p-group, metabelian, nilpotent (class 2), monomial
Aliases: C23.200C24, C24.547C23, C22.392+ (1+4), C22.242- (1+4), (C4×D4)⋊19C4, C42⋊17(C2×C4), C42⋊5C4⋊3C2, C42⋊8C4⋊12C2, C23.8Q8⋊4C2, C22.91(C23×C4), (C23×C4).44C22, (C2×C42).12C22, C23.7Q8⋊14C2, C23.222(C4○D4), C23.34D4⋊10C2, (C22×C4).465C23, C23.124(C22×C4), C24.C22⋊2C2, C23.23D4.3C2, C23.63C23⋊3C2, C22.3(C42⋊C2), C2.2(C22.32C24), (C22×D4).475C22, C2.11(C22.11C24), C2.C42.37C22, C2.8(C23.33C23), C2.2(C22.33C24), C4⋊C4⋊41(C2×C4), (C2×C4×D4).29C2, (C4×C22⋊C4)⋊7C2, C22⋊C4⋊38(C2×C4), (C22×C4)⋊22(C2×C4), (C2×D4).211(C2×C4), C22.85(C2×C4○D4), (C2×C4⋊C4).174C22, (C2×C2.C42)⋊9C2, (C2×C4).223(C22×C4), C2.22(C2×C42⋊C2), (C2×C22⋊C4).25C22, SmallGroup(128,1050)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Subgroups: 524 in 284 conjugacy classes, 140 normal (34 characteristic)
C1, C2 [×3], C2 [×4], C2 [×6], C4 [×18], C22 [×3], C22 [×8], C22 [×22], C2×C4 [×10], C2×C4 [×50], D4 [×8], C23, C23 [×8], C23 [×10], C42 [×4], C42 [×2], C22⋊C4 [×8], C22⋊C4 [×8], C4⋊C4 [×4], C4⋊C4 [×4], C22×C4 [×5], C22×C4 [×16], C22×C4 [×12], C2×D4 [×4], C2×D4 [×4], C24 [×2], C2.C42 [×4], C2.C42 [×10], C2×C42, C2×C42 [×2], C2×C22⋊C4 [×2], C2×C22⋊C4 [×6], C2×C4⋊C4, C2×C4⋊C4 [×4], C4×D4 [×8], C23×C4 [×2], C23×C4 [×2], C22×D4, C2×C2.C42, C4×C22⋊C4, C23.7Q8, C23.34D4, C42⋊8C4, C42⋊5C4, C23.8Q8 [×2], C23.23D4 [×2], C23.63C23 [×2], C24.C22 [×2], C2×C4×D4, C23.200C24
Quotients:
C1, C2 [×15], C4 [×8], C22 [×35], C2×C4 [×28], C23 [×15], C22×C4 [×14], C4○D4 [×4], C24, C42⋊C2 [×4], C23×C4, C2×C4○D4 [×2], 2+ (1+4) [×3], 2- (1+4), C2×C42⋊C2, C22.11C24, C23.33C23, C22.32C24 [×2], C22.33C24 [×2], C23.200C24
Generators and relations
G = < a,b,c,d,e,f,g | a2=b2=c2=e2=f2=1, d2=c, g2=b, ab=ba, ac=ca, ede=gdg-1=ad=da, fef=ae=ea, af=fa, ag=ga, bc=cb, fdf=bd=db, be=eb, bf=fb, bg=gb, cd=dc, ce=ec, cf=fc, cg=gc, eg=ge, fg=gf >
(1 11)(2 12)(3 9)(4 10)(5 38)(6 39)(7 40)(8 37)(13 41)(14 42)(15 43)(16 44)(17 45)(18 46)(19 47)(20 48)(21 49)(22 50)(23 51)(24 52)(25 53)(26 54)(27 55)(28 56)(29 57)(30 58)(31 59)(32 60)(33 63)(34 64)(35 61)(36 62)
(1 51)(2 52)(3 49)(4 50)(5 36)(6 33)(7 34)(8 35)(9 21)(10 22)(11 23)(12 24)(13 25)(14 26)(15 27)(16 28)(17 29)(18 30)(19 31)(20 32)(37 61)(38 62)(39 63)(40 64)(41 53)(42 54)(43 55)(44 56)(45 57)(46 58)(47 59)(48 60)
(1 3)(2 4)(5 7)(6 8)(9 11)(10 12)(13 15)(14 16)(17 19)(18 20)(21 23)(22 24)(25 27)(26 28)(29 31)(30 32)(33 35)(34 36)(37 39)(38 40)(41 43)(42 44)(45 47)(46 48)(49 51)(50 52)(53 55)(54 56)(57 59)(58 60)(61 63)(62 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 47)(2 20)(3 45)(4 18)(5 42)(6 15)(7 44)(8 13)(9 17)(10 46)(11 19)(12 48)(14 38)(16 40)(21 29)(22 58)(23 31)(24 60)(25 35)(26 62)(27 33)(28 64)(30 50)(32 52)(34 56)(36 54)(37 41)(39 43)(49 57)(51 59)(53 61)(55 63)
(2 52)(4 50)(5 62)(6 39)(7 64)(8 37)(10 22)(12 24)(14 26)(16 28)(17 45)(18 58)(19 47)(20 60)(29 57)(30 46)(31 59)(32 48)(33 63)(34 40)(35 61)(36 38)(42 54)(44 56)
(1 55 51 43)(2 28 52 16)(3 53 49 41)(4 26 50 14)(5 46 36 58)(6 19 33 31)(7 48 34 60)(8 17 35 29)(9 25 21 13)(10 54 22 42)(11 27 23 15)(12 56 24 44)(18 62 30 38)(20 64 32 40)(37 45 61 57)(39 47 63 59)
G:=sub<Sym(64)| (1,11)(2,12)(3,9)(4,10)(5,38)(6,39)(7,40)(8,37)(13,41)(14,42)(15,43)(16,44)(17,45)(18,46)(19,47)(20,48)(21,49)(22,50)(23,51)(24,52)(25,53)(26,54)(27,55)(28,56)(29,57)(30,58)(31,59)(32,60)(33,63)(34,64)(35,61)(36,62), (1,51)(2,52)(3,49)(4,50)(5,36)(6,33)(7,34)(8,35)(9,21)(10,22)(11,23)(12,24)(13,25)(14,26)(15,27)(16,28)(17,29)(18,30)(19,31)(20,32)(37,61)(38,62)(39,63)(40,64)(41,53)(42,54)(43,55)(44,56)(45,57)(46,58)(47,59)(48,60), (1,3)(2,4)(5,7)(6,8)(9,11)(10,12)(13,15)(14,16)(17,19)(18,20)(21,23)(22,24)(25,27)(26,28)(29,31)(30,32)(33,35)(34,36)(37,39)(38,40)(41,43)(42,44)(45,47)(46,48)(49,51)(50,52)(53,55)(54,56)(57,59)(58,60)(61,63)(62,64), (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), (1,47)(2,20)(3,45)(4,18)(5,42)(6,15)(7,44)(8,13)(9,17)(10,46)(11,19)(12,48)(14,38)(16,40)(21,29)(22,58)(23,31)(24,60)(25,35)(26,62)(27,33)(28,64)(30,50)(32,52)(34,56)(36,54)(37,41)(39,43)(49,57)(51,59)(53,61)(55,63), (2,52)(4,50)(5,62)(6,39)(7,64)(8,37)(10,22)(12,24)(14,26)(16,28)(17,45)(18,58)(19,47)(20,60)(29,57)(30,46)(31,59)(32,48)(33,63)(34,40)(35,61)(36,38)(42,54)(44,56), (1,55,51,43)(2,28,52,16)(3,53,49,41)(4,26,50,14)(5,46,36,58)(6,19,33,31)(7,48,34,60)(8,17,35,29)(9,25,21,13)(10,54,22,42)(11,27,23,15)(12,56,24,44)(18,62,30,38)(20,64,32,40)(37,45,61,57)(39,47,63,59)>;
G:=Group( (1,11)(2,12)(3,9)(4,10)(5,38)(6,39)(7,40)(8,37)(13,41)(14,42)(15,43)(16,44)(17,45)(18,46)(19,47)(20,48)(21,49)(22,50)(23,51)(24,52)(25,53)(26,54)(27,55)(28,56)(29,57)(30,58)(31,59)(32,60)(33,63)(34,64)(35,61)(36,62), (1,51)(2,52)(3,49)(4,50)(5,36)(6,33)(7,34)(8,35)(9,21)(10,22)(11,23)(12,24)(13,25)(14,26)(15,27)(16,28)(17,29)(18,30)(19,31)(20,32)(37,61)(38,62)(39,63)(40,64)(41,53)(42,54)(43,55)(44,56)(45,57)(46,58)(47,59)(48,60), (1,3)(2,4)(5,7)(6,8)(9,11)(10,12)(13,15)(14,16)(17,19)(18,20)(21,23)(22,24)(25,27)(26,28)(29,31)(30,32)(33,35)(34,36)(37,39)(38,40)(41,43)(42,44)(45,47)(46,48)(49,51)(50,52)(53,55)(54,56)(57,59)(58,60)(61,63)(62,64), (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), (1,47)(2,20)(3,45)(4,18)(5,42)(6,15)(7,44)(8,13)(9,17)(10,46)(11,19)(12,48)(14,38)(16,40)(21,29)(22,58)(23,31)(24,60)(25,35)(26,62)(27,33)(28,64)(30,50)(32,52)(34,56)(36,54)(37,41)(39,43)(49,57)(51,59)(53,61)(55,63), (2,52)(4,50)(5,62)(6,39)(7,64)(8,37)(10,22)(12,24)(14,26)(16,28)(17,45)(18,58)(19,47)(20,60)(29,57)(30,46)(31,59)(32,48)(33,63)(34,40)(35,61)(36,38)(42,54)(44,56), (1,55,51,43)(2,28,52,16)(3,53,49,41)(4,26,50,14)(5,46,36,58)(6,19,33,31)(7,48,34,60)(8,17,35,29)(9,25,21,13)(10,54,22,42)(11,27,23,15)(12,56,24,44)(18,62,30,38)(20,64,32,40)(37,45,61,57)(39,47,63,59) );
G=PermutationGroup([(1,11),(2,12),(3,9),(4,10),(5,38),(6,39),(7,40),(8,37),(13,41),(14,42),(15,43),(16,44),(17,45),(18,46),(19,47),(20,48),(21,49),(22,50),(23,51),(24,52),(25,53),(26,54),(27,55),(28,56),(29,57),(30,58),(31,59),(32,60),(33,63),(34,64),(35,61),(36,62)], [(1,51),(2,52),(3,49),(4,50),(5,36),(6,33),(7,34),(8,35),(9,21),(10,22),(11,23),(12,24),(13,25),(14,26),(15,27),(16,28),(17,29),(18,30),(19,31),(20,32),(37,61),(38,62),(39,63),(40,64),(41,53),(42,54),(43,55),(44,56),(45,57),(46,58),(47,59),(48,60)], [(1,3),(2,4),(5,7),(6,8),(9,11),(10,12),(13,15),(14,16),(17,19),(18,20),(21,23),(22,24),(25,27),(26,28),(29,31),(30,32),(33,35),(34,36),(37,39),(38,40),(41,43),(42,44),(45,47),(46,48),(49,51),(50,52),(53,55),(54,56),(57,59),(58,60),(61,63),(62,64)], [(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)], [(1,47),(2,20),(3,45),(4,18),(5,42),(6,15),(7,44),(8,13),(9,17),(10,46),(11,19),(12,48),(14,38),(16,40),(21,29),(22,58),(23,31),(24,60),(25,35),(26,62),(27,33),(28,64),(30,50),(32,52),(34,56),(36,54),(37,41),(39,43),(49,57),(51,59),(53,61),(55,63)], [(2,52),(4,50),(5,62),(6,39),(7,64),(8,37),(10,22),(12,24),(14,26),(16,28),(17,45),(18,58),(19,47),(20,60),(29,57),(30,46),(31,59),(32,48),(33,63),(34,40),(35,61),(36,38),(42,54),(44,56)], [(1,55,51,43),(2,28,52,16),(3,53,49,41),(4,26,50,14),(5,46,36,58),(6,19,33,31),(7,48,34,60),(8,17,35,29),(9,25,21,13),(10,54,22,42),(11,27,23,15),(12,56,24,44),(18,62,30,38),(20,64,32,40),(37,45,61,57),(39,47,63,59)])
Matrix representation ►G ⊆ GL8(𝔽5)
| 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 4 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 4 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 4 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 4 |
| 4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 4 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 4 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 4 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 4 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 4 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 4 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 4 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 4 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 4 |
| 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 2 | 2 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 3 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 3 | 3 | 2 | 0 |
| 0 | 0 | 0 | 0 | 4 | 4 | 3 | 3 |
| 0 | 0 | 0 | 0 | 4 | 2 | 0 | 3 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 3 |
| 4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 4 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 4 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 4 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 4 | 4 | 1 | 1 |
| 0 | 0 | 0 | 0 | 2 | 2 | 0 | 4 |
| 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 4 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 3 | 4 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 4 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 3 | 1 | 0 |
| 0 | 0 | 0 | 0 | 4 | 2 | 3 | 4 |
| 3 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 3 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 3 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 3 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 2 | 2 | 4 | 0 |
| 0 | 0 | 0 | 0 | 2 | 2 | 0 | 4 |
G:=sub<GL(8,GF(5))| [1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4],[4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4],[0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,0,0,2,3,0,0,0,0,0,0,0,0,3,4,4,0,0,0,0,0,3,4,2,0,0,0,0,0,2,3,0,0,0,0,0,0,0,3,3,3],[4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,1,4,2,0,0,0,0,1,0,4,2,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,4],[1,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,1,3,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,1,0,0,4,0,0,0,0,0,4,3,2,0,0,0,0,0,0,1,3,0,0,0,0,0,0,0,4],[3,0,0,0,0,0,0,0,0,3,0,0,0,0,0,0,0,0,3,0,0,0,0,0,0,0,0,3,0,0,0,0,0,0,0,0,1,0,2,2,0,0,0,0,0,1,2,2,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4] >;
44 conjugacy classes
| class | 1 | 2A | ··· | 2G | 2H | 2I | 2J | 2K | 2L | 2M | 4A | ··· | 4H | 4I | ··· | 4AD |
| order | 1 | 2 | ··· | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 4 | ··· | 4 |
| size | 1 | 1 | ··· | 1 | 2 | 2 | 2 | 2 | 4 | 4 | 2 | ··· | 2 | 4 | ··· | 4 |
44 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | - | ||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C4 | C4○D4 | 2+ (1+4) | 2- (1+4) |
| kernel | C23.200C24 | C2×C2.C42 | C4×C22⋊C4 | C23.7Q8 | C23.34D4 | C42⋊8C4 | C42⋊5C4 | C23.8Q8 | C23.23D4 | C23.63C23 | C24.C22 | C2×C4×D4 | C4×D4 | C23 | C22 | C22 |
| # reps | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 1 | 16 | 8 | 3 | 1 |
In GAP, Magma, Sage, TeX
C_2^3._{200}C_2^4 % in TeX
G:=Group("C2^3.200C2^4"); // GroupNames label
G:=SmallGroup(128,1050);
// by ID
G=gap.SmallGroup(128,1050);
# by ID
G:=PCGroup([7,-2,2,2,2,-2,2,2,448,253,758,219,675,136]);
// Polycyclic
G:=Group<a,b,c,d,e,f,g|a^2=b^2=c^2=e^2=f^2=1,d^2=c,g^2=b,a*b=b*a,a*c=c*a,e*d*e=g*d*g^-1=a*d=d*a,f*e*f=a*e=e*a,a*f=f*a,a*g=g*a,b*c=c*b,f*d*f=b*d=d*b,b*e=e*b,b*f=f*b,b*g=g*b,c*d=d*c,c*e=e*c,c*f=f*c,c*g=g*c,e*g=g*e,f*g=g*f>;
// generators/relations