3,44,40,28)(4,25,33,41)(5,46,34,30)(6,27,35,43)(7,48,36,32)(8,29,37,45)(9,59,50,18)(10,23,51,64)(11,61,52,20)(12,17,53,58)(13,63,54,22)(14,19,55,60)(15,57,56,24)(16,21,49,62), (1,61,34,24)(2,62,35,17)(3,63,36,18)(4,64,37,19)(5,57,38,20)(6,58,39,21)(7,59,40,22)(8,60,33,23)(9,44,54,32)(10,45,55,25)(11,46,56,26)(12,47,49,27)(13,48,50,28)(14,41,51,29)(15,42,52,30)(16,43,53,31), (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), (1,23,61,8,34,60,24,33)(2,7,62,59,35,40,17,22)(3,58,63,39,36,21,18,6)(4,38,64,20,37,5,19,57)(9,47,44,49,54,27,32,12)(10,56,45,26,55,11,25,46)(13,43,48,53,50,31,28,16)(14,52,41,30,51,15,29,42) );
G=PermutationGroup([[(1,42,38,26),(2,31,39,47),(3,44,40,28),(4,25,33,41),(5,46,34,30),(6,27,35,43),(7,48,36,32),(8,29,37,45),(9,59,50,18),(10,23,51,64),(11,61,52,20),(12,17,53,58),(13,63,54,22),(14,19,55,60),(15,57,56,24),(16,21,49,62)], [(1,61,34,24),(2,62,35,17),(3,63,36,18),(4,64,37,19),(5,57,38,20),(6,58,39,21),(7,59,40,22),(8,60,33,23),(9,44,54,32),(10,45,55,25),(11,46,56,26),(12,47,49,27),(13,48,50,28),(14,41,51,29),(15,42,52,30),(16,43,53,31)], [(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)], [(1,23,61,8,34,60,24,33),(2,7,62,59,35,40,17,22),(3,58,63,39,36,21,18,6),(4,38,64,20,37,5,19,57),(9,47,44,49,54,27,32,12),(10,56,45,26,55,11,25,46),(13,43,48,53,50,31,28,16),(14,52,41,30,51,15,29,42)]])
38 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 4A | 4B | 4C | 4D | 4E | ··· | 4L | 4M | ··· | 4S | 8A | ··· | 8H | 8I | 8J | 8K | 8L |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 4 | ··· | 4 | 8 | ··· | 8 | 8 | 8 | 8 | 8 |
| size | 1 | 1 | 1 | 1 | 4 | 4 | 4 | 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 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 4 |
| type | + | + | + | + | + | + | + | + | ||||||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C4 | C4 | C4 | C4 | D4 | D4 | M4(2) | M4(2) | C4○D8 | C42⋊C22 |
| kernel | C42.374D4 | D4⋊C8 | Q8⋊C8 | C42.12C4 | C42.6C4 | C4×C4○D4 | C42⋊C2 | C4×D4 | C4×Q8 | C2×C4○D4 | C42 | C22×C4 | D4 | Q8 | C4 | C2 |
| # reps | 1 | 2 | 2 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 8 | 2 |
Matrix representation of C42.374D4 ►in GL4(𝔽17) generated by
| 13 | 0 | 0 | 0 |
| 0 | 13 | 0 | 0 |
| 0 | 0 | 16 | 0 |
| 0 | 0 | 16 | 1 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 13 | 0 |
| 0 | 0 | 0 | 13 |
| 3 | 14 | 0 | 0 |
| 3 | 3 | 0 | 0 |
| 0 | 0 | 1 | 15 |
| 0 | 0 | 7 | 16 |
| 3 | 14 | 0 | 0 |
| 14 | 14 | 0 | 0 |
| 0 | 0 | 16 | 2 |
| 0 | 0 | 6 | 1 |
G:=sub<GL(4,GF(17))| [13,0,0,0,0,13,0,0,0,0,16,16,0,0,0,1],[1,0,0,0,0,1,0,0,0,0,13,0,0,0,0,13],[3,3,0,0,14,3,0,0,0,0,1,7,0,0,15,16],[3,14,0,0,14,14,0,0,0,0,16,6,0,0,2,1] >;
C42.374D4 in GAP, Magma, Sage, TeX
C_4^2._{374}D_4 % in TeX
G:=Group("C4^2.374D4"); // GroupNames label
G:=SmallGroup(128,220);
// by ID
G=gap.SmallGroup(128,220);
# by ID
G:=PCGroup([7,-2,2,2,-2,2,-2,2,112,141,1430,184,1123,570,136,172]);
// 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=d*a*d^-1=a*b^2,b*c=c*b,b*d=d*b,d*c*d^-1=a^2*b^-1*c^3>;
// generators/relations