p-group, metabelian, nilpotent (class 2), monomial
Aliases: C23.687C24, C24.452C23, C22.3502- 1+4, C22.4602+ 1+4, C23.101(C4○D4), (C22×C4).600C23, (C23×C4).496C22, (C2×C42).714C22, C23.7Q8.75C2, C23.8Q8.64C2, C23.11D4.56C2, C23.84C23⋊15C2, C24.C22.76C2, C23.65C23⋊154C2, C23.63C23⋊186C2, C23.83C23⋊121C2, C23.81C23⋊127C2, C2.104(C22.32C24), C2.C42.391C22, C2.70(C22.50C24), C2.49(C22.57C24), C2.114(C22.47C24), C2.106(C22.33C24), C2.118(C22.46C24), (C2×C4).230(C4○D4), (C2×C4⋊C4).497C22, C22.548(C2×C4○D4), (C2×C22⋊C4).323C22, SmallGroup(128,1519)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C23.687C24
G = < a,b,c,d,e,f,g | a2=b2=c2=f2=1, d2=cb=bc, e2=g2=ba=ab, ac=ca, ede-1=ad=da, geg-1=ae=ea, af=fa, ag=ga, fdf=bd=db, be=eb, bf=fb, bg=gb, cd=dc, fef=ce=ec, cf=fc, cg=gc, gdg-1=abd, fg=gf >
Subgroups: 372 in 196 conjugacy classes, 88 normal (82 characteristic)
C1, C2, C2, C4, C22, C22, C2×C4, C2×C4, C23, C23, C23, C42, C22⋊C4, C4⋊C4, C22×C4, C22×C4, C24, C2.C42, C2×C42, C2×C22⋊C4, C2×C4⋊C4, C23×C4, C23.7Q8, C23.8Q8, C23.63C23, C24.C22, C23.65C23, C23.11D4, C23.81C23, C23.83C23, C23.84C23, C23.687C24
Quotients: C1, C2, C22, C23, C4○D4, C24, C2×C4○D4, 2+ 1+4, 2- 1+4, C22.32C24, C22.33C24, C22.46C24, C22.47C24, C22.50C24, C22.57C24, C23.687C24
►On 64 points
Generators in S64
(1 23)(2 24)(3 21)(4 22)(5 37)(6 38)(7 39)(8 40)(9 41)(10 42)(11 43)(12 44)(13 45)(14 46)(15 47)(16 48)(17 49)(18 50)(19 51)(20 52)(25 53)(26 54)(27 55)(28 56)(29 57)(30 58)(31 59)(32 60)(33 61)(34 62)(35 63)(36 64)
(1 5)(2 6)(3 7)(4 8)(9 55)(10 56)(11 53)(12 54)(13 59)(14 60)(15 57)(16 58)(17 63)(18 64)(19 61)(20 62)(21 39)(22 40)(23 37)(24 38)(25 43)(26 44)(27 41)(28 42)(29 47)(30 48)(31 45)(32 46)(33 51)(34 52)(35 49)(36 50)
(1 7)(2 8)(3 5)(4 6)(9 53)(10 54)(11 55)(12 56)(13 57)(14 58)(15 59)(16 60)(17 61)(18 62)(19 63)(20 64)(21 37)(22 38)(23 39)(24 40)(25 41)(26 42)(27 43)(28 44)(29 45)(30 46)(31 47)(32 48)(33 49)(34 50)(35 51)(36 52)
(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 37 59)(2 14 38 32)(3 47 39 57)(4 16 40 30)(5 31 23 13)(6 60 24 46)(7 29 21 15)(8 58 22 48)(9 35 27 17)(10 64 28 50)(11 33 25 19)(12 62 26 52)(18 42 36 56)(20 44 34 54)(41 63 55 49)(43 61 53 51)
(2 6)(4 8)(10 56)(12 54)(13 57)(14 16)(15 59)(17 61)(18 20)(19 63)(22 40)(24 38)(26 44)(28 42)(29 45)(30 32)(31 47)(33 49)(34 36)(35 51)(46 48)(50 52)(58 60)(62 64)
(1 41 37 55)(2 56 38 42)(3 43 39 53)(4 54 40 44)(5 27 23 9)(6 10 24 28)(7 25 21 11)(8 12 22 26)(13 63 31 49)(14 50 32 64)(15 61 29 51)(16 52 30 62)(17 45 35 59)(18 60 36 46)(19 47 33 57)(20 58 34 48)
G:=sub<Sym(64)| (1,23)(2,24)(3,21)(4,22)(5,37)(6,38)(7,39)(8,40)(9,41)(10,42)(11,43)(12,44)(13,45)(14,46)(15,47)(16,48)(17,49)(18,50)(19,51)(20,52)(25,53)(26,54)(27,55)(28,56)(29,57)(30,58)(31,59)(32,60)(33,61)(34,62)(35,63)(36,64), (1,5)(2,6)(3,7)(4,8)(9,55)(10,56)(11,53)(12,54)(13,59)(14,60)(15,57)(16,58)(17,63)(18,64)(19,61)(20,62)(21,39)(22,40)(23,37)(24,38)(25,43)(26,44)(27,41)(28,42)(29,47)(30,48)(31,45)(32,46)(33,51)(34,52)(35,49)(36,50), (1,7)(2,8)(3,5)(4,6)(9,53)(10,54)(11,55)(12,56)(13,57)(14,58)(15,59)(16,60)(17,61)(18,62)(19,63)(20,64)(21,37)(22,38)(23,39)(24,40)(25,41)(26,42)(27,43)(28,44)(29,45)(30,46)(31,47)(32,48)(33,49)(34,50)(35,51)(36,52), (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), (1,45,37,59)(2,14,38,32)(3,47,39,57)(4,16,40,30)(5,31,23,13)(6,60,24,46)(7,29,21,15)(8,58,22,48)(9,35,27,17)(10,64,28,50)(11,33,25,19)(12,62,26,52)(18,42,36,56)(20,44,34,54)(41,63,55,49)(43,61,53,51), (2,6)(4,8)(10,56)(12,54)(13,57)(14,16)(15,59)(17,61)(18,20)(19,63)(22,40)(24,38)(26,44)(28,42)(29,45)(30,32)(31,47)(33,49)(34,36)(35,51)(46,48)(50,52)(58,60)(62,64), (1,41,37,55)(2,56,38,42)(3,43,39,53)(4,54,40,44)(5,27,23,9)(6,10,24,28)(7,25,21,11)(8,12,22,26)(13,63,31,49)(14,50,32,64)(15,61,29,51)(16,52,30,62)(17,45,35,59)(18,60,36,46)(19,47,33,57)(20,58,34,48)>;
G:=Group( (1,23)(2,24)(3,21)(4,22)(5,37)(6,38)(7,39)(8,40)(9,41)(10,42)(11,43)(12,44)(13,45)(14,46)(15,47)(16,48)(17,49)(18,50)(19,51)(20,52)(25,53)(26,54)(27,55)(28,56)(29,57)(30,58)(31,59)(32,60)(33,61)(34,62)(35,63)(36,64), (1,5)(2,6)(3,7)(4,8)(9,55)(10,56)(11,53)(12,54)(13,59)(14,60)(15,57)(16,58)(17,63)(18,64)(19,61)(20,62)(21,39)(22,40)(23,37)(24,38)(25,43)(26,44)(27,41)(28,42)(29,47)(30,48)(31,45)(32,46)(33,51)(34,52)(35,49)(36,50), (1,7)(2,8)(3,5)(4,6)(9,53)(10,54)(11,55)(12,56)(13,57)(14,58)(15,59)(16,60)(17,61)(18,62)(19,63)(20,64)(21,37)(22,38)(23,39)(24,40)(25,41)(26,42)(27,43)(28,44)(29,45)(30,46)(31,47)(32,48)(33,49)(34,50)(35,51)(36,52), (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), (1,45,37,59)(2,14,38,32)(3,47,39,57)(4,16,40,30)(5,31,23,13)(6,60,24,46)(7,29,21,15)(8,58,22,48)(9,35,27,17)(10,64,28,50)(11,33,25,19)(12,62,26,52)(18,42,36,56)(20,44,34,54)(41,63,55,49)(43,61,53,51), (2,6)(4,8)(10,56)(12,54)(13,57)(14,16)(15,59)(17,61)(18,20)(19,63)(22,40)(24,38)(26,44)(28,42)(29,45)(30,32)(31,47)(33,49)(34,36)(35,51)(46,48)(50,52)(58,60)(62,64), (1,41,37,55)(2,56,38,42)(3,43,39,53)(4,54,40,44)(5,27,23,9)(6,10,24,28)(7,25,21,11)(8,12,22,26)(13,63,31,49)(14,50,32,64)(15,61,29,51)(16,52,30,62)(17,45,35,59)(18,60,36,46)(19,47,33,57)(20,58,34,48) );
G=PermutationGroup([[(1,23),(2,24),(3,21),(4,22),(5,37),(6,38),(7,39),(8,40),(9,41),(10,42),(11,43),(12,44),(13,45),(14,46),(15,47),(16,48),(17,49),(18,50),(19,51),(20,52),(25,53),(26,54),(27,55),(28,56),(29,57),(30,58),(31,59),(32,60),(33,61),(34,62),(35,63),(36,64)], [(1,5),(2,6),(3,7),(4,8),(9,55),(10,56),(11,53),(12,54),(13,59),(14,60),(15,57),(16,58),(17,63),(18,64),(19,61),(20,62),(21,39),(22,40),(23,37),(24,38),(25,43),(26,44),(27,41),(28,42),(29,47),(30,48),(31,45),(32,46),(33,51),(34,52),(35,49),(36,50)], [(1,7),(2,8),(3,5),(4,6),(9,53),(10,54),(11,55),(12,56),(13,57),(14,58),(15,59),(16,60),(17,61),(18,62),(19,63),(20,64),(21,37),(22,38),(23,39),(24,40),(25,41),(26,42),(27,43),(28,44),(29,45),(30,46),(31,47),(32,48),(33,49),(34,50),(35,51),(36,52)], [(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)], [(1,45,37,59),(2,14,38,32),(3,47,39,57),(4,16,40,30),(5,31,23,13),(6,60,24,46),(7,29,21,15),(8,58,22,48),(9,35,27,17),(10,64,28,50),(11,33,25,19),(12,62,26,52),(18,42,36,56),(20,44,34,54),(41,63,55,49),(43,61,53,51)], [(2,6),(4,8),(10,56),(12,54),(13,57),(14,16),(15,59),(17,61),(18,20),(19,63),(22,40),(24,38),(26,44),(28,42),(29,45),(30,32),(31,47),(33,49),(34,36),(35,51),(46,48),(50,52),(58,60),(62,64)], [(1,41,37,55),(2,56,38,42),(3,43,39,53),(4,54,40,44),(5,27,23,9),(6,10,24,28),(7,25,21,11),(8,12,22,26),(13,63,31,49),(14,50,32,64),(15,61,29,51),(16,52,30,62),(17,45,35,59),(18,60,36,46),(19,47,33,57),(20,58,34,48)]])
32 conjugacy classes
| class | 1 | 2A | ··· | 2G | 2H | 2I | 4A | ··· | 4P | 4Q | ··· | 4V |
| order | 1 | 2 | ··· | 2 | 2 | 2 | 4 | ··· | 4 | 4 | ··· | 4 |
| size | 1 | 1 | ··· | 1 | 4 | 4 | 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 | C4○D4 | C4○D4 | 2+ 1+4 | 2- 1+4 |
| kernel | C23.687C24 | C23.7Q8 | C23.8Q8 | C23.63C23 | C24.C22 | C23.65C23 | C23.11D4 | C23.81C23 | C23.83C23 | C23.84C23 | C2×C4 | C23 | C22 | C22 |
| # reps | 1 | 1 | 2 | 3 | 2 | 1 | 2 | 2 | 1 | 1 | 8 | 4 | 2 | 2 |
Matrix representation of C23.687C24 ►in 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 | 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 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 3 | 0 | 0 | 0 |
| 0 | 0 | 0 | 3 | 0 | 0 |
| 0 | 0 | 0 | 0 | 4 | 4 |
| 0 | 0 | 0 | 0 | 2 | 1 |
| 3 | 0 | 0 | 0 | 0 | 0 |
| 0 | 2 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 2 | 0 |
| 0 | 0 | 0 | 0 | 0 | 2 |
| 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 | 1 | 0 |
| 0 | 0 | 0 | 0 | 3 | 4 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 4 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 2 | 0 |
| 0 | 0 | 0 | 0 | 1 | 3 |
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],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,4,0,0,0,0,0,0,4],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[0,1,0,0,0,0,1,0,0,0,0,0,0,0,3,0,0,0,0,0,0,3,0,0,0,0,0,0,4,2,0,0,0,0,4,1],[3,0,0,0,0,0,0,2,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,2,0,0,0,0,0,0,2],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,4,0,0,0,0,0,0,1,3,0,0,0,0,0,4],[0,4,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,2,1,0,0,0,0,0,3] >;
C23.687C24 in GAP, Magma, Sage, TeX
C_2^3._{687}C_2^4 % in TeX
G:=Group("C2^3.687C2^4"); // GroupNames label
G:=SmallGroup(128,1519);
// by ID
G=gap.SmallGroup(128,1519);
# by ID
G:=PCGroup([7,-2,2,2,2,-2,2,2,672,253,344,758,723,1571,346,192]);
// Polycyclic
G:=Group<a,b,c,d,e,f,g|a^2=b^2=c^2=f^2=1,d^2=c*b=b*c,e^2=g^2=b*a=a*b,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,f*d*f=b*d=d*b,b*e=e*b,b*f=f*b,b*g=g*b,c*d=d*c,f*e*f=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