p-group, metabelian, nilpotent (class 3), monomial
Aliases: C42.389D4, C4⋊C8.9C4, (C2×C4).117D8, C22⋊C8.9C4, (C2×C4).51Q16, C42.39(C2×C4), (C2×C4).13C42, (C2×C4).91SD16, C4.9(C8.C4), (C22×C4).24Q8, C23.44(C4⋊C4), (C22×C4).723D4, C4.43(D4⋊C4), C22.2(C4.Q8), C22.2(C2.D8), C4.23(Q8⋊C4), C4⋊M4(2).7C2, C2.8(C4.C42), C2.8(C22.4Q16), (C2×C42).136C22, C2.9(M4(2)⋊4C4), C42.12C4.10C2, C22.51(C2.C42), (C2×C4⋊C8).7C2, (C2×C4).97(C4⋊C4), (C22×C4).156(C2×C4), (C2×C4).374(C22⋊C4), SmallGroup(128,33)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C42.389D4
G = < a,b,c,d | a4=b4=1, c4=a2, d2=a, ab=ba, ac=ca, ad=da, cbc-1=dbd-1=a2b-1, dcd-1=a-1bc3 >
Subgroups: 128 in 74 conjugacy classes, 42 normal (34 characteristic)
C1, C2, C2, C4, C4, C4, C22, C22, C8, C2×C4, C2×C4, C23, C42, C2×C8, M4(2), C22×C4, C4×C8, C22⋊C8, C4⋊C8, C4⋊C8, C2×C42, C22×C8, C2×M4(2), C2×C4⋊C8, C4⋊M4(2), C42.12C4, C42.389D4
Quotients: C1, C2, C4, C22, C2×C4, D4, Q8, C42, C22⋊C4, C4⋊C4, D8, SD16, Q16, C2.C42, D4⋊C4, Q8⋊C4, C4.Q8, C2.D8, C8.C4, C22.4Q16, C4.C42, M4(2)⋊4C4, C42.389D4
►On 64 points
(1 7 5 3)(2 8 6 4)(9 15 13 11)(10 16 14 12)(17 23 21 19)(18 24 22 20)(25 27 29 31)(26 28 30 32)(33 35 37 39)(34 36 38 40)(41 43 45 47)(42 44 46 48)(49 51 53 55)(50 52 54 56)(57 63 61 59)(58 64 62 60)
(1 60 14 21)(2 18 15 57)(3 62 16 23)(4 20 9 59)(5 64 10 17)(6 22 11 61)(7 58 12 19)(8 24 13 63)(25 44 39 52)(26 49 40 41)(27 46 33 54)(28 51 34 43)(29 48 35 56)(30 53 36 45)(31 42 37 50)(32 55 38 47)
(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 7 29 5 31 3 25)(2 47 8 41 6 43 4 45)(9 53 15 55 13 49 11 51)(10 37 16 39 14 33 12 35)(17 46 23 48 21 42 19 44)(18 34 24 36 22 38 20 40)(26 57 28 63 30 61 32 59)(50 58 52 64 54 62 56 60)
G:=sub<Sym(64)| (1,7,5,3)(2,8,6,4)(9,15,13,11)(10,16,14,12)(17,23,21,19)(18,24,22,20)(25,27,29,31)(26,28,30,32)(33,35,37,39)(34,36,38,40)(41,43,45,47)(42,44,46,48)(49,51,53,55)(50,52,54,56)(57,63,61,59)(58,64,62,60), (1,60,14,21)(2,18,15,57)(3,62,16,23)(4,20,9,59)(5,64,10,17)(6,22,11,61)(7,58,12,19)(8,24,13,63)(25,44,39,52)(26,49,40,41)(27,46,33,54)(28,51,34,43)(29,48,35,56)(30,53,36,45)(31,42,37,50)(32,55,38,47), (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), (1,27,7,29,5,31,3,25)(2,47,8,41,6,43,4,45)(9,53,15,55,13,49,11,51)(10,37,16,39,14,33,12,35)(17,46,23,48,21,42,19,44)(18,34,24,36,22,38,20,40)(26,57,28,63,30,61,32,59)(50,58,52,64,54,62,56,60)>;
G:=Group( (1,7,5,3)(2,8,6,4)(9,15,13,11)(10,16,14,12)(17,23,21,19)(18,24,22,20)(25,27,29,31)(26,28,30,32)(33,35,37,39)(34,36,38,40)(41,43,45,47)(42,44,46,48)(49,51,53,55)(50,52,54,56)(57,63,61,59)(58,64,62,60), (1,60,14,21)(2,18,15,57)(3,62,16,23)(4,20,9,59)(5,64,10,17)(6,22,11,61)(7,58,12,19)(8,24,13,63)(25,44,39,52)(26,49,40,41)(27,46,33,54)(28,51,34,43)(29,48,35,56)(30,53,36,45)(31,42,37,50)(32,55,38,47), (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), (1,27,7,29,5,31,3,25)(2,47,8,41,6,43,4,45)(9,53,15,55,13,49,11,51)(10,37,16,39,14,33,12,35)(17,46,23,48,21,42,19,44)(18,34,24,36,22,38,20,40)(26,57,28,63,30,61,32,59)(50,58,52,64,54,62,56,60) );
G=PermutationGroup([[(1,7,5,3),(2,8,6,4),(9,15,13,11),(10,16,14,12),(17,23,21,19),(18,24,22,20),(25,27,29,31),(26,28,30,32),(33,35,37,39),(34,36,38,40),(41,43,45,47),(42,44,46,48),(49,51,53,55),(50,52,54,56),(57,63,61,59),(58,64,62,60)], [(1,60,14,21),(2,18,15,57),(3,62,16,23),(4,20,9,59),(5,64,10,17),(6,22,11,61),(7,58,12,19),(8,24,13,63),(25,44,39,52),(26,49,40,41),(27,46,33,54),(28,51,34,43),(29,48,35,56),(30,53,36,45),(31,42,37,50),(32,55,38,47)], [(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)], [(1,27,7,29,5,31,3,25),(2,47,8,41,6,43,4,45),(9,53,15,55,13,49,11,51),(10,37,16,39,14,33,12,35),(17,46,23,48,21,42,19,44),(18,34,24,36,22,38,20,40),(26,57,28,63,30,61,32,59),(50,58,52,64,54,62,56,60)]])
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 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 |
| type | + | + | + | + | + | + | - | + | - | |||||
| image | C1 | C2 | C2 | C2 | C4 | C4 | D4 | D4 | Q8 | D8 | SD16 | Q16 | C8.C4 | M4(2)⋊4C4 |
| kernel | C42.389D4 | C2×C4⋊C8 | C4⋊M4(2) | C42.12C4 | C22⋊C8 | C4⋊C8 | C42 | C22×C4 | C22×C4 | C2×C4 | C2×C4 | C2×C4 | C4 | C2 |
| # reps | 1 | 1 | 1 | 1 | 4 | 8 | 2 | 1 | 1 | 2 | 4 | 2 | 8 | 2 |
Matrix representation of C42.389D4 ►in GL4(𝔽17) generated by
| 4 | 0 | 0 | 0 |
| 0 | 4 | 0 | 0 |
| 0 | 0 | 16 | 0 |
| 0 | 0 | 0 | 16 |
| 4 | 0 | 0 | 0 |
| 0 | 4 | 0 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 0 | 16 | 0 |
| 8 | 0 | 0 | 0 |
| 14 | 2 | 0 | 0 |
| 0 | 0 | 10 | 16 |
| 0 | 0 | 16 | 7 |
| 8 | 16 | 0 | 0 |
| 9 | 9 | 0 | 0 |
| 0 | 0 | 16 | 10 |
| 0 | 0 | 10 | 1 |
G:=sub<GL(4,GF(17))| [4,0,0,0,0,4,0,0,0,0,16,0,0,0,0,16],[4,0,0,0,0,4,0,0,0,0,0,16,0,0,1,0],[8,14,0,0,0,2,0,0,0,0,10,16,0,0,16,7],[8,9,0,0,16,9,0,0,0,0,16,10,0,0,10,1] >;
C42.389D4 in GAP, Magma, Sage, TeX
C_4^2._{389}D_4 % in TeX
G:=Group("C4^2.389D4"); // GroupNames label
G:=SmallGroup(128,33);
// by ID
G=gap.SmallGroup(128,33);
# by ID
G:=PCGroup([7,-2,2,-2,2,2,-2,2,56,85,120,758,723,184,248,3924,102]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^4=1,c^4=a^2,d^2=a,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c^-1=d*b*d^-1=a^2*b^-1,d*c*d^-1=a^-1*b*c^3>;
// generators/relations