p-group, metabelian, nilpotent (class 2), monomial
Aliases: C23.565C24, C24.379C23, C22.2542- (1+4), C22.3392+ (1+4), (C22×C4)⋊15Q8, C23.69(C2×Q8), C23⋊Q8.18C2, (C23×C4).439C22, (C2×C42).629C22, (C22×C4).170C23, C2.11(C23⋊2Q8), C23.Q8.25C2, C23.7Q8.62C2, C22.139(C22×Q8), C23.34D4.24C2, (C22×Q8).170C22, C23.78C23⋊35C2, C23.67C23⋊76C2, C23.83C23⋊73C2, C2.54(C22.32C24), C23.63C23⋊122C2, C2.C42.279C22, C2.65(C22.36C24), C2.26(C23.41C23), C2.42(C23.37C23), (C2×C4).169(C2×Q8), (C4×C22⋊C4).74C2, (C2×C4).185(C4○D4), (C2×C4⋊C4).386C22, C22.432(C2×C4○D4), (C2×C22⋊C4).522C22, SmallGroup(128,1397)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Subgroups: 420 in 212 conjugacy classes, 96 normal (22 characteristic)
C1, C2 [×3], C2 [×4], C2 [×2], C4 [×18], C22 [×3], C22 [×4], C22 [×10], C2×C4 [×8], C2×C4 [×42], Q8 [×8], C23, C23 [×2], C23 [×6], C42 [×2], C22⋊C4 [×8], C4⋊C4 [×10], C22×C4 [×2], C22×C4 [×16], C22×C4 [×2], C2×Q8 [×8], C24, C2.C42 [×4], C2.C42 [×12], C2×C42 [×2], C2×C22⋊C4 [×2], C2×C22⋊C4 [×4], C2×C4⋊C4 [×8], C23×C4, C22×Q8 [×2], C4×C22⋊C4, C23.7Q8, C23.34D4, C23.63C23 [×2], C23.67C23 [×2], C23⋊Q8 [×2], C23.78C23 [×2], C23.Q8 [×2], C23.83C23 [×2], C23.565C24
Quotients:
C1, C2 [×15], C22 [×35], Q8 [×4], C23 [×15], C2×Q8 [×6], C4○D4 [×4], C24, C22×Q8, C2×C4○D4 [×2], 2+ (1+4) [×3], 2- (1+4), C23.37C23, C22.32C24 [×2], C22.36C24 [×2], C23⋊2Q8, C23.41C23, C23.565C24
Generators and relations
G = < a,b,c,d,e,f,g | a2=b2=c2=d2=1, e2=f2=c, g2=b, ab=ba, ac=ca, ede-1=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 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)
(2 12)(4 10)(5 62)(6 33)(7 64)(8 35)(14 42)(16 44)(17 29)(18 58)(19 31)(20 60)(22 50)(24 52)(26 54)(28 56)(30 46)(32 48)(34 40)(36 38)(37 61)(39 63)(45 57)(47 59)
(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 45 3 47)(2 48 4 46)(5 16 7 14)(6 15 8 13)(9 19 11 17)(10 18 12 20)(21 31 23 29)(22 30 24 32)(25 33 27 35)(26 36 28 34)(37 41 39 43)(38 44 40 42)(49 59 51 57)(50 58 52 60)(53 63 55 61)(54 62 56 64)
(1 55 51 43)(2 56 52 44)(3 53 49 41)(4 54 50 42)(5 46 36 58)(6 47 33 59)(7 48 34 60)(8 45 35 57)(9 25 21 13)(10 26 22 14)(11 27 23 15)(12 28 24 16)(17 61 29 37)(18 62 30 38)(19 63 31 39)(20 64 32 40)
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), (2,12)(4,10)(5,62)(6,33)(7,64)(8,35)(14,42)(16,44)(17,29)(18,58)(19,31)(20,60)(22,50)(24,52)(26,54)(28,56)(30,46)(32,48)(34,40)(36,38)(37,61)(39,63)(45,57)(47,59), (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), (1,45,3,47)(2,48,4,46)(5,16,7,14)(6,15,8,13)(9,19,11,17)(10,18,12,20)(21,31,23,29)(22,30,24,32)(25,33,27,35)(26,36,28,34)(37,41,39,43)(38,44,40,42)(49,59,51,57)(50,58,52,60)(53,63,55,61)(54,62,56,64), (1,55,51,43)(2,56,52,44)(3,53,49,41)(4,54,50,42)(5,46,36,58)(6,47,33,59)(7,48,34,60)(8,45,35,57)(9,25,21,13)(10,26,22,14)(11,27,23,15)(12,28,24,16)(17,61,29,37)(18,62,30,38)(19,63,31,39)(20,64,32,40)>;
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), (2,12)(4,10)(5,62)(6,33)(7,64)(8,35)(14,42)(16,44)(17,29)(18,58)(19,31)(20,60)(22,50)(24,52)(26,54)(28,56)(30,46)(32,48)(34,40)(36,38)(37,61)(39,63)(45,57)(47,59), (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), (1,45,3,47)(2,48,4,46)(5,16,7,14)(6,15,8,13)(9,19,11,17)(10,18,12,20)(21,31,23,29)(22,30,24,32)(25,33,27,35)(26,36,28,34)(37,41,39,43)(38,44,40,42)(49,59,51,57)(50,58,52,60)(53,63,55,61)(54,62,56,64), (1,55,51,43)(2,56,52,44)(3,53,49,41)(4,54,50,42)(5,46,36,58)(6,47,33,59)(7,48,34,60)(8,45,35,57)(9,25,21,13)(10,26,22,14)(11,27,23,15)(12,28,24,16)(17,61,29,37)(18,62,30,38)(19,63,31,39)(20,64,32,40) );
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)], [(2,12),(4,10),(5,62),(6,33),(7,64),(8,35),(14,42),(16,44),(17,29),(18,58),(19,31),(20,60),(22,50),(24,52),(26,54),(28,56),(30,46),(32,48),(34,40),(36,38),(37,61),(39,63),(45,57),(47,59)], [(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)], [(1,45,3,47),(2,48,4,46),(5,16,7,14),(6,15,8,13),(9,19,11,17),(10,18,12,20),(21,31,23,29),(22,30,24,32),(25,33,27,35),(26,36,28,34),(37,41,39,43),(38,44,40,42),(49,59,51,57),(50,58,52,60),(53,63,55,61),(54,62,56,64)], [(1,55,51,43),(2,56,52,44),(3,53,49,41),(4,54,50,42),(5,46,36,58),(6,47,33,59),(7,48,34,60),(8,45,35,57),(9,25,21,13),(10,26,22,14),(11,27,23,15),(12,28,24,16),(17,61,29,37),(18,62,30,38),(19,63,31,39),(20,64,32,40)])
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 |
| 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 |
| 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 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 2 | 4 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 | 4 | 0 |
| 0 | 0 | 0 | 0 | 2 | 3 | 0 | 1 |
| 3 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 4 | 2 | 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 | 1 | 0 | 3 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 | 3 | 4 |
| 0 | 0 | 0 | 0 | 0 | 0 | 4 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 4 |
| 2 | 2 | 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 | 2 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 3 | 2 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 2 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 |
| 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 2 | 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 | 2 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 4 | 3 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 2 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 3 |
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],[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],[1,3,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,1,2,1,2,0,0,0,0,0,4,0,3,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,1],[3,4,0,0,0,0,0,0,0,2,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,3,3,4,0,0,0,0,0,0,4,0,4],[2,0,0,0,0,0,0,0,2,3,0,0,0,0,0,0,0,0,3,0,0,0,0,0,0,0,0,2,0,0,0,0,0,0,0,0,3,1,0,0,0,0,0,0,2,2,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0],[2,0,0,0,0,0,0,0,0,2,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,2,4,0,0,0,0,0,0,0,3,0,0,0,0,0,0,0,0,2,0,0,0,0,0,0,0,0,3] >;
32 conjugacy classes
| class | 1 | 2A | ··· | 2G | 2H | 2I | 4A | 4B | 4C | 4D | 4E | ··· | 4N | 4O | ··· | 4V |
| order | 1 | 2 | ··· | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 4 | ··· | 4 |
| size | 1 | 1 | ··· | 1 | 4 | 4 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 8 | ··· | 8 |
32 irreducible representations
| dim | 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 | Q8 | C4○D4 | 2+ (1+4) | 2- (1+4) |
| kernel | C23.565C24 | C4×C22⋊C4 | C23.7Q8 | C23.34D4 | C23.63C23 | C23.67C23 | C23⋊Q8 | C23.78C23 | C23.Q8 | C23.83C23 | C22×C4 | C2×C4 | C22 | C22 |
| # reps | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 8 | 3 | 1 |
In GAP, Magma, Sage, TeX
C_2^3._{565}C_2^4 % in TeX
G:=Group("C2^3.565C2^4"); // GroupNames label
G:=SmallGroup(128,1397);
// by ID
G=gap.SmallGroup(128,1397);
# by ID
G:=PCGroup([7,-2,2,2,2,-2,2,2,253,456,758,723,352,185,136]);
// Polycyclic
G:=Group<a,b,c,d,e,f,g|a^2=b^2=c^2=d^2=1,e^2=f^2=c,g^2=b,a*b=b*a,a*c=c*a,e*d*e^-1=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