p-group, metabelian, nilpotent (class 3), monomial
Aliases: D4.2M4(2), C42.642C23, C8⋊C8⋊4C2, Q8⋊C8⋊39C2, C8⋊1C8⋊16C2, C4.Q8.9C4, D4⋊C8.13C2, C8⋊4Q8⋊29C2, (C8×D4).16C2, C2.6(C8○D8), (C2×C8).307D4, D4⋊C4.4C4, C4.34(C8○D4), Q8⋊C4.4C4, (C2×SD16).5C4, (C4×SD16).1C2, C2.12(C8⋊9D4), C4⋊C8.276C22, (C4×C8).314C22, C22.133(C4×D4), C4.28(C2×M4(2)), (C4×Q8).11C22, C4.143(C8⋊C22), (C4×D4).273C22, C2.5(SD16⋊C4), C4.137(C8.C22), (C2×C8).32(C2×C4), C4⋊C4.136(C2×C4), (C2×Q8).52(C2×C4), (C2×D4).154(C2×C4), (C2×C4).1478(C2×D4), (C2×C4).503(C4○D4), (C2×C4).334(C22×C4), SmallGroup(128,317)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for D4.M4(2)
G = < a,b,c,d | a4=b2=c8=1, d2=a2, bab=cac-1=dad-1=a-1, cbc-1=dbd-1=ab, dcd-1=a2c5 >
Subgroups: 152 in 84 conjugacy classes, 42 normal (40 characteristic)
C1, C2, C2, C4, C4, C22, C22, C8, C2×C4, C2×C4, D4, D4, Q8, C23, C42, C42, C22⋊C4, C4⋊C4, C4⋊C4, C2×C8, C2×C8, SD16, C22×C4, C2×D4, C2×Q8, C4×C8, C8⋊C4, C22⋊C8, D4⋊C4, Q8⋊C4, C4⋊C8, C4⋊C8, C4.Q8, C4×D4, C4×Q8, C22×C8, C2×SD16, C8⋊C8, D4⋊C8, Q8⋊C8, C8⋊1C8, C8×D4, C4×SD16, C8⋊4Q8, D4.M4(2)
Quotients: C1, C2, C4, C22, C2×C4, D4, C23, M4(2), C22×C4, C2×D4, C4○D4, C4×D4, C2×M4(2), C8○D4, C8⋊C22, C8.C22, C8⋊9D4, SD16⋊C4, C8○D8, D4.M4(2)
►On 64 points
(1 28 49 43)(2 44 50 29)(3 30 51 45)(4 46 52 31)(5 32 53 47)(6 48 54 25)(7 26 55 41)(8 42 56 27)(9 62 39 19)(10 20 40 63)(11 64 33 21)(12 22 34 57)(13 58 35 23)(14 24 36 59)(15 60 37 17)(16 18 38 61)
(1 47)(2 54)(3 41)(4 56)(5 43)(6 50)(7 45)(8 52)(9 13)(10 24)(11 15)(12 18)(14 20)(16 22)(17 64)(19 58)(21 60)(23 62)(25 29)(26 51)(27 31)(28 53)(30 55)(32 49)(33 37)(34 61)(35 39)(36 63)(38 57)(40 59)(42 46)(44 48)
(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 49 58)(2 63 50 20)(3 17 51 60)(4 57 52 22)(5 19 53 62)(6 59 54 24)(7 21 55 64)(8 61 56 18)(9 32 39 47)(10 44 40 29)(11 26 33 41)(12 46 34 31)(13 28 35 43)(14 48 36 25)(15 30 37 45)(16 42 38 27)
G:=sub<Sym(64)| (1,28,49,43)(2,44,50,29)(3,30,51,45)(4,46,52,31)(5,32,53,47)(6,48,54,25)(7,26,55,41)(8,42,56,27)(9,62,39,19)(10,20,40,63)(11,64,33,21)(12,22,34,57)(13,58,35,23)(14,24,36,59)(15,60,37,17)(16,18,38,61), (1,47)(2,54)(3,41)(4,56)(5,43)(6,50)(7,45)(8,52)(9,13)(10,24)(11,15)(12,18)(14,20)(16,22)(17,64)(19,58)(21,60)(23,62)(25,29)(26,51)(27,31)(28,53)(30,55)(32,49)(33,37)(34,61)(35,39)(36,63)(38,57)(40,59)(42,46)(44,48), (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), (1,23,49,58)(2,63,50,20)(3,17,51,60)(4,57,52,22)(5,19,53,62)(6,59,54,24)(7,21,55,64)(8,61,56,18)(9,32,39,47)(10,44,40,29)(11,26,33,41)(12,46,34,31)(13,28,35,43)(14,48,36,25)(15,30,37,45)(16,42,38,27)>;
G:=Group( (1,28,49,43)(2,44,50,29)(3,30,51,45)(4,46,52,31)(5,32,53,47)(6,48,54,25)(7,26,55,41)(8,42,56,27)(9,62,39,19)(10,20,40,63)(11,64,33,21)(12,22,34,57)(13,58,35,23)(14,24,36,59)(15,60,37,17)(16,18,38,61), (1,47)(2,54)(3,41)(4,56)(5,43)(6,50)(7,45)(8,52)(9,13)(10,24)(11,15)(12,18)(14,20)(16,22)(17,64)(19,58)(21,60)(23,62)(25,29)(26,51)(27,31)(28,53)(30,55)(32,49)(33,37)(34,61)(35,39)(36,63)(38,57)(40,59)(42,46)(44,48), (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), (1,23,49,58)(2,63,50,20)(3,17,51,60)(4,57,52,22)(5,19,53,62)(6,59,54,24)(7,21,55,64)(8,61,56,18)(9,32,39,47)(10,44,40,29)(11,26,33,41)(12,46,34,31)(13,28,35,43)(14,48,36,25)(15,30,37,45)(16,42,38,27) );
G=PermutationGroup([[(1,28,49,43),(2,44,50,29),(3,30,51,45),(4,46,52,31),(5,32,53,47),(6,48,54,25),(7,26,55,41),(8,42,56,27),(9,62,39,19),(10,20,40,63),(11,64,33,21),(12,22,34,57),(13,58,35,23),(14,24,36,59),(15,60,37,17),(16,18,38,61)], [(1,47),(2,54),(3,41),(4,56),(5,43),(6,50),(7,45),(8,52),(9,13),(10,24),(11,15),(12,18),(14,20),(16,22),(17,64),(19,58),(21,60),(23,62),(25,29),(26,51),(27,31),(28,53),(30,55),(32,49),(33,37),(34,61),(35,39),(36,63),(38,57),(40,59),(42,46),(44,48)], [(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)], [(1,23,49,58),(2,63,50,20),(3,17,51,60),(4,57,52,22),(5,19,53,62),(6,59,54,24),(7,21,55,64),(8,61,56,18),(9,32,39,47),(10,44,40,29),(11,26,33,41),(12,46,34,31),(13,28,35,43),(14,48,36,25),(15,30,37,45),(16,42,38,27)]])
38 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 4K | 4L | 8A | 8B | 8C | 8D | 8E | ··· | 8R | 8S | 8T |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 8 | 8 | 8 | 8 | 8 | ··· | 8 | 8 | 8 |
| size | 1 | 1 | 1 | 1 | 4 | 4 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 4 | 4 | 8 | 8 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 8 | 8 |
38 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | - | ||||||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C4 | C4 | C4 | C4 | D4 | C4○D4 | M4(2) | C8○D4 | C8○D8 | C8⋊C22 | C8.C22 |
| kernel | D4.M4(2) | C8⋊C8 | D4⋊C8 | Q8⋊C8 | C8⋊1C8 | C8×D4 | C4×SD16 | C8⋊4Q8 | D4⋊C4 | Q8⋊C4 | C4.Q8 | C2×SD16 | C2×C8 | C2×C4 | D4 | C4 | C2 | C4 | C4 |
| # reps | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 8 | 1 | 1 |
Matrix representation of D4.M4(2) ►in GL4(𝔽17) generated by
| 0 | 1 | 0 | 0 |
| 16 | 0 | 0 | 0 |
| 0 | 0 | 16 | 0 |
| 0 | 0 | 0 | 16 |
| 0 | 16 | 0 | 0 |
| 16 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 3 | 16 |
| 10 | 7 | 0 | 0 |
| 7 | 7 | 0 | 0 |
| 0 | 0 | 5 | 8 |
| 0 | 0 | 8 | 12 |
| 5 | 12 | 0 | 0 |
| 12 | 12 | 0 | 0 |
| 0 | 0 | 10 | 16 |
| 0 | 0 | 14 | 7 |
G:=sub<GL(4,GF(17))| [0,16,0,0,1,0,0,0,0,0,16,0,0,0,0,16],[0,16,0,0,16,0,0,0,0,0,1,3,0,0,0,16],[10,7,0,0,7,7,0,0,0,0,5,8,0,0,8,12],[5,12,0,0,12,12,0,0,0,0,10,14,0,0,16,7] >;
D4.M4(2) in GAP, Magma, Sage, TeX
D_4.M_4(2)
% in TeX
G:=Group("D4.M4(2)"); // GroupNames label
G:=SmallGroup(128,317);
// by ID
G=gap.SmallGroup(128,317);
# by ID
G:=PCGroup([7,-2,2,2,-2,2,-2,2,112,141,2102,100,1123,570,136,172]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^2=c^8=1,d^2=a^2,b*a*b=c*a*c^-1=d*a*d^-1=a^-1,c*b*c^-1=d*b*d^-1=a*b,d*c*d^-1=a^2*c^5>;
// generators/relations