p-group, metabelian, nilpotent (class 2), monomial
Aliases: C24.49C23, C23.553C24, C22.3282+ (1+4), C22.2442- (1+4), C23⋊Q8⋊34C2, C23.73(C4○D4), (C2×C42).85C22, C23.8Q8⋊91C2, C23.11D4⋊70C2, (C22×C4).163C23, (C23×C4).146C22, C23.84C23⋊8C2, C23.10D4.36C2, C23.23D4.48C2, (C22×D4).205C22, (C22×Q8).163C22, C24.C22⋊110C2, C23.67C23⋊75C2, C23.81C23⋊69C2, C2.51(C22.32C24), C23.63C23⋊120C2, C2.C42.270C22, C2.60(C22.36C24), C2.50(C22.33C24), C2.105(C23.36C23), (C4×C22⋊C4)⋊97C2, (C2×C4).178(C4○D4), (C2×C4⋊C4).378C22, C22.425(C2×C4○D4), (C2×C22⋊C4).474C22, SmallGroup(128,1385)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Subgroups: 468 in 221 conjugacy classes, 88 normal (82 characteristic)
C1, C2 [×7], C2 [×3], C4 [×15], C22 [×7], C22 [×17], C2×C4 [×4], C2×C4 [×41], D4 [×4], Q8 [×4], C23, C23 [×2], C23 [×13], C42 [×2], C22⋊C4 [×16], C4⋊C4 [×6], C22×C4 [×13], C22×C4 [×4], C2×D4 [×4], C2×Q8 [×4], C24 [×2], C2.C42 [×14], C2×C42 [×2], C2×C22⋊C4 [×11], C2×C4⋊C4 [×5], C23×C4, C22×D4, C22×Q8, C4×C22⋊C4, C23.8Q8, C23.23D4, C23.63C23, C24.C22 [×2], C23.67C23, C23⋊Q8 [×2], C23.10D4 [×2], C23.11D4 [×2], C23.81C23, C23.84C23, C23.553C24
Quotients:
C1, C2 [×15], C22 [×35], C23 [×15], C4○D4 [×6], C24, C2×C4○D4 [×3], 2+ (1+4) [×3], 2- (1+4), C23.36C23, C22.32C24 [×3], C22.33C24, C22.36C24 [×2], C23.553C24
Generators and relations
G = < a,b,c,d,e,f,g | a2=b2=c2=e2=1, d2=ca=ac, f2=a, g2=b, ab=ba, ede=ad=da, ae=ea, gfg-1=af=fa, ag=ga, bc=cb, fdf-1=bd=db, be=eb, bf=fb, bg=gb, cd=dc, fef-1=ce=ec, cf=fc, cg=gc, dg=gd, eg=ge >
(1 36)(2 33)(3 34)(4 35)(5 55)(6 56)(7 53)(8 54)(9 16)(10 13)(11 14)(12 15)(17 24)(18 21)(19 22)(20 23)(25 32)(26 29)(27 30)(28 31)(37 44)(38 41)(39 42)(40 43)(45 52)(46 49)(47 50)(48 51)(57 62)(58 63)(59 64)(60 61)
(1 58)(2 59)(3 60)(4 57)(5 31)(6 32)(7 29)(8 30)(9 38)(10 39)(11 40)(12 37)(13 42)(14 43)(15 44)(16 41)(17 49)(18 50)(19 51)(20 52)(21 47)(22 48)(23 45)(24 46)(25 56)(26 53)(27 54)(28 55)(33 64)(34 61)(35 62)(36 63)
(1 34)(2 35)(3 36)(4 33)(5 53)(6 54)(7 55)(8 56)(9 14)(10 15)(11 16)(12 13)(17 22)(18 23)(19 24)(20 21)(25 30)(26 31)(27 32)(28 29)(37 42)(38 43)(39 44)(40 41)(45 50)(46 51)(47 52)(48 49)(57 64)(58 61)(59 62)(60 63)
(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 17)(2 21)(3 19)(4 23)(5 39)(6 43)(7 37)(8 41)(9 27)(10 31)(11 25)(12 29)(13 28)(14 32)(15 26)(16 30)(18 33)(20 35)(22 34)(24 36)(38 54)(40 56)(42 55)(44 53)(45 57)(46 63)(47 59)(48 61)(49 58)(50 64)(51 60)(52 62)
(1 9 36 16)(2 39 33 42)(3 11 34 14)(4 37 35 44)(5 23 55 20)(6 46 56 49)(7 21 53 18)(8 48 54 51)(10 64 13 59)(12 62 15 57)(17 32 24 25)(19 30 22 27)(26 50 29 47)(28 52 31 45)(38 63 41 58)(40 61 43 60)
(1 26 58 53)(2 27 59 54)(3 28 60 55)(4 25 57 56)(5 34 31 61)(6 35 32 62)(7 36 29 63)(8 33 30 64)(9 47 38 21)(10 48 39 22)(11 45 40 23)(12 46 37 24)(13 51 42 19)(14 52 43 20)(15 49 44 17)(16 50 41 18)
G:=sub<Sym(64)| (1,36)(2,33)(3,34)(4,35)(5,55)(6,56)(7,53)(8,54)(9,16)(10,13)(11,14)(12,15)(17,24)(18,21)(19,22)(20,23)(25,32)(26,29)(27,30)(28,31)(37,44)(38,41)(39,42)(40,43)(45,52)(46,49)(47,50)(48,51)(57,62)(58,63)(59,64)(60,61), (1,58)(2,59)(3,60)(4,57)(5,31)(6,32)(7,29)(8,30)(9,38)(10,39)(11,40)(12,37)(13,42)(14,43)(15,44)(16,41)(17,49)(18,50)(19,51)(20,52)(21,47)(22,48)(23,45)(24,46)(25,56)(26,53)(27,54)(28,55)(33,64)(34,61)(35,62)(36,63), (1,34)(2,35)(3,36)(4,33)(5,53)(6,54)(7,55)(8,56)(9,14)(10,15)(11,16)(12,13)(17,22)(18,23)(19,24)(20,21)(25,30)(26,31)(27,32)(28,29)(37,42)(38,43)(39,44)(40,41)(45,50)(46,51)(47,52)(48,49)(57,64)(58,61)(59,62)(60,63), (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), (1,17)(2,21)(3,19)(4,23)(5,39)(6,43)(7,37)(8,41)(9,27)(10,31)(11,25)(12,29)(13,28)(14,32)(15,26)(16,30)(18,33)(20,35)(22,34)(24,36)(38,54)(40,56)(42,55)(44,53)(45,57)(46,63)(47,59)(48,61)(49,58)(50,64)(51,60)(52,62), (1,9,36,16)(2,39,33,42)(3,11,34,14)(4,37,35,44)(5,23,55,20)(6,46,56,49)(7,21,53,18)(8,48,54,51)(10,64,13,59)(12,62,15,57)(17,32,24,25)(19,30,22,27)(26,50,29,47)(28,52,31,45)(38,63,41,58)(40,61,43,60), (1,26,58,53)(2,27,59,54)(3,28,60,55)(4,25,57,56)(5,34,31,61)(6,35,32,62)(7,36,29,63)(8,33,30,64)(9,47,38,21)(10,48,39,22)(11,45,40,23)(12,46,37,24)(13,51,42,19)(14,52,43,20)(15,49,44,17)(16,50,41,18)>;
G:=Group( (1,36)(2,33)(3,34)(4,35)(5,55)(6,56)(7,53)(8,54)(9,16)(10,13)(11,14)(12,15)(17,24)(18,21)(19,22)(20,23)(25,32)(26,29)(27,30)(28,31)(37,44)(38,41)(39,42)(40,43)(45,52)(46,49)(47,50)(48,51)(57,62)(58,63)(59,64)(60,61), (1,58)(2,59)(3,60)(4,57)(5,31)(6,32)(7,29)(8,30)(9,38)(10,39)(11,40)(12,37)(13,42)(14,43)(15,44)(16,41)(17,49)(18,50)(19,51)(20,52)(21,47)(22,48)(23,45)(24,46)(25,56)(26,53)(27,54)(28,55)(33,64)(34,61)(35,62)(36,63), (1,34)(2,35)(3,36)(4,33)(5,53)(6,54)(7,55)(8,56)(9,14)(10,15)(11,16)(12,13)(17,22)(18,23)(19,24)(20,21)(25,30)(26,31)(27,32)(28,29)(37,42)(38,43)(39,44)(40,41)(45,50)(46,51)(47,52)(48,49)(57,64)(58,61)(59,62)(60,63), (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), (1,17)(2,21)(3,19)(4,23)(5,39)(6,43)(7,37)(8,41)(9,27)(10,31)(11,25)(12,29)(13,28)(14,32)(15,26)(16,30)(18,33)(20,35)(22,34)(24,36)(38,54)(40,56)(42,55)(44,53)(45,57)(46,63)(47,59)(48,61)(49,58)(50,64)(51,60)(52,62), (1,9,36,16)(2,39,33,42)(3,11,34,14)(4,37,35,44)(5,23,55,20)(6,46,56,49)(7,21,53,18)(8,48,54,51)(10,64,13,59)(12,62,15,57)(17,32,24,25)(19,30,22,27)(26,50,29,47)(28,52,31,45)(38,63,41,58)(40,61,43,60), (1,26,58,53)(2,27,59,54)(3,28,60,55)(4,25,57,56)(5,34,31,61)(6,35,32,62)(7,36,29,63)(8,33,30,64)(9,47,38,21)(10,48,39,22)(11,45,40,23)(12,46,37,24)(13,51,42,19)(14,52,43,20)(15,49,44,17)(16,50,41,18) );
G=PermutationGroup([(1,36),(2,33),(3,34),(4,35),(5,55),(6,56),(7,53),(8,54),(9,16),(10,13),(11,14),(12,15),(17,24),(18,21),(19,22),(20,23),(25,32),(26,29),(27,30),(28,31),(37,44),(38,41),(39,42),(40,43),(45,52),(46,49),(47,50),(48,51),(57,62),(58,63),(59,64),(60,61)], [(1,58),(2,59),(3,60),(4,57),(5,31),(6,32),(7,29),(8,30),(9,38),(10,39),(11,40),(12,37),(13,42),(14,43),(15,44),(16,41),(17,49),(18,50),(19,51),(20,52),(21,47),(22,48),(23,45),(24,46),(25,56),(26,53),(27,54),(28,55),(33,64),(34,61),(35,62),(36,63)], [(1,34),(2,35),(3,36),(4,33),(5,53),(6,54),(7,55),(8,56),(9,14),(10,15),(11,16),(12,13),(17,22),(18,23),(19,24),(20,21),(25,30),(26,31),(27,32),(28,29),(37,42),(38,43),(39,44),(40,41),(45,50),(46,51),(47,52),(48,49),(57,64),(58,61),(59,62),(60,63)], [(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)], [(1,17),(2,21),(3,19),(4,23),(5,39),(6,43),(7,37),(8,41),(9,27),(10,31),(11,25),(12,29),(13,28),(14,32),(15,26),(16,30),(18,33),(20,35),(22,34),(24,36),(38,54),(40,56),(42,55),(44,53),(45,57),(46,63),(47,59),(48,61),(49,58),(50,64),(51,60),(52,62)], [(1,9,36,16),(2,39,33,42),(3,11,34,14),(4,37,35,44),(5,23,55,20),(6,46,56,49),(7,21,53,18),(8,48,54,51),(10,64,13,59),(12,62,15,57),(17,32,24,25),(19,30,22,27),(26,50,29,47),(28,52,31,45),(38,63,41,58),(40,61,43,60)], [(1,26,58,53),(2,27,59,54),(3,28,60,55),(4,25,57,56),(5,34,31,61),(6,35,32,62),(7,36,29,63),(8,33,30,64),(9,47,38,21),(10,48,39,22),(11,45,40,23),(12,46,37,24),(13,51,42,19),(14,52,43,20),(15,49,44,17),(16,50,41,18)])
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 | 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 |
| 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 | 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 | 3 | 0 | 0 | 0 | 0 | 0 | 0 |
| 2 | 0 | 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 | 0 | 0 | 3 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 2 |
| 0 | 0 | 0 | 0 | 3 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 2 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 3 | 4 | 0 | 0 | 0 | 0 |
| 0 | 0 | 3 | 2 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 3 | 0 | 0 |
| 0 | 0 | 0 | 0 | 2 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 3 |
| 0 | 0 | 0 | 0 | 0 | 0 | 2 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 4 | 2 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 4 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 4 | 0 | 0 |
| 3 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 3 | 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 | 4 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 4 |
| 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 |
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],[4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[0,2,0,0,0,0,0,0,3,0,0,0,0,0,0,0,0,0,3,0,0,0,0,0,0,0,0,3,0,0,0,0,0,0,0,0,0,0,3,0,0,0,0,0,0,0,0,2,0,0,0,0,3,0,0,0,0,0,0,0,0,2,0,0],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,3,3,0,0,0,0,0,0,4,2,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,0,3,0,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,0,3,0],[0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,2,1,0,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0],[3,0,0,0,0,0,0,0,0,3,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,4,0] >;
32 conjugacy classes
| class | 1 | 2A | ··· | 2G | 2H | 2I | 2J | 4A | 4B | 4C | 4D | 4E | ··· | 4N | 4O | ··· | 4U |
| order | 1 | 2 | ··· | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 4 | ··· | 4 |
| size | 1 | 1 | ··· | 1 | 4 | 4 | 8 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 8 | ··· | 8 |
32 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | - | ||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C4○D4 | C4○D4 | 2+ (1+4) | 2- (1+4) |
| kernel | C23.553C24 | C4×C22⋊C4 | C23.8Q8 | C23.23D4 | C23.63C23 | C24.C22 | C23.67C23 | C23⋊Q8 | C23.10D4 | C23.11D4 | C23.81C23 | C23.84C23 | C2×C4 | C23 | C22 | C22 |
| # reps | 1 | 1 | 1 | 1 | 1 | 2 | 1 | 2 | 2 | 2 | 1 | 1 | 8 | 4 | 3 | 1 |
In GAP, Magma, Sage, TeX
C_2^3._{553}C_2^4 % in TeX
G:=Group("C2^3.553C2^4"); // GroupNames label
G:=SmallGroup(128,1385);
// by ID
G=gap.SmallGroup(128,1385);
# by ID
G:=PCGroup([7,-2,2,2,2,-2,2,2,560,253,758,723,100,185,136]);
// Polycyclic
G:=Group<a,b,c,d,e,f,g|a^2=b^2=c^2=e^2=1,d^2=c*a=a*c,f^2=a,g^2=b,a*b=b*a,e*d*e=a*d=d*a,a*e=e*a,g*f*g^-1=a*f=f*a,a*g=g*a,b*c=c*b,f*d*f^-1=b*d=d*b,b*e=e*b,b*f=f*b,b*g=g*b,c*d=d*c,f*e*f^-1=c*e=e*c,c*f=f*c,c*g=g*c,d*g=g*d,e*g=g*e>;
// generators/relations