p-group, metabelian, nilpotent (class 3), monomial
Aliases: Q8⋊4M4(2), D4⋊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), (C2×C4).315(C22×C4), (C22×C4).188(C2×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 Q8⋊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 [×3], C2 [×3], C4 [×4], C4 [×2], C4 [×7], C22, C22 [×7], C8 [×6], C2×C4 [×6], C2×C4 [×17], D4 [×2], D4 [×5], Q8 [×2], Q8, C23, C23, C42 [×4], C42 [×3], C22⋊C4 [×3], C4⋊C4 [×2], C4⋊C4 [×2], C2×C8 [×4], M4(2) [×6], C22×C4 [×3], C22×C4 [×3], C2×D4, C2×D4, C2×Q8, C4○D4 [×4], C4×C8 [×2], C8⋊C4, C4⋊C8 [×2], C4⋊C8, C2×C42, C2×C42, C42⋊C2, C42⋊C2, C4×D4 [×2], C4×D4 [×2], C4×Q8 [×2], C2×M4(2) [×2], C2×C4○D4, D4⋊C8 [×2], Q8⋊C8 [×2], C4×M4(2), C4⋊M4(2), C4×C4○D4, Q8⋊4M4(2)
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C2×C4 [×6], D4 [×4], C23, C22⋊C4 [×4], M4(2) [×4], C22×C4, C2×D4 [×2], C4≀C2 [×2], C2×C22⋊C4, C2×M4(2) [×2], C8⋊C22, C8.C22, C24.4C4, C23.36D4, C2×C4≀C2, Q8⋊4M4(2)
►On 64 points
(1 52 59 22)(2 23 60 53)(3 54 61 24)(4 17 62 55)(5 56 63 18)(6 19 64 49)(7 50 57 20)(8 21 58 51)(9 45 40 27)(10 28 33 46)(11 47 34 29)(12 30 35 48)(13 41 36 31)(14 32 37 42)(15 43 38 25)(16 26 39 44)
(1 25 59 43)(2 39 60 16)(3 27 61 45)(4 33 62 10)(5 29 63 47)(6 35 64 12)(7 31 57 41)(8 37 58 14)(9 54 40 24)(11 56 34 18)(13 50 36 20)(15 52 38 22)(17 28 55 46)(19 30 49 48)(21 32 51 42)(23 26 53 44)
(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 25)(2 30)(3 27)(4 32)(5 29)(6 26)(7 31)(8 28)(9 54)(10 51)(11 56)(12 53)(13 50)(14 55)(15 52)(16 49)(17 37)(18 34)(19 39)(20 36)(21 33)(22 38)(23 35)(24 40)(41 57)(42 62)(43 59)(44 64)(45 61)(46 58)(47 63)(48 60)
G:=sub<Sym(64)| (1,52,59,22)(2,23,60,53)(3,54,61,24)(4,17,62,55)(5,56,63,18)(6,19,64,49)(7,50,57,20)(8,21,58,51)(9,45,40,27)(10,28,33,46)(11,47,34,29)(12,30,35,48)(13,41,36,31)(14,32,37,42)(15,43,38,25)(16,26,39,44), (1,25,59,43)(2,39,60,16)(3,27,61,45)(4,33,62,10)(5,29,63,47)(6,35,64,12)(7,31,57,41)(8,37,58,14)(9,54,40,24)(11,56,34,18)(13,50,36,20)(15,52,38,22)(17,28,55,46)(19,30,49,48)(21,32,51,42)(23,26,53,44), (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), (1,25)(2,30)(3,27)(4,32)(5,29)(6,26)(7,31)(8,28)(9,54)(10,51)(11,56)(12,53)(13,50)(14,55)(15,52)(16,49)(17,37)(18,34)(19,39)(20,36)(21,33)(22,38)(23,35)(24,40)(41,57)(42,62)(43,59)(44,64)(45,61)(46,58)(47,63)(48,60)>;
G:=Group( (1,52,59,22)(2,23,60,53)(3,54,61,24)(4,17,62,55)(5,56,63,18)(6,19,64,49)(7,50,57,20)(8,21,58,51)(9,45,40,27)(10,28,33,46)(11,47,34,29)(12,30,35,48)(13,41,36,31)(14,32,37,42)(15,43,38,25)(16,26,39,44), (1,25,59,43)(2,39,60,16)(3,27,61,45)(4,33,62,10)(5,29,63,47)(6,35,64,12)(7,31,57,41)(8,37,58,14)(9,54,40,24)(11,56,34,18)(13,50,36,20)(15,52,38,22)(17,28,55,46)(19,30,49,48)(21,32,51,42)(23,26,53,44), (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), (1,25)(2,30)(3,27)(4,32)(5,29)(6,26)(7,31)(8,28)(9,54)(10,51)(11,56)(12,53)(13,50)(14,55)(15,52)(16,49)(17,37)(18,34)(19,39)(20,36)(21,33)(22,38)(23,35)(24,40)(41,57)(42,62)(43,59)(44,64)(45,61)(46,58)(47,63)(48,60) );
G=PermutationGroup([(1,52,59,22),(2,23,60,53),(3,54,61,24),(4,17,62,55),(5,56,63,18),(6,19,64,49),(7,50,57,20),(8,21,58,51),(9,45,40,27),(10,28,33,46),(11,47,34,29),(12,30,35,48),(13,41,36,31),(14,32,37,42),(15,43,38,25),(16,26,39,44)], [(1,25,59,43),(2,39,60,16),(3,27,61,45),(4,33,62,10),(5,29,63,47),(6,35,64,12),(7,31,57,41),(8,37,58,14),(9,54,40,24),(11,56,34,18),(13,50,36,20),(15,52,38,22),(17,28,55,46),(19,30,49,48),(21,32,51,42),(23,26,53,44)], [(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)], [(1,25),(2,30),(3,27),(4,32),(5,29),(6,26),(7,31),(8,28),(9,54),(10,51),(11,56),(12,53),(13,50),(14,55),(15,52),(16,49),(17,37),(18,34),(19,39),(20,36),(21,33),(22,38),(23,35),(24,40),(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 | Q8⋊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 Q8⋊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] >;
Q8⋊4M4(2) in GAP, Magma, Sage, TeX
Q_8\rtimes_4M_{4(2}) % in TeX
G:=Group("Q8:4M4(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