p-group, metabelian, nilpotent (class 3), monomial
Aliases: D4⋊4M4(2), Q8⋊4M4(2), C42.402D4, C42.610C23, C4.33C4≀C2, D4⋊C8⋊28C2, Q8⋊C8⋊32C2, (C4×D4).17C4, (C4×Q8).17C4, C42.63(C2×C4), C4⋊C8.195C22, (C4×M4(2))⋊14C2, (C4×C8).312C22, (C22×C4).205D4, C4.21(C2×M4(2)), C4.133(C8⋊C22), C4⋊M4(2)⋊16C2, C42⋊C2.18C4, (C4×D4).266C22, (C4×Q8).253C22, C4.127(C8.C22), C23.47(C22⋊C4), (C2×C42).166C22, C2.15(C24.4C4), C2.6(C23.36D4), C2.9(C2×C4≀C2), (C4×C4○D4).4C2, C4⋊C4.182(C2×C4), (C2×C4○D4).16C4, (C2×D4).194(C2×C4), (C2×C4).1138(C2×D4), (C2×Q8).177(C2×C4), (C22×C4).188(C2×C4), (C2×C4).315(C22×C4), (C2×C4).316(C22⋊C4), C22.165(C2×C22⋊C4), SmallGroup(128,221)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for D4⋊4M4(2)
G = < a,b,c,d | a4=c8=d2=1, b2=a2, bab-1=cac-1=a-1, ad=da, cbc-1=ab, dbd=a2b, dcd=c5 >
Subgroups: 244 in 133 conjugacy classes, 52 normal (38 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, M4(2), C22×C4, C22×C4, C2×D4, C2×D4, C2×Q8, C4○D4, C4×C8, C8⋊C4, C4⋊C8, C4⋊C8, C2×C42, C2×C42, C42⋊C2, C42⋊C2, C4×D4, C4×D4, C4×Q8, C2×M4(2), C2×C4○D4, D4⋊C8, Q8⋊C8, C4×M4(2), C4⋊M4(2), C4×C4○D4, D4⋊4M4(2)
Quotients: C1, C2, C4, C22, C2×C4, D4, C23, C22⋊C4, M4(2), C22×C4, C2×D4, C4≀C2, C2×C22⋊C4, C2×M4(2), C8⋊C22, C8.C22, C24.4C4, C23.36D4, C2×C4≀C2, D4⋊4M4(2)
►On 64 points
(1 52 59 24)(2 17 60 53)(3 54 61 18)(4 19 62 55)(5 56 63 20)(6 21 64 49)(7 50 57 22)(8 23 58 51)(9 45 30 39)(10 40 31 46)(11 47 32 33)(12 34 25 48)(13 41 26 35)(14 36 27 42)(15 43 28 37)(16 38 29 44)
(1 37 59 43)(2 29 60 16)(3 39 61 45)(4 31 62 10)(5 33 63 47)(6 25 64 12)(7 35 57 41)(8 27 58 14)(9 54 30 18)(11 56 32 20)(13 50 26 22)(15 52 28 24)(17 38 53 44)(19 40 55 46)(21 34 49 48)(23 36 51 42)
(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 37)(2 34)(3 39)(4 36)(5 33)(6 38)(7 35)(8 40)(9 54)(10 51)(11 56)(12 53)(13 50)(14 55)(15 52)(16 49)(17 25)(18 30)(19 27)(20 32)(21 29)(22 26)(23 31)(24 28)(41 57)(42 62)(43 59)(44 64)(45 61)(46 58)(47 63)(48 60)
G:=sub<Sym(64)| (1,52,59,24)(2,17,60,53)(3,54,61,18)(4,19,62,55)(5,56,63,20)(6,21,64,49)(7,50,57,22)(8,23,58,51)(9,45,30,39)(10,40,31,46)(11,47,32,33)(12,34,25,48)(13,41,26,35)(14,36,27,42)(15,43,28,37)(16,38,29,44), (1,37,59,43)(2,29,60,16)(3,39,61,45)(4,31,62,10)(5,33,63,47)(6,25,64,12)(7,35,57,41)(8,27,58,14)(9,54,30,18)(11,56,32,20)(13,50,26,22)(15,52,28,24)(17,38,53,44)(19,40,55,46)(21,34,49,48)(23,36,51,42), (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), (1,37)(2,34)(3,39)(4,36)(5,33)(6,38)(7,35)(8,40)(9,54)(10,51)(11,56)(12,53)(13,50)(14,55)(15,52)(16,49)(17,25)(18,30)(19,27)(20,32)(21,29)(22,26)(23,31)(24,28)(41,57)(42,62)(43,59)(44,64)(45,61)(46,58)(47,63)(48,60)>;
G:=Group( (1,52,59,24)(2,17,60,53)(3,54,61,18)(4,19,62,55)(5,56,63,20)(6,21,64,49)(7,50,57,22)(8,23,58,51)(9,45,30,39)(10,40,31,46)(11,47,32,33)(12,34,25,48)(13,41,26,35)(14,36,27,42)(15,43,28,37)(16,38,29,44), (1,37,59,43)(2,29,60,16)(3,39,61,45)(4,31,62,10)(5,33,63,47)(6,25,64,12)(7,35,57,41)(8,27,58,14)(9,54,30,18)(11,56,32,20)(13,50,26,22)(15,52,28,24)(17,38,53,44)(19,40,55,46)(21,34,49,48)(23,36,51,42), (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), (1,37)(2,34)(3,39)(4,36)(5,33)(6,38)(7,35)(8,40)(9,54)(10,51)(11,56)(12,53)(13,50)(14,55)(15,52)(16,49)(17,25)(18,30)(19,27)(20,32)(21,29)(22,26)(23,31)(24,28)(41,57)(42,62)(43,59)(44,64)(45,61)(46,58)(47,63)(48,60) );
G=PermutationGroup([[(1,52,59,24),(2,17,60,53),(3,54,61,18),(4,19,62,55),(5,56,63,20),(6,21,64,49),(7,50,57,22),(8,23,58,51),(9,45,30,39),(10,40,31,46),(11,47,32,33),(12,34,25,48),(13,41,26,35),(14,36,27,42),(15,43,28,37),(16,38,29,44)], [(1,37,59,43),(2,29,60,16),(3,39,61,45),(4,31,62,10),(5,33,63,47),(6,25,64,12),(7,35,57,41),(8,27,58,14),(9,54,30,18),(11,56,32,20),(13,50,26,22),(15,52,28,24),(17,38,53,44),(19,40,55,46),(21,34,49,48),(23,36,51,42)], [(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)], [(1,37),(2,34),(3,39),(4,36),(5,33),(6,38),(7,35),(8,40),(9,54),(10,51),(11,56),(12,53),(13,50),(14,55),(15,52),(16,49),(17,25),(18,30),(19,27),(20,32),(21,29),(22,26),(23,31),(24,28),(41,57),(42,62),(43,59),(44,64),(45,61),(46,58),(47,63),(48,60)]])
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 | 4 |
| type | + | + | + | + | + | + | + | + | + | - | |||||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C4 | C4 | C4 | C4 | D4 | D4 | M4(2) | M4(2) | C4≀C2 | C8⋊C22 | C8.C22 |
| kernel | D4⋊4M4(2) | D4⋊C8 | Q8⋊C8 | C4×M4(2) | C4⋊M4(2) | C4×C4○D4 | C42⋊C2 | C4×D4 | C4×Q8 | C2×C4○D4 | C42 | C22×C4 | D4 | Q8 | C4 | C4 | C4 |
| # reps | 1 | 2 | 2 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 8 | 1 | 1 |
Matrix representation of D4⋊4M4(2) ►in GL4(𝔽17) generated by
| 0 | 16 | 0 | 0 |
| 1 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 4 | 0 | 0 |
| 4 | 0 | 0 | 0 |
| 0 | 0 | 16 | 0 |
| 0 | 0 | 0 | 16 |
| 6 | 6 | 0 | 0 |
| 6 | 11 | 0 | 0 |
| 0 | 0 | 10 | 15 |
| 0 | 0 | 14 | 7 |
| 0 | 13 | 0 | 0 |
| 4 | 0 | 0 | 0 |
| 0 | 0 | 16 | 0 |
| 0 | 0 | 7 | 1 |
G:=sub<GL(4,GF(17))| [0,1,0,0,16,0,0,0,0,0,1,0,0,0,0,1],[0,4,0,0,4,0,0,0,0,0,16,0,0,0,0,16],[6,6,0,0,6,11,0,0,0,0,10,14,0,0,15,7],[0,4,0,0,13,0,0,0,0,0,16,7,0,0,0,1] >;
D4⋊4M4(2) in GAP, Magma, Sage, TeX
D_4\rtimes_4M_4(2)
% in TeX
G:=Group("D4:4M4(2)"); // GroupNames label
G:=SmallGroup(128,221);
// by ID
G=gap.SmallGroup(128,221);
# by ID
G:=PCGroup([7,-2,2,2,-2,2,-2,2,112,141,1430,387,1123,570,136,172]);
// Polycyclic
G:=Group<a,b,c,d|a^4=c^8=d^2=1,b^2=a^2,b*a*b^-1=c*a*c^-1=a^-1,a*d=d*a,c*b*c^-1=a*b,d*b*d=a^2*b,d*c*d=c^5>;
// generators/relations