p-group, metabelian, nilpotent (class 2), monomial
Aliases: C24.36C23, C23.496C24, C22.2782+ (1+4), C22.2042- (1+4), (C22×C4).118C23, (C2×C42).584C22, C23.4Q8.11C2, C23.11D4.23C2, C23.65C23⋊96C2, C23.83C23⋊52C2, C24.C22.39C2, C23.63C23⋊101C2, C2.C42.495C22, C2.98(C23.36C23), C2.72(C22.46C24), C2.73(C22.47C24), C2.35(C22.33C24), C2.28(C22.53C24), C2.49(C22.50C24), (C4×C4⋊C4)⋊109C2, (C2×C4).405(C4○D4), (C2×C4⋊C4).337C22, C22.372(C2×C4○D4), (C2×C22⋊C4).199C22, SmallGroup(128,1328)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Subgroups: 356 in 199 conjugacy classes, 92 normal (82 characteristic)
C1, C2 [×7], C2, C4 [×19], C22 [×7], C22 [×7], C2×C4 [×10], C2×C4 [×37], C23, C23 [×7], C42 [×7], C22⋊C4 [×12], C4⋊C4 [×17], C22×C4 [×14], C24, C2.C42 [×14], C2×C42 [×5], C2×C22⋊C4 [×7], C2×C4⋊C4 [×9], C4×C4⋊C4 [×2], C23.63C23 [×3], C24.C22 [×5], C23.65C23, C23.11D4, C23.4Q8, C23.83C23 [×2], C23.496C24
Quotients:
C1, C2 [×15], C22 [×35], C23 [×15], C4○D4 [×10], C24, C2×C4○D4 [×5], 2+ (1+4), 2- (1+4), C23.36C23 [×2], C22.33C24, C22.46C24, C22.47C24, C22.50C24, C22.53C24, C23.496C24
Generators and relations
G = < a,b,c,d,e,f,g | a2=b2=c2=1, d2=f2=c, e2=cb=bc, g2=ba=ab, ac=ca, ede-1=gdg-1=ad=da, ae=ea, af=fa, ag=ga, fdf-1=bd=db, be=eb, bf=fb, bg=gb, cd=dc, fef-1=ce=ec, cf=fc, cg=gc, eg=ge, fg=gf >
(1 10)(2 11)(3 12)(4 9)(5 22)(6 23)(7 24)(8 21)(13 20)(14 17)(15 18)(16 19)(25 64)(26 61)(27 62)(28 63)(29 46)(30 47)(31 48)(32 45)(33 58)(34 59)(35 60)(36 57)(37 54)(38 55)(39 56)(40 53)(41 50)(42 51)(43 52)(44 49)
(1 29)(2 30)(3 31)(4 32)(5 16)(6 13)(7 14)(8 15)(9 45)(10 46)(11 47)(12 48)(17 24)(18 21)(19 22)(20 23)(25 49)(26 50)(27 51)(28 52)(33 55)(34 56)(35 53)(36 54)(37 57)(38 58)(39 59)(40 60)(41 61)(42 62)(43 63)(44 64)
(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 28 31 50)(2 64 32 42)(3 26 29 52)(4 62 30 44)(5 53 14 33)(6 37 15 59)(7 55 16 35)(8 39 13 57)(9 27 47 49)(10 63 48 41)(11 25 45 51)(12 61 46 43)(17 58 22 40)(18 34 23 54)(19 60 24 38)(20 36 21 56)
(1 17 3 19)(2 21 4 23)(5 46 7 48)(6 11 8 9)(10 14 12 16)(13 47 15 45)(18 32 20 30)(22 29 24 31)(25 37 27 39)(26 58 28 60)(33 63 35 61)(34 44 36 42)(38 52 40 50)(41 55 43 53)(49 57 51 59)(54 62 56 64)
(1 15 46 21)(2 19 47 5)(3 13 48 23)(4 17 45 7)(6 12 20 31)(8 10 18 29)(9 14 32 24)(11 16 30 22)(25 35 44 40)(26 57 41 54)(27 33 42 38)(28 59 43 56)(34 52 39 63)(36 50 37 61)(49 53 64 60)(51 55 62 58)
G:=sub<Sym(64)| (1,10)(2,11)(3,12)(4,9)(5,22)(6,23)(7,24)(8,21)(13,20)(14,17)(15,18)(16,19)(25,64)(26,61)(27,62)(28,63)(29,46)(30,47)(31,48)(32,45)(33,58)(34,59)(35,60)(36,57)(37,54)(38,55)(39,56)(40,53)(41,50)(42,51)(43,52)(44,49), (1,29)(2,30)(3,31)(4,32)(5,16)(6,13)(7,14)(8,15)(9,45)(10,46)(11,47)(12,48)(17,24)(18,21)(19,22)(20,23)(25,49)(26,50)(27,51)(28,52)(33,55)(34,56)(35,53)(36,54)(37,57)(38,58)(39,59)(40,60)(41,61)(42,62)(43,63)(44,64), (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), (1,28,31,50)(2,64,32,42)(3,26,29,52)(4,62,30,44)(5,53,14,33)(6,37,15,59)(7,55,16,35)(8,39,13,57)(9,27,47,49)(10,63,48,41)(11,25,45,51)(12,61,46,43)(17,58,22,40)(18,34,23,54)(19,60,24,38)(20,36,21,56), (1,17,3,19)(2,21,4,23)(5,46,7,48)(6,11,8,9)(10,14,12,16)(13,47,15,45)(18,32,20,30)(22,29,24,31)(25,37,27,39)(26,58,28,60)(33,63,35,61)(34,44,36,42)(38,52,40,50)(41,55,43,53)(49,57,51,59)(54,62,56,64), (1,15,46,21)(2,19,47,5)(3,13,48,23)(4,17,45,7)(6,12,20,31)(8,10,18,29)(9,14,32,24)(11,16,30,22)(25,35,44,40)(26,57,41,54)(27,33,42,38)(28,59,43,56)(34,52,39,63)(36,50,37,61)(49,53,64,60)(51,55,62,58)>;
G:=Group( (1,10)(2,11)(3,12)(4,9)(5,22)(6,23)(7,24)(8,21)(13,20)(14,17)(15,18)(16,19)(25,64)(26,61)(27,62)(28,63)(29,46)(30,47)(31,48)(32,45)(33,58)(34,59)(35,60)(36,57)(37,54)(38,55)(39,56)(40,53)(41,50)(42,51)(43,52)(44,49), (1,29)(2,30)(3,31)(4,32)(5,16)(6,13)(7,14)(8,15)(9,45)(10,46)(11,47)(12,48)(17,24)(18,21)(19,22)(20,23)(25,49)(26,50)(27,51)(28,52)(33,55)(34,56)(35,53)(36,54)(37,57)(38,58)(39,59)(40,60)(41,61)(42,62)(43,63)(44,64), (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), (1,28,31,50)(2,64,32,42)(3,26,29,52)(4,62,30,44)(5,53,14,33)(6,37,15,59)(7,55,16,35)(8,39,13,57)(9,27,47,49)(10,63,48,41)(11,25,45,51)(12,61,46,43)(17,58,22,40)(18,34,23,54)(19,60,24,38)(20,36,21,56), (1,17,3,19)(2,21,4,23)(5,46,7,48)(6,11,8,9)(10,14,12,16)(13,47,15,45)(18,32,20,30)(22,29,24,31)(25,37,27,39)(26,58,28,60)(33,63,35,61)(34,44,36,42)(38,52,40,50)(41,55,43,53)(49,57,51,59)(54,62,56,64), (1,15,46,21)(2,19,47,5)(3,13,48,23)(4,17,45,7)(6,12,20,31)(8,10,18,29)(9,14,32,24)(11,16,30,22)(25,35,44,40)(26,57,41,54)(27,33,42,38)(28,59,43,56)(34,52,39,63)(36,50,37,61)(49,53,64,60)(51,55,62,58) );
G=PermutationGroup([(1,10),(2,11),(3,12),(4,9),(5,22),(6,23),(7,24),(8,21),(13,20),(14,17),(15,18),(16,19),(25,64),(26,61),(27,62),(28,63),(29,46),(30,47),(31,48),(32,45),(33,58),(34,59),(35,60),(36,57),(37,54),(38,55),(39,56),(40,53),(41,50),(42,51),(43,52),(44,49)], [(1,29),(2,30),(3,31),(4,32),(5,16),(6,13),(7,14),(8,15),(9,45),(10,46),(11,47),(12,48),(17,24),(18,21),(19,22),(20,23),(25,49),(26,50),(27,51),(28,52),(33,55),(34,56),(35,53),(36,54),(37,57),(38,58),(39,59),(40,60),(41,61),(42,62),(43,63),(44,64)], [(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)], [(1,28,31,50),(2,64,32,42),(3,26,29,52),(4,62,30,44),(5,53,14,33),(6,37,15,59),(7,55,16,35),(8,39,13,57),(9,27,47,49),(10,63,48,41),(11,25,45,51),(12,61,46,43),(17,58,22,40),(18,34,23,54),(19,60,24,38),(20,36,21,56)], [(1,17,3,19),(2,21,4,23),(5,46,7,48),(6,11,8,9),(10,14,12,16),(13,47,15,45),(18,32,20,30),(22,29,24,31),(25,37,27,39),(26,58,28,60),(33,63,35,61),(34,44,36,42),(38,52,40,50),(41,55,43,53),(49,57,51,59),(54,62,56,64)], [(1,15,46,21),(2,19,47,5),(3,13,48,23),(4,17,45,7),(6,12,20,31),(8,10,18,29),(9,14,32,24),(11,16,30,22),(25,35,44,40),(26,57,41,54),(27,33,42,38),(28,59,43,56),(34,52,39,63),(36,50,37,61),(49,53,64,60),(51,55,62,58)])
Matrix representation ►G ⊆ GL6(𝔽5)
| 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 |
| 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 | 4 | 0 |
| 0 | 0 | 0 | 0 | 0 | 4 |
| 4 | 0 | 0 | 0 | 0 | 0 |
| 0 | 4 | 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 |
| 2 | 0 | 0 | 0 | 0 | 0 |
| 0 | 2 | 0 | 0 | 0 | 0 |
| 0 | 0 | 4 | 4 | 0 | 0 |
| 0 | 0 | 2 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 4 | 0 |
| 0 | 0 | 0 | 0 | 4 | 1 |
| 2 | 2 | 0 | 0 | 0 | 0 |
| 0 | 3 | 0 | 0 | 0 | 0 |
| 0 | 0 | 3 | 3 | 0 | 0 |
| 0 | 0 | 4 | 2 | 0 | 0 |
| 0 | 0 | 0 | 0 | 2 | 1 |
| 0 | 0 | 0 | 0 | 0 | 3 |
| 3 | 0 | 0 | 0 | 0 | 0 |
| 4 | 2 | 0 | 0 | 0 | 0 |
| 0 | 0 | 2 | 2 | 0 | 0 |
| 0 | 0 | 0 | 3 | 0 | 0 |
| 0 | 0 | 0 | 0 | 4 | 2 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 2 | 0 | 0 | 0 |
| 0 | 0 | 0 | 2 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 3 |
| 0 | 0 | 0 | 0 | 0 | 4 |
G:=sub<GL(6,GF(5))| [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],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,4],[4,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[2,0,0,0,0,0,0,2,0,0,0,0,0,0,4,2,0,0,0,0,4,1,0,0,0,0,0,0,4,4,0,0,0,0,0,1],[2,0,0,0,0,0,2,3,0,0,0,0,0,0,3,4,0,0,0,0,3,2,0,0,0,0,0,0,2,0,0,0,0,0,1,3],[3,4,0,0,0,0,0,2,0,0,0,0,0,0,2,0,0,0,0,0,2,3,0,0,0,0,0,0,4,0,0,0,0,0,2,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,2,0,0,0,0,0,0,2,0,0,0,0,0,0,1,0,0,0,0,0,3,4] >;
38 conjugacy classes
| class | 1 | 2A | ··· | 2G | 2H | 4A | ··· | 4H | 4I | ··· | 4Z | 4AA | 4AB | 4AC |
| order | 1 | 2 | ··· | 2 | 2 | 4 | ··· | 4 | 4 | ··· | 4 | 4 | 4 | 4 |
| size | 1 | 1 | ··· | 1 | 8 | 2 | ··· | 2 | 4 | ··· | 4 | 8 | 8 | 8 |
38 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | - | |
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C4○D4 | 2+ (1+4) | 2- (1+4) |
| kernel | C23.496C24 | C4×C4⋊C4 | C23.63C23 | C24.C22 | C23.65C23 | C23.11D4 | C23.4Q8 | C23.83C23 | C2×C4 | C22 | C22 |
| # reps | 1 | 2 | 3 | 5 | 1 | 1 | 1 | 2 | 20 | 1 | 1 |
In GAP, Magma, Sage, TeX
C_2^3._{496}C_2^4 % in TeX
G:=Group("C2^3.496C2^4"); // GroupNames label
G:=SmallGroup(128,1328);
// by ID
G=gap.SmallGroup(128,1328);
# by ID
G:=PCGroup([7,-2,2,2,2,-2,2,2,448,253,680,758,723,352,675,192]);
// Polycyclic
G:=Group<a,b,c,d,e,f,g|a^2=b^2=c^2=1,d^2=f^2=c,e^2=c*b=b*c,g^2=b*a=a*b,a*c=c*a,e*d*e^-1=g*d*g^-1=a*d=d*a,a*e=e*a,a*f=f*a,a*g=g*a,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,e*g=g*e,f*g=g*f>;
// generators/relations