p-group, metabelian, nilpotent (class 2), monomial
Aliases: C24.589C23, C23.523C24, C22.3002+ 1+4, C22.2192- 1+4, (C22×C4).401D4, C23.625(C2×D4), C23.8Q8⋊83C2, C23.4Q8⋊26C2, C23.241(C4○D4), C23.11D4⋊57C2, C23.34D4⋊42C2, (C23×C4).425C22, (C22×C4).133C23, C22.348(C22×D4), C23.23D4.45C2, (C22×D4).194C22, C23.81C23⋊58C2, C23.83C23⋊57C2, C2.36(C22.29C24), C2.C42.249C22, C22.38(C22.D4), C2.37(C22.33C24), C2.36(C23.38C23), (C22×C4⋊C4)⋊30C2, (C2×C4).382(C2×D4), (C2×C4⋊C4).889C22, C22.395(C2×C4○D4), C2.41(C2×C22.D4), (C2×C22⋊C4).214C22, (C2×C22.D4).20C2, SmallGroup(128,1355)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C24.589C23
G = < a,b,c,d,e,f,g | a2=b2=c2=d2=g2=1, e2=b, f2=c, gag=ab=ba, ac=ca, ad=da, ae=ea, af=fa, bc=cb, bd=db, fef-1=be=eb, bf=fb, bg=gb, cd=dc, geg=ce=ec, cf=fc, cg=gc, de=ed, gfg=df=fd, dg=gd >
Subgroups: 516 in 263 conjugacy classes, 100 normal (24 characteristic)
C1, C2, C2, C2, C4, C22, C22, C22, C2×C4, C2×C4, D4, C23, C23, C23, C22⋊C4, C4⋊C4, C22×C4, C22×C4, C22×C4, C2×D4, C24, C2.C42, C2×C22⋊C4, C2×C22⋊C4, C2×C4⋊C4, C2×C4⋊C4, C2×C4⋊C4, C22.D4, C23×C4, C23×C4, C22×D4, C23.34D4, C23.8Q8, C23.23D4, C23.11D4, C23.81C23, C23.4Q8, C23.83C23, C22×C4⋊C4, C2×C22.D4, C24.589C23
Quotients: C1, C2, C22, D4, C23, C2×D4, C4○D4, C24, C22.D4, C22×D4, C2×C4○D4, 2+ 1+4, 2- 1+4, C2×C22.D4, C22.29C24, C23.38C23, C22.33C24, C24.589C23
►On 64 points
Generators in S64
(1 41)(2 42)(3 43)(4 44)(5 64)(6 61)(7 62)(8 63)(9 13)(10 14)(11 15)(12 16)(17 21)(18 22)(19 23)(20 24)(25 29)(26 30)(27 31)(28 32)(33 37)(34 38)(35 39)(36 40)(45 49)(46 50)(47 51)(48 52)(53 57)(54 58)(55 59)(56 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 9)(2 10)(3 11)(4 12)(5 39)(6 40)(7 37)(8 38)(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 55)(26 56)(27 53)(28 54)(29 59)(30 60)(31 57)(32 58)(33 62)(34 63)(35 64)(36 61)
(1 53)(2 54)(3 55)(4 56)(5 52)(6 49)(7 50)(8 51)(9 27)(10 28)(11 25)(12 26)(13 31)(14 32)(15 29)(16 30)(17 36)(18 33)(19 34)(20 35)(21 40)(22 37)(23 38)(24 39)(41 57)(42 58)(43 59)(44 60)(45 61)(46 62)(47 63)(48 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 17 9 45)(2 20 10 48)(3 19 11 47)(4 18 12 46)(5 58 39 32)(6 57 40 31)(7 60 37 30)(8 59 38 29)(13 49 41 21)(14 52 42 24)(15 51 43 23)(16 50 44 22)(25 63 55 34)(26 62 56 33)(27 61 53 36)(28 64 54 35)
(2 10)(4 12)(5 22)(6 51)(7 24)(8 49)(13 15)(14 44)(16 42)(17 36)(18 62)(19 34)(20 64)(21 38)(23 40)(26 56)(28 54)(29 31)(30 58)(32 60)(33 46)(35 48)(37 52)(39 50)(41 43)(45 61)(47 63)(57 59)
G:=sub<Sym(64)| (1,41)(2,42)(3,43)(4,44)(5,64)(6,61)(7,62)(8,63)(9,13)(10,14)(11,15)(12,16)(17,21)(18,22)(19,23)(20,24)(25,29)(26,30)(27,31)(28,32)(33,37)(34,38)(35,39)(36,40)(45,49)(46,50)(47,51)(48,52)(53,57)(54,58)(55,59)(56,60), (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), (1,9)(2,10)(3,11)(4,12)(5,39)(6,40)(7,37)(8,38)(13,41)(14,42)(15,43)(16,44)(17,45)(18,46)(19,47)(20,48)(21,49)(22,50)(23,51)(24,52)(25,55)(26,56)(27,53)(28,54)(29,59)(30,60)(31,57)(32,58)(33,62)(34,63)(35,64)(36,61), (1,53)(2,54)(3,55)(4,56)(5,52)(6,49)(7,50)(8,51)(9,27)(10,28)(11,25)(12,26)(13,31)(14,32)(15,29)(16,30)(17,36)(18,33)(19,34)(20,35)(21,40)(22,37)(23,38)(24,39)(41,57)(42,58)(43,59)(44,60)(45,61)(46,62)(47,63)(48,64), (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), (1,17,9,45)(2,20,10,48)(3,19,11,47)(4,18,12,46)(5,58,39,32)(6,57,40,31)(7,60,37,30)(8,59,38,29)(13,49,41,21)(14,52,42,24)(15,51,43,23)(16,50,44,22)(25,63,55,34)(26,62,56,33)(27,61,53,36)(28,64,54,35), (2,10)(4,12)(5,22)(6,51)(7,24)(8,49)(13,15)(14,44)(16,42)(17,36)(18,62)(19,34)(20,64)(21,38)(23,40)(26,56)(28,54)(29,31)(30,58)(32,60)(33,46)(35,48)(37,52)(39,50)(41,43)(45,61)(47,63)(57,59)>;
G:=Group( (1,41)(2,42)(3,43)(4,44)(5,64)(6,61)(7,62)(8,63)(9,13)(10,14)(11,15)(12,16)(17,21)(18,22)(19,23)(20,24)(25,29)(26,30)(27,31)(28,32)(33,37)(34,38)(35,39)(36,40)(45,49)(46,50)(47,51)(48,52)(53,57)(54,58)(55,59)(56,60), (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), (1,9)(2,10)(3,11)(4,12)(5,39)(6,40)(7,37)(8,38)(13,41)(14,42)(15,43)(16,44)(17,45)(18,46)(19,47)(20,48)(21,49)(22,50)(23,51)(24,52)(25,55)(26,56)(27,53)(28,54)(29,59)(30,60)(31,57)(32,58)(33,62)(34,63)(35,64)(36,61), (1,53)(2,54)(3,55)(4,56)(5,52)(6,49)(7,50)(8,51)(9,27)(10,28)(11,25)(12,26)(13,31)(14,32)(15,29)(16,30)(17,36)(18,33)(19,34)(20,35)(21,40)(22,37)(23,38)(24,39)(41,57)(42,58)(43,59)(44,60)(45,61)(46,62)(47,63)(48,64), (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), (1,17,9,45)(2,20,10,48)(3,19,11,47)(4,18,12,46)(5,58,39,32)(6,57,40,31)(7,60,37,30)(8,59,38,29)(13,49,41,21)(14,52,42,24)(15,51,43,23)(16,50,44,22)(25,63,55,34)(26,62,56,33)(27,61,53,36)(28,64,54,35), (2,10)(4,12)(5,22)(6,51)(7,24)(8,49)(13,15)(14,44)(16,42)(17,36)(18,62)(19,34)(20,64)(21,38)(23,40)(26,56)(28,54)(29,31)(30,58)(32,60)(33,46)(35,48)(37,52)(39,50)(41,43)(45,61)(47,63)(57,59) );
G=PermutationGroup([[(1,41),(2,42),(3,43),(4,44),(5,64),(6,61),(7,62),(8,63),(9,13),(10,14),(11,15),(12,16),(17,21),(18,22),(19,23),(20,24),(25,29),(26,30),(27,31),(28,32),(33,37),(34,38),(35,39),(36,40),(45,49),(46,50),(47,51),(48,52),(53,57),(54,58),(55,59),(56,60)], [(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)], [(1,9),(2,10),(3,11),(4,12),(5,39),(6,40),(7,37),(8,38),(13,41),(14,42),(15,43),(16,44),(17,45),(18,46),(19,47),(20,48),(21,49),(22,50),(23,51),(24,52),(25,55),(26,56),(27,53),(28,54),(29,59),(30,60),(31,57),(32,58),(33,62),(34,63),(35,64),(36,61)], [(1,53),(2,54),(3,55),(4,56),(5,52),(6,49),(7,50),(8,51),(9,27),(10,28),(11,25),(12,26),(13,31),(14,32),(15,29),(16,30),(17,36),(18,33),(19,34),(20,35),(21,40),(22,37),(23,38),(24,39),(41,57),(42,58),(43,59),(44,60),(45,61),(46,62),(47,63),(48,64)], [(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)], [(1,17,9,45),(2,20,10,48),(3,19,11,47),(4,18,12,46),(5,58,39,32),(6,57,40,31),(7,60,37,30),(8,59,38,29),(13,49,41,21),(14,52,42,24),(15,51,43,23),(16,50,44,22),(25,63,55,34),(26,62,56,33),(27,61,53,36),(28,64,54,35)], [(2,10),(4,12),(5,22),(6,51),(7,24),(8,49),(13,15),(14,44),(16,42),(17,36),(18,62),(19,34),(20,64),(21,38),(23,40),(26,56),(28,54),(29,31),(30,58),(32,60),(33,46),(35,48),(37,52),(39,50),(41,43),(45,61),(47,63),(57,59)]])
32 conjugacy classes
| class | 1 | 2A | ··· | 2G | 2H | 2I | 2J | 2K | 2L | 4A | ··· | 4L | 4M | ··· | 4S |
| order | 1 | 2 | ··· | 2 | 2 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 4 | ··· | 4 |
| size | 1 | 1 | ··· | 1 | 2 | 2 | 2 | 2 | 8 | 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 | D4 | C4○D4 | 2+ 1+4 | 2- 1+4 |
| kernel | C24.589C23 | C23.34D4 | C23.8Q8 | C23.23D4 | C23.11D4 | C23.81C23 | C23.4Q8 | C23.83C23 | C22×C4⋊C4 | C2×C22.D4 | C22×C4 | C23 | C22 | C22 |
| # reps | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 1 | 1 | 4 | 8 | 2 | 2 |
Matrix representation of C24.589C23 ►in GL8(𝔽5)
| 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 | 3 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 3 | 0 | 1 |
| 0 | 0 | 0 | 0 | 0 | 2 | 1 | 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 | 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 | 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 |
| 2 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 2 | 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 | 0 | 0 | 1 | 0 | 4 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 3 | 3 |
| 0 | 0 | 0 | 0 | 2 | 0 | 4 | 0 |
| 0 | 0 | 0 | 0 | 3 | 3 | 1 | 0 |
| 1 | 3 | 0 | 0 | 0 | 0 | 0 | 0 |
| 1 | 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 | 0 | 0 | 2 | 0 | 3 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 1 | 1 |
| 0 | 0 | 0 | 0 | 0 | 0 | 3 | 0 |
| 0 | 0 | 0 | 0 | 0 | 4 | 2 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 1 | 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 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 4 | 4 | 0 | 0 |
| 0 | 0 | 0 | 0 | 2 | 0 | 4 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 |
G:=sub<GL(8,GF(5))| [4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,3,1,3,2,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4],[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],[4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4],[2,2,0,0,0,0,0,0,1,3,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,2,3,0,0,0,0,0,0,0,3,0,0,0,0,4,3,4,1,0,0,0,0,0,3,0,0],[1,1,0,0,0,0,0,0,3,4,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,0,0,0,0,0,4,0,0,0,0,3,1,3,2,0,0,0,0,0,1,0,0],[1,1,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,1,4,2,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,1] >;
C24.589C23 in GAP, Magma, Sage, TeX
C_2^4._{589}C_2^3 % in TeX
G:=Group("C2^4.589C2^3"); // GroupNames label
G:=SmallGroup(128,1355);
// by ID
G=gap.SmallGroup(128,1355);
# by ID
G:=PCGroup([7,-2,2,2,2,-2,2,2,112,253,232,758,723,185]);
// Polycyclic
G:=Group<a,b,c,d,e,f,g|a^2=b^2=c^2=d^2=g^2=1,e^2=b,f^2=c,g*a*g=a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,a*f=f*a,b*c=c*b,b*d=d*b,f*e*f^-1=b*e=e*b,b*f=f*b,b*g=g*b,c*d=d*c,g*e*g=c*e=e*c,c*f=f*c,c*g=g*c,d*e=e*d,g*f*g=d*f=f*d,d*g=g*d>;
// generators/relations