p-group, metabelian, nilpotent (class 2), monomial
Aliases: C24.75C23, C23.633C24, C22.4062+ (1+4), (C2×Q8)⋊15D4, C23⋊2D4⋊44C2, C2.36(Q8⋊6D4), C23.10D4⋊98C2, C2.57(C23⋊3D4), (C2×C42).684C22, (C22×C4).891C23, C22.442(C22×D4), C24.3C22⋊90C2, C2.7(C24⋊C22), (C22×D4).259C22, (C22×Q8).202C22, C24.C22⋊149C2, C23.67C23⋊94C2, C2.76(C22.32C24), C2.C42.339C22, C2.37(C22.49C24), (C2×C4).127(C2×D4), (C2×C4.4D4)⋊32C2, (C2×C4).213(C4○D4), (C2×C4⋊C4).446C22, C22.495(C2×C4○D4), (C2×C22⋊C4).296C22, SmallGroup(128,1465)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Subgroups: 692 in 296 conjugacy classes, 96 normal (14 characteristic)
C1, C2 [×3], C2 [×4], C2 [×4], C4 [×16], C22 [×3], C22 [×4], C22 [×28], C2×C4 [×10], C2×C4 [×28], D4 [×24], Q8 [×4], C23, C23 [×28], C42 [×5], C22⋊C4 [×28], C4⋊C4 [×3], C22×C4, C22×C4 [×10], C2×D4 [×22], C2×Q8 [×4], C2×Q8 [×2], C24 [×4], C2.C42 [×6], C2×C42, C2×C42 [×2], C2×C22⋊C4 [×16], C2×C4⋊C4, C2×C4⋊C4 [×2], C4.4D4 [×8], C22×D4 [×6], C22×Q8, C24.C22 [×4], C24.3C22 [×2], C23.67C23, C23⋊2D4 [×4], C23.10D4 [×2], C2×C4.4D4 [×2], C23.633C24
Quotients:
C1, C2 [×15], C22 [×35], D4 [×4], C23 [×15], C2×D4 [×6], C4○D4 [×4], C24, C22×D4, C2×C4○D4 [×2], 2+ (1+4) [×4], C23⋊3D4, C22.32C24 [×2], Q8⋊6D4 [×2], C22.49C24, C24⋊C22, C23.633C24
Generators and relations
G = < a,b,c,d,e,f,g | a2=b2=c2=1, d2=g2=ba=ab, e2=b, f2=a, ac=ca, ede-1=ad=da, geg-1=ae=ea, 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, gdg-1=abd, fg=gf >
(1 9)(2 10)(3 11)(4 12)(5 37)(6 38)(7 39)(8 40)(13 45)(14 46)(15 47)(16 48)(17 49)(18 50)(19 51)(20 52)(21 43)(22 44)(23 41)(24 42)(25 55)(26 56)(27 53)(28 54)(29 59)(30 60)(31 57)(32 58)(33 64)(34 61)(35 62)(36 63)
(1 11)(2 12)(3 9)(4 10)(5 39)(6 40)(7 37)(8 38)(13 47)(14 48)(15 45)(16 46)(17 51)(18 52)(19 49)(20 50)(21 41)(22 42)(23 43)(24 44)(25 53)(26 54)(27 55)(28 56)(29 57)(30 58)(31 59)(32 60)(33 62)(34 63)(35 64)(36 61)
(1 25)(2 26)(3 27)(4 28)(5 22)(6 23)(7 24)(8 21)(9 55)(10 56)(11 53)(12 54)(13 59)(14 60)(15 57)(16 58)(17 63)(18 64)(19 61)(20 62)(29 45)(30 46)(31 47)(32 48)(33 50)(34 51)(35 52)(36 49)(37 44)(38 41)(39 42)(40 43)
(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 11 13)(2 16 12 46)(3 45 9 15)(4 14 10 48)(5 18 39 52)(6 51 40 17)(7 20 37 50)(8 49 38 19)(21 36 41 61)(22 64 42 35)(23 34 43 63)(24 62 44 33)(25 31 53 59)(26 58 54 30)(27 29 55 57)(28 60 56 32)
(1 17 9 49)(2 52 10 20)(3 19 11 51)(4 50 12 18)(5 32 37 58)(6 57 38 31)(7 30 39 60)(8 59 40 29)(13 43 45 21)(14 24 46 42)(15 41 47 23)(16 22 48 44)(25 63 55 36)(26 35 56 62)(27 61 53 34)(28 33 54 64)
(1 45 3 47)(2 48 4 46)(5 64 7 62)(6 63 8 61)(9 13 11 15)(10 16 12 14)(17 21 19 23)(18 24 20 22)(25 29 27 31)(26 32 28 30)(33 39 35 37)(34 38 36 40)(41 49 43 51)(42 52 44 50)(53 57 55 59)(54 60 56 58)
G:=sub<Sym(64)| (1,9)(2,10)(3,11)(4,12)(5,37)(6,38)(7,39)(8,40)(13,45)(14,46)(15,47)(16,48)(17,49)(18,50)(19,51)(20,52)(21,43)(22,44)(23,41)(24,42)(25,55)(26,56)(27,53)(28,54)(29,59)(30,60)(31,57)(32,58)(33,64)(34,61)(35,62)(36,63), (1,11)(2,12)(3,9)(4,10)(5,39)(6,40)(7,37)(8,38)(13,47)(14,48)(15,45)(16,46)(17,51)(18,52)(19,49)(20,50)(21,41)(22,42)(23,43)(24,44)(25,53)(26,54)(27,55)(28,56)(29,57)(30,58)(31,59)(32,60)(33,62)(34,63)(35,64)(36,61), (1,25)(2,26)(3,27)(4,28)(5,22)(6,23)(7,24)(8,21)(9,55)(10,56)(11,53)(12,54)(13,59)(14,60)(15,57)(16,58)(17,63)(18,64)(19,61)(20,62)(29,45)(30,46)(31,47)(32,48)(33,50)(34,51)(35,52)(36,49)(37,44)(38,41)(39,42)(40,43), (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), (1,47,11,13)(2,16,12,46)(3,45,9,15)(4,14,10,48)(5,18,39,52)(6,51,40,17)(7,20,37,50)(8,49,38,19)(21,36,41,61)(22,64,42,35)(23,34,43,63)(24,62,44,33)(25,31,53,59)(26,58,54,30)(27,29,55,57)(28,60,56,32), (1,17,9,49)(2,52,10,20)(3,19,11,51)(4,50,12,18)(5,32,37,58)(6,57,38,31)(7,30,39,60)(8,59,40,29)(13,43,45,21)(14,24,46,42)(15,41,47,23)(16,22,48,44)(25,63,55,36)(26,35,56,62)(27,61,53,34)(28,33,54,64), (1,45,3,47)(2,48,4,46)(5,64,7,62)(6,63,8,61)(9,13,11,15)(10,16,12,14)(17,21,19,23)(18,24,20,22)(25,29,27,31)(26,32,28,30)(33,39,35,37)(34,38,36,40)(41,49,43,51)(42,52,44,50)(53,57,55,59)(54,60,56,58)>;
G:=Group( (1,9)(2,10)(3,11)(4,12)(5,37)(6,38)(7,39)(8,40)(13,45)(14,46)(15,47)(16,48)(17,49)(18,50)(19,51)(20,52)(21,43)(22,44)(23,41)(24,42)(25,55)(26,56)(27,53)(28,54)(29,59)(30,60)(31,57)(32,58)(33,64)(34,61)(35,62)(36,63), (1,11)(2,12)(3,9)(4,10)(5,39)(6,40)(7,37)(8,38)(13,47)(14,48)(15,45)(16,46)(17,51)(18,52)(19,49)(20,50)(21,41)(22,42)(23,43)(24,44)(25,53)(26,54)(27,55)(28,56)(29,57)(30,58)(31,59)(32,60)(33,62)(34,63)(35,64)(36,61), (1,25)(2,26)(3,27)(4,28)(5,22)(6,23)(7,24)(8,21)(9,55)(10,56)(11,53)(12,54)(13,59)(14,60)(15,57)(16,58)(17,63)(18,64)(19,61)(20,62)(29,45)(30,46)(31,47)(32,48)(33,50)(34,51)(35,52)(36,49)(37,44)(38,41)(39,42)(40,43), (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), (1,47,11,13)(2,16,12,46)(3,45,9,15)(4,14,10,48)(5,18,39,52)(6,51,40,17)(7,20,37,50)(8,49,38,19)(21,36,41,61)(22,64,42,35)(23,34,43,63)(24,62,44,33)(25,31,53,59)(26,58,54,30)(27,29,55,57)(28,60,56,32), (1,17,9,49)(2,52,10,20)(3,19,11,51)(4,50,12,18)(5,32,37,58)(6,57,38,31)(7,30,39,60)(8,59,40,29)(13,43,45,21)(14,24,46,42)(15,41,47,23)(16,22,48,44)(25,63,55,36)(26,35,56,62)(27,61,53,34)(28,33,54,64), (1,45,3,47)(2,48,4,46)(5,64,7,62)(6,63,8,61)(9,13,11,15)(10,16,12,14)(17,21,19,23)(18,24,20,22)(25,29,27,31)(26,32,28,30)(33,39,35,37)(34,38,36,40)(41,49,43,51)(42,52,44,50)(53,57,55,59)(54,60,56,58) );
G=PermutationGroup([(1,9),(2,10),(3,11),(4,12),(5,37),(6,38),(7,39),(8,40),(13,45),(14,46),(15,47),(16,48),(17,49),(18,50),(19,51),(20,52),(21,43),(22,44),(23,41),(24,42),(25,55),(26,56),(27,53),(28,54),(29,59),(30,60),(31,57),(32,58),(33,64),(34,61),(35,62),(36,63)], [(1,11),(2,12),(3,9),(4,10),(5,39),(6,40),(7,37),(8,38),(13,47),(14,48),(15,45),(16,46),(17,51),(18,52),(19,49),(20,50),(21,41),(22,42),(23,43),(24,44),(25,53),(26,54),(27,55),(28,56),(29,57),(30,58),(31,59),(32,60),(33,62),(34,63),(35,64),(36,61)], [(1,25),(2,26),(3,27),(4,28),(5,22),(6,23),(7,24),(8,21),(9,55),(10,56),(11,53),(12,54),(13,59),(14,60),(15,57),(16,58),(17,63),(18,64),(19,61),(20,62),(29,45),(30,46),(31,47),(32,48),(33,50),(34,51),(35,52),(36,49),(37,44),(38,41),(39,42),(40,43)], [(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)], [(1,47,11,13),(2,16,12,46),(3,45,9,15),(4,14,10,48),(5,18,39,52),(6,51,40,17),(7,20,37,50),(8,49,38,19),(21,36,41,61),(22,64,42,35),(23,34,43,63),(24,62,44,33),(25,31,53,59),(26,58,54,30),(27,29,55,57),(28,60,56,32)], [(1,17,9,49),(2,52,10,20),(3,19,11,51),(4,50,12,18),(5,32,37,58),(6,57,38,31),(7,30,39,60),(8,59,40,29),(13,43,45,21),(14,24,46,42),(15,41,47,23),(16,22,48,44),(25,63,55,36),(26,35,56,62),(27,61,53,34),(28,33,54,64)], [(1,45,3,47),(2,48,4,46),(5,64,7,62),(6,63,8,61),(9,13,11,15),(10,16,12,14),(17,21,19,23),(18,24,20,22),(25,29,27,31),(26,32,28,30),(33,39,35,37),(34,38,36,40),(41,49,43,51),(42,52,44,50),(53,57,55,59),(54,60,56,58)])
Matrix representation ►G ⊆ GL6(𝔽5)
| 4 | 0 | 0 | 0 | 0 | 0 |
| 0 | 4 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 4 | 0 | 0 | 0 |
| 0 | 0 | 0 | 4 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 4 | 0 |
| 0 | 0 | 0 | 0 | 0 | 4 |
| 2 | 0 | 0 | 0 | 0 | 0 |
| 0 | 3 | 0 | 0 | 0 | 0 |
| 0 | 0 | 3 | 3 | 0 | 0 |
| 0 | 0 | 0 | 2 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 2 | 0 | 0 | 0 |
| 0 | 0 | 0 | 2 | 0 | 0 |
| 0 | 0 | 0 | 0 | 4 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 3 | 0 | 0 | 0 | 0 | 0 |
| 0 | 3 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 3 | 4 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 4 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 3 | 0 | 0 | 0 |
| 0 | 0 | 4 | 2 | 0 | 0 |
| 0 | 0 | 0 | 0 | 4 | 0 |
| 0 | 0 | 0 | 0 | 0 | 4 |
G:=sub<GL(6,GF(5))| [4,0,0,0,0,0,0,4,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,4,0,0,0,0,0,0,4],[2,0,0,0,0,0,0,3,0,0,0,0,0,0,3,0,0,0,0,0,3,2,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[0,1,0,0,0,0,1,0,0,0,0,0,0,0,2,0,0,0,0,0,0,2,0,0,0,0,0,0,4,0,0,0,0,0,0,1],[3,0,0,0,0,0,0,3,0,0,0,0,0,0,1,3,0,0,0,0,0,4,0,0,0,0,0,0,0,1,0,0,0,0,1,0],[0,4,0,0,0,0,1,0,0,0,0,0,0,0,3,4,0,0,0,0,0,2,0,0,0,0,0,0,4,0,0,0,0,0,0,4] >;
32 conjugacy classes
| class | 1 | 2A | ··· | 2G | 2H | 2I | 2J | 2K | 4A | ··· | 4R | 4S | 4T |
| order | 1 | 2 | ··· | 2 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 4 | 4 |
| size | 1 | 1 | ··· | 1 | 8 | 8 | 8 | 8 | 4 | ··· | 4 | 8 | 8 |
32 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 4 |
| type | + | + | + | + | + | + | + | + | + | |
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | D4 | C4○D4 | 2+ (1+4) |
| kernel | C23.633C24 | C24.C22 | C24.3C22 | C23.67C23 | C23⋊2D4 | C23.10D4 | C2×C4.4D4 | C2×Q8 | C2×C4 | C22 |
| # reps | 1 | 4 | 2 | 1 | 4 | 2 | 2 | 4 | 8 | 4 |
In GAP, Magma, Sage, TeX
C_2^3._{633}C_2^4 % in TeX
G:=Group("C2^3.633C2^4"); // GroupNames label
G:=SmallGroup(128,1465);
// by ID
G=gap.SmallGroup(128,1465);
# by ID
G:=PCGroup([7,-2,2,2,2,-2,2,2,336,253,232,758,723,100,1571,346,192]);
// Polycyclic
G:=Group<a,b,c,d,e,f,g|a^2=b^2=c^2=1,d^2=g^2=b*a=a*b,e^2=b,f^2=a,a*c=c*a,e*d*e^-1=a*d=d*a,g*e*g^-1=a*e=e*a,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,g*d*g^-1=a*b*d,f*g=g*f>;
// generators/relations