p-group, metabelian, nilpotent (class 3), monomial, rational
Aliases: M4(2)⋊5Q8, C42.253D4, C42.719C23, C8⋊Q8⋊6C2, C8⋊1(C2×Q8), C8⋊3Q8⋊5C2, C8⋊2Q8⋊28C2, C4.42(C4⋊Q8), C4.16(C22×Q8), C4⋊C4.104C23, C4.22(C8⋊C22), (C2×C8).274C23, (C4×C8).179C22, (C2×C4).363C24, C23.686(C2×D4), (C22×C4).473D4, (C4×M4(2)).4C2, C4⋊Q8.287C22, C22.18(C4⋊Q8), C2.D8.94C22, C4.Q8.24C22, C4.22(C8.C22), C8⋊C4.124C22, (C2×C42).860C22, C22.623(C22×D4), M4(2)⋊C4.14C2, (C22×C4).1047C23, C42.C2.118C22, C42⋊C2.148C22, (C2×M4(2)).277C22, C23.37C23.33C2, C2.33(C2×C4⋊Q8), (C2×C4⋊Q8).52C2, (C2×C4).863(C2×D4), C2.44(C2×C8⋊C22), (C2×C4).107(C2×Q8), C2.44(C2×C8.C22), (C2×C4⋊C4).634C22, SmallGroup(128,1897)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Subgroups: 324 in 190 conjugacy classes, 112 normal (26 characteristic)
C1, C2 [×3], C2 [×2], C4 [×8], C4 [×10], C22, C22 [×2], C22 [×2], C8 [×8], C2×C4 [×6], C2×C4 [×4], C2×C4 [×16], Q8 [×12], C23, C42 [×4], C42 [×2], C22⋊C4 [×2], C4⋊C4 [×8], C4⋊C4 [×12], C2×C8 [×4], M4(2) [×8], C22×C4 [×3], C22×C4 [×2], C2×Q8 [×10], C4×C8 [×2], C8⋊C4 [×2], C4.Q8 [×8], C2.D8 [×8], C2×C42, C2×C4⋊C4 [×2], C2×C4⋊C4, C42⋊C2 [×2], C4×Q8 [×2], C22⋊Q8 [×2], C42.C2 [×2], C4⋊Q8 [×2], C4⋊Q8 [×4], C4⋊Q8 [×2], C2×M4(2) [×2], C22×Q8, C4×M4(2), M4(2)⋊C4 [×4], C8⋊3Q8 [×2], C8⋊2Q8 [×2], C8⋊Q8 [×4], C2×C4⋊Q8, C23.37C23, M4(2)⋊5Q8
Quotients:
C1, C2 [×15], C22 [×35], D4 [×4], Q8 [×8], C23 [×15], C2×D4 [×6], C2×Q8 [×12], C24, C4⋊Q8 [×4], C8⋊C22 [×2], C8.C22 [×2], C22×D4, C22×Q8 [×2], C2×C4⋊Q8, C2×C8⋊C22, C2×C8.C22, M4(2)⋊5Q8
Generators and relations
G = < a,b,c,d | a8=b2=c4=1, d2=c2, bab=a5, ac=ca, dad-1=a-1, bc=cb, bd=db, dcd-1=c-1 >
►On 64 points
(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 44)(2 41)(3 46)(4 43)(5 48)(6 45)(7 42)(8 47)(9 22)(10 19)(11 24)(12 21)(13 18)(14 23)(15 20)(16 17)(25 53)(26 50)(27 55)(28 52)(29 49)(30 54)(31 51)(32 56)(33 64)(34 61)(35 58)(36 63)(37 60)(38 57)(39 62)(40 59)
(1 9 44 22)(2 10 45 23)(3 11 46 24)(4 12 47 17)(5 13 48 18)(6 14 41 19)(7 15 42 20)(8 16 43 21)(25 63 49 40)(26 64 50 33)(27 57 51 34)(28 58 52 35)(29 59 53 36)(30 60 54 37)(31 61 55 38)(32 62 56 39)
(1 52 44 28)(2 51 45 27)(3 50 46 26)(4 49 47 25)(5 56 48 32)(6 55 41 31)(7 54 42 30)(8 53 43 29)(9 58 22 35)(10 57 23 34)(11 64 24 33)(12 63 17 40)(13 62 18 39)(14 61 19 38)(15 60 20 37)(16 59 21 36)
G:=sub<Sym(64)| (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), (1,44)(2,41)(3,46)(4,43)(5,48)(6,45)(7,42)(8,47)(9,22)(10,19)(11,24)(12,21)(13,18)(14,23)(15,20)(16,17)(25,53)(26,50)(27,55)(28,52)(29,49)(30,54)(31,51)(32,56)(33,64)(34,61)(35,58)(36,63)(37,60)(38,57)(39,62)(40,59), (1,9,44,22)(2,10,45,23)(3,11,46,24)(4,12,47,17)(5,13,48,18)(6,14,41,19)(7,15,42,20)(8,16,43,21)(25,63,49,40)(26,64,50,33)(27,57,51,34)(28,58,52,35)(29,59,53,36)(30,60,54,37)(31,61,55,38)(32,62,56,39), (1,52,44,28)(2,51,45,27)(3,50,46,26)(4,49,47,25)(5,56,48,32)(6,55,41,31)(7,54,42,30)(8,53,43,29)(9,58,22,35)(10,57,23,34)(11,64,24,33)(12,63,17,40)(13,62,18,39)(14,61,19,38)(15,60,20,37)(16,59,21,36)>;
G:=Group( (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), (1,44)(2,41)(3,46)(4,43)(5,48)(6,45)(7,42)(8,47)(9,22)(10,19)(11,24)(12,21)(13,18)(14,23)(15,20)(16,17)(25,53)(26,50)(27,55)(28,52)(29,49)(30,54)(31,51)(32,56)(33,64)(34,61)(35,58)(36,63)(37,60)(38,57)(39,62)(40,59), (1,9,44,22)(2,10,45,23)(3,11,46,24)(4,12,47,17)(5,13,48,18)(6,14,41,19)(7,15,42,20)(8,16,43,21)(25,63,49,40)(26,64,50,33)(27,57,51,34)(28,58,52,35)(29,59,53,36)(30,60,54,37)(31,61,55,38)(32,62,56,39), (1,52,44,28)(2,51,45,27)(3,50,46,26)(4,49,47,25)(5,56,48,32)(6,55,41,31)(7,54,42,30)(8,53,43,29)(9,58,22,35)(10,57,23,34)(11,64,24,33)(12,63,17,40)(13,62,18,39)(14,61,19,38)(15,60,20,37)(16,59,21,36) );
G=PermutationGroup([(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)], [(1,44),(2,41),(3,46),(4,43),(5,48),(6,45),(7,42),(8,47),(9,22),(10,19),(11,24),(12,21),(13,18),(14,23),(15,20),(16,17),(25,53),(26,50),(27,55),(28,52),(29,49),(30,54),(31,51),(32,56),(33,64),(34,61),(35,58),(36,63),(37,60),(38,57),(39,62),(40,59)], [(1,9,44,22),(2,10,45,23),(3,11,46,24),(4,12,47,17),(5,13,48,18),(6,14,41,19),(7,15,42,20),(8,16,43,21),(25,63,49,40),(26,64,50,33),(27,57,51,34),(28,58,52,35),(29,59,53,36),(30,60,54,37),(31,61,55,38),(32,62,56,39)], [(1,52,44,28),(2,51,45,27),(3,50,46,26),(4,49,47,25),(5,56,48,32),(6,55,41,31),(7,54,42,30),(8,53,43,29),(9,58,22,35),(10,57,23,34),(11,64,24,33),(12,63,17,40),(13,62,18,39),(14,61,19,38),(15,60,20,37),(16,59,21,36)])
Matrix representation ►G ⊆ GL6(𝔽17)
| 16 | 16 | 0 | 0 | 0 | 0 |
| 2 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 2 | 0 | 0 |
| 0 | 0 | 2 | 0 | 0 | 0 |
| 0 | 0 | 7 | 0 | 16 | 9 |
| 0 | 0 | 14 | 6 | 7 | 1 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 16 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 16 | 0 |
| 0 | 0 | 11 | 0 | 13 | 1 |
| 16 | 16 | 0 | 0 | 0 | 0 |
| 2 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 4 | 0 | 0 | 0 |
| 0 | 0 | 0 | 4 | 0 | 0 |
| 0 | 0 | 0 | 0 | 13 | 0 |
| 0 | 0 | 7 | 0 | 0 | 13 |
| 11 | 1 | 0 | 0 | 0 | 0 |
| 14 | 6 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 14 | 0 | 15 | 1 |
| 0 | 0 | 16 | 0 | 0 | 0 |
| 0 | 0 | 15 | 16 | 3 | 0 |
G:=sub<GL(6,GF(17))| [16,2,0,0,0,0,16,1,0,0,0,0,0,0,0,2,7,14,0,0,2,0,0,6,0,0,0,0,16,7,0,0,0,0,9,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,16,0,0,11,0,0,0,1,0,0,0,0,0,0,16,13,0,0,0,0,0,1],[16,2,0,0,0,0,16,1,0,0,0,0,0,0,4,0,0,7,0,0,0,4,0,0,0,0,0,0,13,0,0,0,0,0,0,13],[11,14,0,0,0,0,1,6,0,0,0,0,0,0,0,14,16,15,0,0,0,0,0,16,0,0,1,15,0,3,0,0,0,1,0,0] >;
32 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 4A | ··· | 4H | 4I | 4J | 4K | ··· | 4R | 8A | ··· | 8H |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 4 | 4 | 4 | ··· | 4 | 8 | ··· | 8 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 2 | ··· | 2 | 4 | 4 | 8 | ··· | 8 | 4 | ··· | 4 |
32 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | - | + | + | - |
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | D4 | Q8 | D4 | C8⋊C22 | C8.C22 |
| kernel | M4(2)⋊5Q8 | C4×M4(2) | M4(2)⋊C4 | C8⋊3Q8 | C8⋊2Q8 | C8⋊Q8 | C2×C4⋊Q8 | C23.37C23 | C42 | M4(2) | C22×C4 | C4 | C4 |
| # reps | 1 | 1 | 4 | 2 | 2 | 4 | 1 | 1 | 2 | 8 | 2 | 2 | 2 |
In GAP, Magma, Sage, TeX
M_{4(2)}\rtimes_5Q_8 % in TeX
G:=Group("M4(2):5Q8"); // GroupNames label
G:=SmallGroup(128,1897);
// by ID
G=gap.SmallGroup(128,1897);
# by ID
G:=PCGroup([7,-2,2,2,2,-2,2,-2,112,253,120,758,184,521,4037,124]);
// Polycyclic
G:=Group<a,b,c,d|a^8=b^2=c^4=1,d^2=c^2,b*a*b=a^5,a*c=c*a,d*a*d^-1=a^-1,b*c=c*b,b*d=d*b,d*c*d^-1=c^-1>;
// generators/relations