p-group, metabelian, nilpotent (class 2), monomial
Aliases: C24.87C23, C23.677C24, C22.4502+ 1+4, C22.3432- 1+4, C42⋊5C4⋊36C2, (C2×C42).708C22, (C22×C4).592C23, C23.Q8.39C2, C23.4Q8.28C2, C23.11D4.54C2, C24.C22.74C2, C23.81C23⋊122C2, C23.63C23⋊179C2, C23.83C23⋊113C2, C23.65C23⋊150C2, C2.C42.381C22, C2.48(C22.57C24), C2.60(C22.34C24), C2.67(C22.50C24), C2.100(C22.33C24), C2.114(C22.36C24), C2.106(C22.47C24), C2.115(C22.46C24), (C2×C4).226(C4○D4), (C2×C4⋊C4).487C22, C22.538(C2×C4○D4), (C2×C22⋊C4).315C22, SmallGroup(128,1509)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C23.677C24
G = < a,b,c,d,e,f,g | a2=b2=c2=1, d2=abc, e2=f2=ba=ab, g2=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 >
Subgroups: 356 in 189 conjugacy classes, 88 normal (82 characteristic)
C1, C2, C2, C4, C22, C22, C2×C4, C2×C4, C23, C23, C42, C22⋊C4, C4⋊C4, C22×C4, C24, C2.C42, C2×C42, C2×C22⋊C4, C2×C4⋊C4, C42⋊5C4, C23.63C23, C24.C22, C23.65C23, C23.Q8, C23.11D4, C23.81C23, C23.4Q8, C23.83C23, C23.677C24
Quotients: C1, C2, C22, C23, C4○D4, C24, C2×C4○D4, 2+ 1+4, 2- 1+4, C22.33C24, C22.34C24, C22.36C24, C22.46C24, C22.47C24, C22.50C24, C22.57C24, C23.677C24
►On 64 points
Generators in S64
(1 61)(2 62)(3 63)(4 64)(5 17)(6 18)(7 19)(8 20)(9 16)(10 13)(11 14)(12 15)(21 34)(22 35)(23 36)(24 33)(25 42)(26 43)(27 44)(28 41)(29 38)(30 39)(31 40)(32 37)(45 58)(46 59)(47 60)(48 57)(49 56)(50 53)(51 54)(52 55)
(1 36)(2 33)(3 34)(4 35)(5 50)(6 51)(7 52)(8 49)(9 32)(10 29)(11 30)(12 31)(13 38)(14 39)(15 40)(16 37)(17 53)(18 54)(19 55)(20 56)(21 63)(22 64)(23 61)(24 62)(25 46)(26 47)(27 48)(28 45)(41 58)(42 59)(43 60)(44 57)
(1 21)(2 22)(3 23)(4 24)(5 55)(6 56)(7 53)(8 54)(9 39)(10 40)(11 37)(12 38)(13 31)(14 32)(15 29)(16 30)(17 52)(18 49)(19 50)(20 51)(25 57)(26 58)(27 59)(28 60)(33 64)(34 61)(35 62)(36 63)(41 47)(42 48)(43 45)(44 46)
(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 25 23 59)(2 43 24 47)(3 27 21 57)(4 41 22 45)(5 37 53 9)(6 29 54 13)(7 39 55 11)(8 31 56 15)(10 18 38 51)(12 20 40 49)(14 19 30 52)(16 17 32 50)(26 33 60 62)(28 35 58 64)(34 48 63 44)(36 46 61 42)
(1 47 23 43)(2 27 24 57)(3 45 21 41)(4 25 22 59)(5 38 53 10)(6 14 54 30)(7 40 55 12)(8 16 56 32)(9 49 37 20)(11 51 39 18)(13 17 29 50)(15 19 31 52)(26 61 60 36)(28 63 58 34)(33 48 62 44)(35 46 64 42)
(1 17 61 5)(2 51 62 54)(3 19 63 7)(4 49 64 56)(6 24 18 33)(8 22 20 35)(9 46 16 59)(10 43 13 26)(11 48 14 57)(12 41 15 28)(21 52 34 55)(23 50 36 53)(25 37 42 32)(27 39 44 30)(29 60 38 47)(31 58 40 45)
G:=sub<Sym(64)| (1,61)(2,62)(3,63)(4,64)(5,17)(6,18)(7,19)(8,20)(9,16)(10,13)(11,14)(12,15)(21,34)(22,35)(23,36)(24,33)(25,42)(26,43)(27,44)(28,41)(29,38)(30,39)(31,40)(32,37)(45,58)(46,59)(47,60)(48,57)(49,56)(50,53)(51,54)(52,55), (1,36)(2,33)(3,34)(4,35)(5,50)(6,51)(7,52)(8,49)(9,32)(10,29)(11,30)(12,31)(13,38)(14,39)(15,40)(16,37)(17,53)(18,54)(19,55)(20,56)(21,63)(22,64)(23,61)(24,62)(25,46)(26,47)(27,48)(28,45)(41,58)(42,59)(43,60)(44,57), (1,21)(2,22)(3,23)(4,24)(5,55)(6,56)(7,53)(8,54)(9,39)(10,40)(11,37)(12,38)(13,31)(14,32)(15,29)(16,30)(17,52)(18,49)(19,50)(20,51)(25,57)(26,58)(27,59)(28,60)(33,64)(34,61)(35,62)(36,63)(41,47)(42,48)(43,45)(44,46), (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), (1,25,23,59)(2,43,24,47)(3,27,21,57)(4,41,22,45)(5,37,53,9)(6,29,54,13)(7,39,55,11)(8,31,56,15)(10,18,38,51)(12,20,40,49)(14,19,30,52)(16,17,32,50)(26,33,60,62)(28,35,58,64)(34,48,63,44)(36,46,61,42), (1,47,23,43)(2,27,24,57)(3,45,21,41)(4,25,22,59)(5,38,53,10)(6,14,54,30)(7,40,55,12)(8,16,56,32)(9,49,37,20)(11,51,39,18)(13,17,29,50)(15,19,31,52)(26,61,60,36)(28,63,58,34)(33,48,62,44)(35,46,64,42), (1,17,61,5)(2,51,62,54)(3,19,63,7)(4,49,64,56)(6,24,18,33)(8,22,20,35)(9,46,16,59)(10,43,13,26)(11,48,14,57)(12,41,15,28)(21,52,34,55)(23,50,36,53)(25,37,42,32)(27,39,44,30)(29,60,38,47)(31,58,40,45)>;
G:=Group( (1,61)(2,62)(3,63)(4,64)(5,17)(6,18)(7,19)(8,20)(9,16)(10,13)(11,14)(12,15)(21,34)(22,35)(23,36)(24,33)(25,42)(26,43)(27,44)(28,41)(29,38)(30,39)(31,40)(32,37)(45,58)(46,59)(47,60)(48,57)(49,56)(50,53)(51,54)(52,55), (1,36)(2,33)(3,34)(4,35)(5,50)(6,51)(7,52)(8,49)(9,32)(10,29)(11,30)(12,31)(13,38)(14,39)(15,40)(16,37)(17,53)(18,54)(19,55)(20,56)(21,63)(22,64)(23,61)(24,62)(25,46)(26,47)(27,48)(28,45)(41,58)(42,59)(43,60)(44,57), (1,21)(2,22)(3,23)(4,24)(5,55)(6,56)(7,53)(8,54)(9,39)(10,40)(11,37)(12,38)(13,31)(14,32)(15,29)(16,30)(17,52)(18,49)(19,50)(20,51)(25,57)(26,58)(27,59)(28,60)(33,64)(34,61)(35,62)(36,63)(41,47)(42,48)(43,45)(44,46), (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), (1,25,23,59)(2,43,24,47)(3,27,21,57)(4,41,22,45)(5,37,53,9)(6,29,54,13)(7,39,55,11)(8,31,56,15)(10,18,38,51)(12,20,40,49)(14,19,30,52)(16,17,32,50)(26,33,60,62)(28,35,58,64)(34,48,63,44)(36,46,61,42), (1,47,23,43)(2,27,24,57)(3,45,21,41)(4,25,22,59)(5,38,53,10)(6,14,54,30)(7,40,55,12)(8,16,56,32)(9,49,37,20)(11,51,39,18)(13,17,29,50)(15,19,31,52)(26,61,60,36)(28,63,58,34)(33,48,62,44)(35,46,64,42), (1,17,61,5)(2,51,62,54)(3,19,63,7)(4,49,64,56)(6,24,18,33)(8,22,20,35)(9,46,16,59)(10,43,13,26)(11,48,14,57)(12,41,15,28)(21,52,34,55)(23,50,36,53)(25,37,42,32)(27,39,44,30)(29,60,38,47)(31,58,40,45) );
G=PermutationGroup([[(1,61),(2,62),(3,63),(4,64),(5,17),(6,18),(7,19),(8,20),(9,16),(10,13),(11,14),(12,15),(21,34),(22,35),(23,36),(24,33),(25,42),(26,43),(27,44),(28,41),(29,38),(30,39),(31,40),(32,37),(45,58),(46,59),(47,60),(48,57),(49,56),(50,53),(51,54),(52,55)], [(1,36),(2,33),(3,34),(4,35),(5,50),(6,51),(7,52),(8,49),(9,32),(10,29),(11,30),(12,31),(13,38),(14,39),(15,40),(16,37),(17,53),(18,54),(19,55),(20,56),(21,63),(22,64),(23,61),(24,62),(25,46),(26,47),(27,48),(28,45),(41,58),(42,59),(43,60),(44,57)], [(1,21),(2,22),(3,23),(4,24),(5,55),(6,56),(7,53),(8,54),(9,39),(10,40),(11,37),(12,38),(13,31),(14,32),(15,29),(16,30),(17,52),(18,49),(19,50),(20,51),(25,57),(26,58),(27,59),(28,60),(33,64),(34,61),(35,62),(36,63),(41,47),(42,48),(43,45),(44,46)], [(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)], [(1,25,23,59),(2,43,24,47),(3,27,21,57),(4,41,22,45),(5,37,53,9),(6,29,54,13),(7,39,55,11),(8,31,56,15),(10,18,38,51),(12,20,40,49),(14,19,30,52),(16,17,32,50),(26,33,60,62),(28,35,58,64),(34,48,63,44),(36,46,61,42)], [(1,47,23,43),(2,27,24,57),(3,45,21,41),(4,25,22,59),(5,38,53,10),(6,14,54,30),(7,40,55,12),(8,16,56,32),(9,49,37,20),(11,51,39,18),(13,17,29,50),(15,19,31,52),(26,61,60,36),(28,63,58,34),(33,48,62,44),(35,46,64,42)], [(1,17,61,5),(2,51,62,54),(3,19,63,7),(4,49,64,56),(6,24,18,33),(8,22,20,35),(9,46,16,59),(10,43,13,26),(11,48,14,57),(12,41,15,28),(21,52,34,55),(23,50,36,53),(25,37,42,32),(27,39,44,30),(29,60,38,47),(31,58,40,45)]])
32 conjugacy classes
| class | 1 | 2A | ··· | 2G | 2H | 4A | ··· | 4R | 4S | ··· | 4W |
| order | 1 | 2 | ··· | 2 | 2 | 4 | ··· | 4 | 4 | ··· | 4 |
| size | 1 | 1 | ··· | 1 | 8 | 4 | ··· | 4 | 8 | ··· | 8 |
32 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | - | |
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C4○D4 | 2+ 1+4 | 2- 1+4 |
| kernel | C23.677C24 | C42⋊5C4 | C23.63C23 | C24.C22 | C23.65C23 | C23.Q8 | C23.11D4 | C23.81C23 | C23.4Q8 | C23.83C23 | C2×C4 | C22 | C22 |
| # reps | 1 | 1 | 2 | 3 | 3 | 1 | 2 | 1 | 1 | 1 | 12 | 2 | 2 |
Matrix representation of C23.677C24 ►in GL6(𝔽5)
| 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 |
| 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 |
| 3 | 0 | 0 | 0 | 0 | 0 |
| 0 | 3 | 0 | 0 | 0 | 0 |
| 0 | 0 | 4 | 3 | 0 | 0 |
| 0 | 0 | 1 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 4 | 0 |
| 4 | 4 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 3 | 1 | 0 | 0 |
| 0 | 0 | 0 | 2 | 0 | 0 |
| 0 | 0 | 0 | 0 | 3 | 0 |
| 0 | 0 | 0 | 0 | 0 | 3 |
| 3 | 3 | 0 | 0 | 0 | 0 |
| 4 | 2 | 0 | 0 | 0 | 0 |
| 0 | 0 | 2 | 0 | 0 | 0 |
| 0 | 0 | 0 | 2 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 3 |
| 0 | 0 | 0 | 0 | 3 | 0 |
| 4 | 0 | 0 | 0 | 0 | 0 |
| 0 | 4 | 0 | 0 | 0 | 0 |
| 0 | 0 | 3 | 0 | 0 | 0 |
| 0 | 0 | 2 | 2 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 1 | 0 |
G:=sub<GL(6,GF(5))| [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],[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],[3,0,0,0,0,0,0,3,0,0,0,0,0,0,4,1,0,0,0,0,3,1,0,0,0,0,0,0,0,4,0,0,0,0,1,0],[4,0,0,0,0,0,4,1,0,0,0,0,0,0,3,0,0,0,0,0,1,2,0,0,0,0,0,0,3,0,0,0,0,0,0,3],[3,4,0,0,0,0,3,2,0,0,0,0,0,0,2,0,0,0,0,0,0,2,0,0,0,0,0,0,0,3,0,0,0,0,3,0],[4,0,0,0,0,0,0,4,0,0,0,0,0,0,3,2,0,0,0,0,0,2,0,0,0,0,0,0,0,1,0,0,0,0,1,0] >;
C23.677C24 in GAP, Magma, Sage, TeX
C_2^3._{677}C_2^4 % in TeX
G:=Group("C2^3.677C2^4"); // GroupNames label
G:=SmallGroup(128,1509);
// by ID
G=gap.SmallGroup(128,1509);
# by ID
G:=PCGroup([7,-2,2,2,2,-2,2,2,784,253,344,758,723,268,1571,346,80]);
// Polycyclic
G:=Group<a,b,c,d,e,f,g|a^2=b^2=c^2=1,d^2=a*b*c,e^2=f^2=b*a=a*b,g^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