p-group, metabelian, nilpotent (class 2), monomial
Aliases: C24.81C23, C23.654C24, C22.4272+ 1+4, C22.3222- 1+4, C42⋊8C4⋊60C2, (C2×C42).99C22, C23⋊Q8.24C2, (C22×C4).575C23, C23.4Q8.24C2, C23.11D4.45C2, (C22×Q8).208C22, C23.67C23⋊97C2, C24.C22.63C2, C23.81C23⋊112C2, C23.83C23⋊104C2, C23.63C23⋊168C2, C2.C42.358C22, C2.106(C22.45C24), C2.39(C22.53C24), C2.33(C22.56C24), C2.51(C22.35C24), C2.98(C22.46C24), C2.97(C22.36C24), (C2×C4).453(C4○D4), (C2×C4⋊C4).465C22, C22.515(C2×C4○D4), (C2×C22⋊C4).69C22, SmallGroup(128,1486)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C23.654C24
G = < a,b,c,d,e,f,g | a2=b2=c2=1, d2=c, e2=g2=a, f2=b, ab=ba, 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: 372 in 193 conjugacy classes, 88 normal (82 characteristic)
C1, C2, C2, C4, C22, C22, C2×C4, C2×C4, Q8, C23, C23, C42, C22⋊C4, C4⋊C4, C22×C4, C2×Q8, C24, C2.C42, C2×C42, C2×C22⋊C4, C2×C4⋊C4, C22×Q8, C42⋊8C4, C23.63C23, C24.C22, C23.67C23, C23⋊Q8, C23.11D4, C23.81C23, C23.4Q8, C23.83C23, C23.654C24
Quotients: C1, C2, C22, C23, C4○D4, C24, C2×C4○D4, 2+ 1+4, 2- 1+4, C22.35C24, C22.36C24, C22.45C24, C22.46C24, C22.53C24, C22.56C24, C23.654C24
►On 64 points
Generators in S64
(1 19)(2 20)(3 17)(4 18)(5 46)(6 47)(7 48)(8 45)(9 56)(10 53)(11 54)(12 55)(13 60)(14 57)(15 58)(16 59)(21 33)(22 34)(23 35)(24 36)(25 39)(26 40)(27 37)(28 38)(29 43)(30 44)(31 41)(32 42)(49 64)(50 61)(51 62)(52 63)
(1 57)(2 58)(3 59)(4 60)(5 28)(6 25)(7 26)(8 27)(9 29)(10 30)(11 31)(12 32)(13 18)(14 19)(15 20)(16 17)(21 49)(22 50)(23 51)(24 52)(33 64)(34 61)(35 62)(36 63)(37 45)(38 46)(39 47)(40 48)(41 54)(42 55)(43 56)(44 53)
(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 19 38)(2 39 20 25)(3 26 17 40)(4 37 18 27)(5 14 46 57)(6 58 47 15)(7 16 48 59)(8 60 45 13)(9 24 56 36)(10 33 53 21)(11 22 54 34)(12 35 55 23)(29 52 43 63)(30 64 44 49)(31 50 41 61)(32 62 42 51)
(1 47 57 39)(2 40 58 48)(3 45 59 37)(4 38 60 46)(5 18 28 13)(6 14 25 19)(7 20 26 15)(8 16 27 17)(9 23 29 51)(10 52 30 24)(11 21 31 49)(12 50 32 22)(33 41 64 54)(34 55 61 42)(35 43 62 56)(36 53 63 44)
(1 50 19 61)(2 35 20 23)(3 52 17 63)(4 33 18 21)(5 11 46 54)(6 42 47 32)(7 9 48 56)(8 44 45 30)(10 27 53 37)(12 25 55 39)(13 49 60 64)(14 34 57 22)(15 51 58 62)(16 36 59 24)(26 29 40 43)(28 31 38 41)
G:=sub<Sym(64)| (1,19)(2,20)(3,17)(4,18)(5,46)(6,47)(7,48)(8,45)(9,56)(10,53)(11,54)(12,55)(13,60)(14,57)(15,58)(16,59)(21,33)(22,34)(23,35)(24,36)(25,39)(26,40)(27,37)(28,38)(29,43)(30,44)(31,41)(32,42)(49,64)(50,61)(51,62)(52,63), (1,57)(2,58)(3,59)(4,60)(5,28)(6,25)(7,26)(8,27)(9,29)(10,30)(11,31)(12,32)(13,18)(14,19)(15,20)(16,17)(21,49)(22,50)(23,51)(24,52)(33,64)(34,61)(35,62)(36,63)(37,45)(38,46)(39,47)(40,48)(41,54)(42,55)(43,56)(44,53), (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), (1,28,19,38)(2,39,20,25)(3,26,17,40)(4,37,18,27)(5,14,46,57)(6,58,47,15)(7,16,48,59)(8,60,45,13)(9,24,56,36)(10,33,53,21)(11,22,54,34)(12,35,55,23)(29,52,43,63)(30,64,44,49)(31,50,41,61)(32,62,42,51), (1,47,57,39)(2,40,58,48)(3,45,59,37)(4,38,60,46)(5,18,28,13)(6,14,25,19)(7,20,26,15)(8,16,27,17)(9,23,29,51)(10,52,30,24)(11,21,31,49)(12,50,32,22)(33,41,64,54)(34,55,61,42)(35,43,62,56)(36,53,63,44), (1,50,19,61)(2,35,20,23)(3,52,17,63)(4,33,18,21)(5,11,46,54)(6,42,47,32)(7,9,48,56)(8,44,45,30)(10,27,53,37)(12,25,55,39)(13,49,60,64)(14,34,57,22)(15,51,58,62)(16,36,59,24)(26,29,40,43)(28,31,38,41)>;
G:=Group( (1,19)(2,20)(3,17)(4,18)(5,46)(6,47)(7,48)(8,45)(9,56)(10,53)(11,54)(12,55)(13,60)(14,57)(15,58)(16,59)(21,33)(22,34)(23,35)(24,36)(25,39)(26,40)(27,37)(28,38)(29,43)(30,44)(31,41)(32,42)(49,64)(50,61)(51,62)(52,63), (1,57)(2,58)(3,59)(4,60)(5,28)(6,25)(7,26)(8,27)(9,29)(10,30)(11,31)(12,32)(13,18)(14,19)(15,20)(16,17)(21,49)(22,50)(23,51)(24,52)(33,64)(34,61)(35,62)(36,63)(37,45)(38,46)(39,47)(40,48)(41,54)(42,55)(43,56)(44,53), (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), (1,28,19,38)(2,39,20,25)(3,26,17,40)(4,37,18,27)(5,14,46,57)(6,58,47,15)(7,16,48,59)(8,60,45,13)(9,24,56,36)(10,33,53,21)(11,22,54,34)(12,35,55,23)(29,52,43,63)(30,64,44,49)(31,50,41,61)(32,62,42,51), (1,47,57,39)(2,40,58,48)(3,45,59,37)(4,38,60,46)(5,18,28,13)(6,14,25,19)(7,20,26,15)(8,16,27,17)(9,23,29,51)(10,52,30,24)(11,21,31,49)(12,50,32,22)(33,41,64,54)(34,55,61,42)(35,43,62,56)(36,53,63,44), (1,50,19,61)(2,35,20,23)(3,52,17,63)(4,33,18,21)(5,11,46,54)(6,42,47,32)(7,9,48,56)(8,44,45,30)(10,27,53,37)(12,25,55,39)(13,49,60,64)(14,34,57,22)(15,51,58,62)(16,36,59,24)(26,29,40,43)(28,31,38,41) );
G=PermutationGroup([[(1,19),(2,20),(3,17),(4,18),(5,46),(6,47),(7,48),(8,45),(9,56),(10,53),(11,54),(12,55),(13,60),(14,57),(15,58),(16,59),(21,33),(22,34),(23,35),(24,36),(25,39),(26,40),(27,37),(28,38),(29,43),(30,44),(31,41),(32,42),(49,64),(50,61),(51,62),(52,63)], [(1,57),(2,58),(3,59),(4,60),(5,28),(6,25),(7,26),(8,27),(9,29),(10,30),(11,31),(12,32),(13,18),(14,19),(15,20),(16,17),(21,49),(22,50),(23,51),(24,52),(33,64),(34,61),(35,62),(36,63),(37,45),(38,46),(39,47),(40,48),(41,54),(42,55),(43,56),(44,53)], [(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)], [(1,28,19,38),(2,39,20,25),(3,26,17,40),(4,37,18,27),(5,14,46,57),(6,58,47,15),(7,16,48,59),(8,60,45,13),(9,24,56,36),(10,33,53,21),(11,22,54,34),(12,35,55,23),(29,52,43,63),(30,64,44,49),(31,50,41,61),(32,62,42,51)], [(1,47,57,39),(2,40,58,48),(3,45,59,37),(4,38,60,46),(5,18,28,13),(6,14,25,19),(7,20,26,15),(8,16,27,17),(9,23,29,51),(10,52,30,24),(11,21,31,49),(12,50,32,22),(33,41,64,54),(34,55,61,42),(35,43,62,56),(36,53,63,44)], [(1,50,19,61),(2,35,20,23),(3,52,17,63),(4,33,18,21),(5,11,46,54),(6,42,47,32),(7,9,48,56),(8,44,45,30),(10,27,53,37),(12,25,55,39),(13,49,60,64),(14,34,57,22),(15,51,58,62),(16,36,59,24),(26,29,40,43),(28,31,38,41)]])
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 | C23.654C24 | C42⋊8C4 | C23.63C23 | C24.C22 | C23.67C23 | C23⋊Q8 | C23.11D4 | C23.81C23 | C23.4Q8 | C23.83C23 | C2×C4 | C22 | C22 |
| # reps | 1 | 1 | 3 | 3 | 2 | 1 | 2 | 1 | 1 | 1 | 12 | 2 | 2 |
Matrix representation of C23.654C24 ►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 |
| 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 |
| 0 | 3 | 0 | 0 | 0 | 0 |
| 2 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 3 | 2 | 0 | 0 |
| 0 | 0 | 1 | 2 | 0 | 0 |
| 0 | 0 | 0 | 0 | 2 | 0 |
| 0 | 0 | 0 | 0 | 0 | 2 |
| 4 | 0 | 0 | 0 | 0 | 0 |
| 0 | 4 | 0 | 0 | 0 | 0 |
| 0 | 0 | 2 | 3 | 0 | 0 |
| 0 | 0 | 0 | 3 | 0 | 0 |
| 0 | 0 | 0 | 0 | 3 | 1 |
| 0 | 0 | 0 | 0 | 2 | 2 |
| 0 | 3 | 0 | 0 | 0 | 0 |
| 3 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 4 | 0 | 0 | 0 |
| 0 | 0 | 0 | 4 | 0 | 0 |
| 0 | 0 | 0 | 0 | 4 | 3 |
| 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 | 1 | 2 | 0 | 0 |
| 0 | 0 | 0 | 0 | 4 | 0 |
| 0 | 0 | 0 | 0 | 0 | 4 |
G:=sub<GL(6,GF(5))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[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],[0,2,0,0,0,0,3,0,0,0,0,0,0,0,3,1,0,0,0,0,2,2,0,0,0,0,0,0,2,0,0,0,0,0,0,2],[4,0,0,0,0,0,0,4,0,0,0,0,0,0,2,0,0,0,0,0,3,3,0,0,0,0,0,0,3,2,0,0,0,0,1,2],[0,3,0,0,0,0,3,0,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,3,1],[0,1,0,0,0,0,1,0,0,0,0,0,0,0,3,1,0,0,0,0,0,2,0,0,0,0,0,0,4,0,0,0,0,0,0,4] >;
C23.654C24 in GAP, Magma, Sage, TeX
C_2^3._{654}C_2^4 % in TeX
G:=Group("C2^3.654C2^4"); // GroupNames label
G:=SmallGroup(128,1486);
// by ID
G=gap.SmallGroup(128,1486);
# by ID
G:=PCGroup([7,-2,2,2,2,-2,2,2,448,253,120,758,723,184,1571,346,80]);
// Polycyclic
G:=Group<a,b,c,d,e,f,g|a^2=b^2=c^2=1,d^2=c,e^2=g^2=a,f^2=b,a*b=b*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