p-group, metabelian, nilpotent (class 2), monomial
Aliases: M4(2)⋊23D4, C42.682C23, (C8×D4)⋊40C2, C4⋊2(C8○D4), C8.87(C2×D4), C4⋊Q8.29C4, C8⋊6D4⋊35C2, C8⋊9D4⋊34C2, C4.100(C4×D4), M4(2)○(C4⋊C8), C4⋊1D4.18C4, C4⋊D4.20C4, C22.20(C4×D4), C4⋊C8.356C22, (C4×M4(2))⋊34C2, (C4×C8).328C22, (C2×C4).654C24, (C2×C8).407C23, C42.211(C2×C4), C4.4D4.16C4, (C4×D4).56C22, C4.200(C22×D4), C23.37(C22×C4), C8⋊C4.156C22, C2.18(Q8○M4(2)), C22⋊C8.231C22, C22.181(C23×C4), (C2×C42).761C22, (C22×C8).435C22, (C22×C4).921C23, (C2×M4(2)).349C22, C22.26C24.26C2, (C2×C4⋊C8)⋊49C2, C2.52(C2×C4×D4), C4⋊C8○(C2×M4(2)), (C2×C8○D4)⋊24C2, C2.19(C2×C8○D4), C4⋊C4.161(C2×C4), C4.305(C2×C4○D4), (C2×D4).174(C2×C4), (C2×C4).1086(C2×D4), C22⋊C4.37(C2×C4), (C2×Q8).157(C2×C4), (C22×C8)⋊C2⋊31C2, (C2×C4).898(C4○D4), (C2×C4).263(C22×C4), (C22×C4).344(C2×C4), (C2×C4○D4).288C22, SmallGroup(128,1667)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for M4(2)⋊23D4
G = < a,b,c,d | a8=b2=c4=d2=1, bab=dad=a5, ac=ca, bc=cb, dbd=a4b, dcd=c-1 >
Subgroups: 364 in 246 conjugacy classes, 142 normal (32 characteristic)
C1, C2, C2, C4, C4, C4, C22, C22, C22, C8, C8, C2×C4, C2×C4, C2×C4, D4, Q8, C23, C23, C42, C22⋊C4, C4⋊C4, C2×C8, C2×C8, M4(2), M4(2), C22×C4, C22×C4, C2×D4, C2×Q8, C4○D4, C4×C8, C8⋊C4, C22⋊C8, C4⋊C8, C4⋊C8, C2×C42, C4×D4, C4⋊D4, C4.4D4, C4⋊1D4, C4⋊Q8, C22×C8, C2×M4(2), C2×M4(2), C8○D4, C2×C4○D4, C4×M4(2), (C22×C8)⋊C2, C2×C4⋊C8, C8×D4, C8⋊9D4, C8⋊6D4, C22.26C24, C2×C8○D4, M4(2)⋊23D4
Quotients: C1, C2, C4, C22, C2×C4, D4, C23, C22×C4, C2×D4, C4○D4, C24, C4×D4, C8○D4, C23×C4, C22×D4, C2×C4○D4, C2×C4×D4, C2×C8○D4, Q8○M4(2), M4(2)⋊23D4
►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 16)(2 13)(3 10)(4 15)(5 12)(6 9)(7 14)(8 11)(17 41)(18 46)(19 43)(20 48)(21 45)(22 42)(23 47)(24 44)(25 58)(26 63)(27 60)(28 57)(29 62)(30 59)(31 64)(32 61)(33 49)(34 54)(35 51)(36 56)(37 53)(38 50)(39 55)(40 52)
(1 25 43 55)(2 26 44 56)(3 27 45 49)(4 28 46 50)(5 29 47 51)(6 30 48 52)(7 31 41 53)(8 32 42 54)(9 59 20 40)(10 60 21 33)(11 61 22 34)(12 62 23 35)(13 63 24 36)(14 64 17 37)(15 57 18 38)(16 58 19 39)
(1 33)(2 38)(3 35)(4 40)(5 37)(6 34)(7 39)(8 36)(9 50)(10 55)(11 52)(12 49)(13 54)(14 51)(15 56)(16 53)(17 29)(18 26)(19 31)(20 28)(21 25)(22 30)(23 27)(24 32)(41 58)(42 63)(43 60)(44 57)(45 62)(46 59)(47 64)(48 61)
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), (1,16)(2,13)(3,10)(4,15)(5,12)(6,9)(7,14)(8,11)(17,41)(18,46)(19,43)(20,48)(21,45)(22,42)(23,47)(24,44)(25,58)(26,63)(27,60)(28,57)(29,62)(30,59)(31,64)(32,61)(33,49)(34,54)(35,51)(36,56)(37,53)(38,50)(39,55)(40,52), (1,25,43,55)(2,26,44,56)(3,27,45,49)(4,28,46,50)(5,29,47,51)(6,30,48,52)(7,31,41,53)(8,32,42,54)(9,59,20,40)(10,60,21,33)(11,61,22,34)(12,62,23,35)(13,63,24,36)(14,64,17,37)(15,57,18,38)(16,58,19,39), (1,33)(2,38)(3,35)(4,40)(5,37)(6,34)(7,39)(8,36)(9,50)(10,55)(11,52)(12,49)(13,54)(14,51)(15,56)(16,53)(17,29)(18,26)(19,31)(20,28)(21,25)(22,30)(23,27)(24,32)(41,58)(42,63)(43,60)(44,57)(45,62)(46,59)(47,64)(48,61)>;
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), (1,16)(2,13)(3,10)(4,15)(5,12)(6,9)(7,14)(8,11)(17,41)(18,46)(19,43)(20,48)(21,45)(22,42)(23,47)(24,44)(25,58)(26,63)(27,60)(28,57)(29,62)(30,59)(31,64)(32,61)(33,49)(34,54)(35,51)(36,56)(37,53)(38,50)(39,55)(40,52), (1,25,43,55)(2,26,44,56)(3,27,45,49)(4,28,46,50)(5,29,47,51)(6,30,48,52)(7,31,41,53)(8,32,42,54)(9,59,20,40)(10,60,21,33)(11,61,22,34)(12,62,23,35)(13,63,24,36)(14,64,17,37)(15,57,18,38)(16,58,19,39), (1,33)(2,38)(3,35)(4,40)(5,37)(6,34)(7,39)(8,36)(9,50)(10,55)(11,52)(12,49)(13,54)(14,51)(15,56)(16,53)(17,29)(18,26)(19,31)(20,28)(21,25)(22,30)(23,27)(24,32)(41,58)(42,63)(43,60)(44,57)(45,62)(46,59)(47,64)(48,61) );
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)], [(1,16),(2,13),(3,10),(4,15),(5,12),(6,9),(7,14),(8,11),(17,41),(18,46),(19,43),(20,48),(21,45),(22,42),(23,47),(24,44),(25,58),(26,63),(27,60),(28,57),(29,62),(30,59),(31,64),(32,61),(33,49),(34,54),(35,51),(36,56),(37,53),(38,50),(39,55),(40,52)], [(1,25,43,55),(2,26,44,56),(3,27,45,49),(4,28,46,50),(5,29,47,51),(6,30,48,52),(7,31,41,53),(8,32,42,54),(9,59,20,40),(10,60,21,33),(11,61,22,34),(12,62,23,35),(13,63,24,36),(14,64,17,37),(15,57,18,38),(16,58,19,39)], [(1,33),(2,38),(3,35),(4,40),(5,37),(6,34),(7,39),(8,36),(9,50),(10,55),(11,52),(12,49),(13,54),(14,51),(15,56),(16,53),(17,29),(18,26),(19,31),(20,28),(21,25),(22,30),(23,27),(24,32),(41,58),(42,63),(43,60),(44,57),(45,62),(46,59),(47,64),(48,61)]])
50 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 2H | 2I | 4A | 4B | 4C | 4D | 4E | ··· | 4J | 4K | ··· | 4P | 8A | ··· | 8P | 8Q | ··· | 8X |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 4 | ··· | 4 | 8 | ··· | 8 | 8 | ··· | 8 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 4 | 4 | 4 | 4 | 1 | 1 | 1 | 1 | 2 | ··· | 2 | 4 | ··· | 4 | 2 | ··· | 2 | 4 | ··· | 4 |
50 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | |||||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C4 | C4 | C4 | C4 | D4 | C4○D4 | C8○D4 | Q8○M4(2) |
| kernel | M4(2)⋊23D4 | C4×M4(2) | (C22×C8)⋊C2 | C2×C4⋊C8 | C8×D4 | C8⋊9D4 | C8⋊6D4 | C22.26C24 | C2×C8○D4 | C4⋊D4 | C4.4D4 | C4⋊1D4 | C4⋊Q8 | M4(2) | C2×C4 | C4 | C2 |
| # reps | 1 | 1 | 2 | 1 | 2 | 4 | 2 | 1 | 2 | 8 | 4 | 2 | 2 | 4 | 4 | 8 | 2 |
Matrix representation of M4(2)⋊23D4 ►in GL4(𝔽17) generated by
| 16 | 0 | 0 | 0 |
| 0 | 16 | 0 | 0 |
| 0 | 0 | 9 | 0 |
| 0 | 0 | 8 | 8 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 16 | 15 |
| 0 | 0 | 0 | 1 |
| 0 | 16 | 0 | 0 |
| 1 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 16 | 0 | 0 |
| 16 | 0 | 0 | 0 |
| 0 | 0 | 13 | 9 |
| 0 | 0 | 4 | 4 |
G:=sub<GL(4,GF(17))| [16,0,0,0,0,16,0,0,0,0,9,8,0,0,0,8],[1,0,0,0,0,1,0,0,0,0,16,0,0,0,15,1],[0,1,0,0,16,0,0,0,0,0,1,0,0,0,0,1],[0,16,0,0,16,0,0,0,0,0,13,4,0,0,9,4] >;
M4(2)⋊23D4 in GAP, Magma, Sage, TeX
M_4(2)\rtimes_{23}D_4 % in TeX
G:=Group("M4(2):23D4"); // GroupNames label
G:=SmallGroup(128,1667);
// by ID
G=gap.SmallGroup(128,1667);
# by ID
G:=PCGroup([7,-2,2,2,2,-2,2,-2,224,253,184,1018,521,124]);
// Polycyclic
G:=Group<a,b,c,d|a^8=b^2=c^4=d^2=1,b*a*b=d*a*d=a^5,a*c=c*a,b*c=c*b,d*b*d=a^4*b,d*c*d=c^-1>;
// generators/relations