direct product, p-group, metabelian, nilpotent (class 2), monomial
Aliases: C23×M4(2), C8⋊4C24, C25.9C4, C4.19C25, (C23×C8)⋊16C2, (C2×C8)⋊17C23, C4.55(C23×C4), C2.13(C24×C4), (C24×C4).15C2, (C23×C4).44C4, C4○(C22×M4(2)), M4(2)○(C22×C4), C24.132(C2×C4), (C2×C4).694C24, (C22×C8)⋊71C22, C22.49(C23×C4), (C23×C4).709C22, C23.235(C22×C4), (C22×C4).1652C23, (C2×C4)○2(C2×M4(2)), (C2×C4)○(C22×M4(2)), (C22×C4)○(C2×M4(2)), (C2×C4).576(C22×C4), (C22×C4).499(C2×C4), (C22×C4)○(C22×M4(2)), SmallGroup(128,2302)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Subgroups: 988 in 860 conjugacy classes, 732 normal (9 characteristic)
C1, C2, C2 [×14], C2 [×8], C4, C4 [×15], C22 [×43], C22 [×56], C8 [×16], C2×C4 [×120], C23 [×43], C23 [×56], C2×C8 [×56], M4(2) [×64], C22×C4 [×140], C24, C24 [×14], C24 [×8], C22×C8 [×28], C2×M4(2) [×112], C23×C4 [×2], C23×C4 [×28], C25, C23×C8 [×2], C22×M4(2) [×28], C24×C4, C23×M4(2)
Quotients:
C1, C2 [×31], C4 [×16], C22 [×155], C2×C4 [×120], C23 [×155], M4(2) [×8], C22×C4 [×140], C24 [×31], C2×M4(2) [×28], C23×C4 [×30], C25, C22×M4(2) [×14], C24×C4, C23×M4(2)
Generators and relations
G = < a,b,c,d,e | a2=b2=c2=d8=e2=1, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, cd=dc, ce=ec, ede=d5 >
►On 64 points
(1 15)(2 16)(3 9)(4 10)(5 11)(6 12)(7 13)(8 14)(17 48)(18 41)(19 42)(20 43)(21 44)(22 45)(23 46)(24 47)(25 62)(26 63)(27 64)(28 57)(29 58)(30 59)(31 60)(32 61)(33 52)(34 53)(35 54)(36 55)(37 56)(38 49)(39 50)(40 51)
(1 24)(2 17)(3 18)(4 19)(5 20)(6 21)(7 22)(8 23)(9 41)(10 42)(11 43)(12 44)(13 45)(14 46)(15 47)(16 48)(25 38)(26 39)(27 40)(28 33)(29 34)(30 35)(31 36)(32 37)(49 62)(50 63)(51 64)(52 57)(53 58)(54 59)(55 60)(56 61)
(1 59)(2 60)(3 61)(4 62)(5 63)(6 64)(7 57)(8 58)(9 32)(10 25)(11 26)(12 27)(13 28)(14 29)(15 30)(16 31)(17 55)(18 56)(19 49)(20 50)(21 51)(22 52)(23 53)(24 54)(33 45)(34 46)(35 47)(36 48)(37 41)(38 42)(39 43)(40 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)
(2 6)(4 8)(10 14)(12 16)(17 21)(19 23)(25 29)(27 31)(34 38)(36 40)(42 46)(44 48)(49 53)(51 55)(58 62)(60 64)
G:=sub<Sym(64)| (1,15)(2,16)(3,9)(4,10)(5,11)(6,12)(7,13)(8,14)(17,48)(18,41)(19,42)(20,43)(21,44)(22,45)(23,46)(24,47)(25,62)(26,63)(27,64)(28,57)(29,58)(30,59)(31,60)(32,61)(33,52)(34,53)(35,54)(36,55)(37,56)(38,49)(39,50)(40,51), (1,24)(2,17)(3,18)(4,19)(5,20)(6,21)(7,22)(8,23)(9,41)(10,42)(11,43)(12,44)(13,45)(14,46)(15,47)(16,48)(25,38)(26,39)(27,40)(28,33)(29,34)(30,35)(31,36)(32,37)(49,62)(50,63)(51,64)(52,57)(53,58)(54,59)(55,60)(56,61), (1,59)(2,60)(3,61)(4,62)(5,63)(6,64)(7,57)(8,58)(9,32)(10,25)(11,26)(12,27)(13,28)(14,29)(15,30)(16,31)(17,55)(18,56)(19,49)(20,50)(21,51)(22,52)(23,53)(24,54)(33,45)(34,46)(35,47)(36,48)(37,41)(38,42)(39,43)(40,44), (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), (2,6)(4,8)(10,14)(12,16)(17,21)(19,23)(25,29)(27,31)(34,38)(36,40)(42,46)(44,48)(49,53)(51,55)(58,62)(60,64)>;
G:=Group( (1,15)(2,16)(3,9)(4,10)(5,11)(6,12)(7,13)(8,14)(17,48)(18,41)(19,42)(20,43)(21,44)(22,45)(23,46)(24,47)(25,62)(26,63)(27,64)(28,57)(29,58)(30,59)(31,60)(32,61)(33,52)(34,53)(35,54)(36,55)(37,56)(38,49)(39,50)(40,51), (1,24)(2,17)(3,18)(4,19)(5,20)(6,21)(7,22)(8,23)(9,41)(10,42)(11,43)(12,44)(13,45)(14,46)(15,47)(16,48)(25,38)(26,39)(27,40)(28,33)(29,34)(30,35)(31,36)(32,37)(49,62)(50,63)(51,64)(52,57)(53,58)(54,59)(55,60)(56,61), (1,59)(2,60)(3,61)(4,62)(5,63)(6,64)(7,57)(8,58)(9,32)(10,25)(11,26)(12,27)(13,28)(14,29)(15,30)(16,31)(17,55)(18,56)(19,49)(20,50)(21,51)(22,52)(23,53)(24,54)(33,45)(34,46)(35,47)(36,48)(37,41)(38,42)(39,43)(40,44), (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), (2,6)(4,8)(10,14)(12,16)(17,21)(19,23)(25,29)(27,31)(34,38)(36,40)(42,46)(44,48)(49,53)(51,55)(58,62)(60,64) );
G=PermutationGroup([(1,15),(2,16),(3,9),(4,10),(5,11),(6,12),(7,13),(8,14),(17,48),(18,41),(19,42),(20,43),(21,44),(22,45),(23,46),(24,47),(25,62),(26,63),(27,64),(28,57),(29,58),(30,59),(31,60),(32,61),(33,52),(34,53),(35,54),(36,55),(37,56),(38,49),(39,50),(40,51)], [(1,24),(2,17),(3,18),(4,19),(5,20),(6,21),(7,22),(8,23),(9,41),(10,42),(11,43),(12,44),(13,45),(14,46),(15,47),(16,48),(25,38),(26,39),(27,40),(28,33),(29,34),(30,35),(31,36),(32,37),(49,62),(50,63),(51,64),(52,57),(53,58),(54,59),(55,60),(56,61)], [(1,59),(2,60),(3,61),(4,62),(5,63),(6,64),(7,57),(8,58),(9,32),(10,25),(11,26),(12,27),(13,28),(14,29),(15,30),(16,31),(17,55),(18,56),(19,49),(20,50),(21,51),(22,52),(23,53),(24,54),(33,45),(34,46),(35,47),(36,48),(37,41),(38,42),(39,43),(40,44)], [(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)], [(2,6),(4,8),(10,14),(12,16),(17,21),(19,23),(25,29),(27,31),(34,38),(36,40),(42,46),(44,48),(49,53),(51,55),(58,62),(60,64)])
Matrix representation ►G ⊆ GL5(𝔽17)
| 1 | 0 | 0 | 0 | 0 |
| 0 | 16 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 16 | 0 |
| 0 | 0 | 0 | 0 | 16 |
| 16 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 1 |
| 16 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 16 | 0 | 0 |
| 0 | 0 | 0 | 16 | 0 |
| 0 | 0 | 0 | 0 | 16 |
| 13 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 16 |
| 0 | 0 | 0 | 4 | 0 |
| 16 | 0 | 0 | 0 | 0 |
| 0 | 16 | 0 | 0 | 0 |
| 0 | 0 | 16 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 16 |
G:=sub<GL(5,GF(17))| [1,0,0,0,0,0,16,0,0,0,0,0,1,0,0,0,0,0,16,0,0,0,0,0,16],[16,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1],[16,0,0,0,0,0,1,0,0,0,0,0,16,0,0,0,0,0,16,0,0,0,0,0,16],[13,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,0,4,0,0,0,16,0],[16,0,0,0,0,0,16,0,0,0,0,0,16,0,0,0,0,0,1,0,0,0,0,0,16] >;
80 conjugacy classes
| class | 1 | 2A | ··· | 2O | 2P | ··· | 2W | 4A | ··· | 4P | 4Q | ··· | 4X | 8A | ··· | 8AF |
| order | 1 | 2 | ··· | 2 | 2 | ··· | 2 | 4 | ··· | 4 | 4 | ··· | 4 | 8 | ··· | 8 |
| size | 1 | 1 | ··· | 1 | 2 | ··· | 2 | 1 | ··· | 1 | 2 | ··· | 2 | 2 | ··· | 2 |
80 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 |
| type | + | + | + | + | |||
| image | C1 | C2 | C2 | C2 | C4 | C4 | M4(2) |
| kernel | C23×M4(2) | C23×C8 | C22×M4(2) | C24×C4 | C23×C4 | C25 | C23 |
| # reps | 1 | 2 | 28 | 1 | 30 | 2 | 16 |
In GAP, Magma, Sage, TeX
C_2^3\times M_{4(2)} % in TeX
G:=Group("C2^3xM4(2)"); // GroupNames label
G:=SmallGroup(128,2302);
// by ID
G=gap.SmallGroup(128,2302);
# by ID
G:=PCGroup([7,-2,2,2,2,2,-2,-2,224,925,124]);
// Polycyclic
G:=Group<a,b,c,d,e|a^2=b^2=c^2=d^8=e^2=1,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,b*c=c*b,b*d=d*b,b*e=e*b,c*d=d*c,c*e=e*c,e*d*e=d^5>;
// generators/relations