p-group, metabelian, nilpotent (class 2), monomial
Aliases: C24.82C23, C23.656C24, C22.4292+ 1+4, C42⋊5C4⋊30C2, C23.93(C4○D4), C23⋊2D4.31C2, (C23×C4).489C22, (C22×C4).576C23, (C2×C42).100C22, C23.7Q8⋊106C2, C23.10D4⋊102C2, C23.23D4⋊104C2, C23.11D4⋊113C2, C24.3C22⋊94C2, (C22×D4).271C22, C24.C22⋊162C2, C2.87(C22.32C24), C23.81C23⋊113C2, C2.28(C22.54C24), C2.C42.360C22, C2.108(C22.45C24), C2.98(C22.47C24), C2.40(C22.49C24), C2.56(C22.34C24), (C2×C4).454(C4○D4), (C2×C4⋊C4).467C22, C22.517(C2×C4○D4), (C2×C22⋊C4).306C22, SmallGroup(128,1488)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C23.656C24
G = < a,b,c,d,e,f,g | a2=b2=c2=e2=1, d2=c, f2=ba=ab, g2=a, ac=ca, ede=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: 548 in 242 conjugacy classes, 88 normal (82 characteristic)
C1, C2, C2, C4, C22, C22, C2×C4, C2×C4, D4, C23, C23, C23, C42, C22⋊C4, C4⋊C4, C22×C4, C22×C4, C2×D4, C24, C2.C42, C2×C42, C2×C22⋊C4, C2×C4⋊C4, C23×C4, C22×D4, C23.7Q8, C42⋊5C4, C23.23D4, C24.C22, C24.3C22, C23⋊2D4, C23.10D4, C23.11D4, C23.81C23, C23.656C24
Quotients: C1, C2, C22, C23, C4○D4, C24, C2×C4○D4, 2+ 1+4, C22.32C24, C22.34C24, C22.45C24, C22.47C24, C22.49C24, C22.54C24, C23.656C24
►On 64 points
Generators in S64
(1 24)(2 21)(3 22)(4 23)(5 46)(6 47)(7 48)(8 45)(9 56)(10 53)(11 54)(12 55)(13 60)(14 57)(15 58)(16 59)(17 33)(18 34)(19 35)(20 36)(25 43)(26 44)(27 41)(28 42)(29 37)(30 38)(31 39)(32 40)(49 64)(50 61)(51 62)(52 63)
(1 57)(2 58)(3 59)(4 60)(5 30)(6 31)(7 32)(8 29)(9 43)(10 44)(11 41)(12 42)(13 23)(14 24)(15 21)(16 22)(17 49)(18 50)(19 51)(20 52)(25 56)(26 53)(27 54)(28 55)(33 64)(34 61)(35 62)(36 63)(37 45)(38 46)(39 47)(40 48)
(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 38)(2 31)(3 40)(4 29)(5 14)(6 58)(7 16)(8 60)(9 20)(10 33)(11 18)(12 35)(13 45)(15 47)(17 53)(19 55)(21 39)(22 32)(23 37)(24 30)(25 63)(26 49)(27 61)(28 51)(34 54)(36 56)(41 50)(42 62)(43 52)(44 64)(46 57)(48 59)
(1 47 14 31)(2 40 15 7)(3 45 16 29)(4 38 13 5)(6 57 39 24)(8 59 37 22)(9 35 25 51)(10 63 26 20)(11 33 27 49)(12 61 28 18)(17 41 64 54)(19 43 62 56)(21 32 58 48)(23 30 60 46)(34 55 50 42)(36 53 52 44)
(1 50 24 61)(2 35 21 19)(3 52 22 63)(4 33 23 17)(5 11 46 54)(6 28 47 42)(7 9 48 56)(8 26 45 44)(10 29 53 37)(12 31 55 39)(13 49 60 64)(14 34 57 18)(15 51 58 62)(16 36 59 20)(25 32 43 40)(27 30 41 38)
G:=sub<Sym(64)| (1,24)(2,21)(3,22)(4,23)(5,46)(6,47)(7,48)(8,45)(9,56)(10,53)(11,54)(12,55)(13,60)(14,57)(15,58)(16,59)(17,33)(18,34)(19,35)(20,36)(25,43)(26,44)(27,41)(28,42)(29,37)(30,38)(31,39)(32,40)(49,64)(50,61)(51,62)(52,63), (1,57)(2,58)(3,59)(4,60)(5,30)(6,31)(7,32)(8,29)(9,43)(10,44)(11,41)(12,42)(13,23)(14,24)(15,21)(16,22)(17,49)(18,50)(19,51)(20,52)(25,56)(26,53)(27,54)(28,55)(33,64)(34,61)(35,62)(36,63)(37,45)(38,46)(39,47)(40,48), (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), (1,38)(2,31)(3,40)(4,29)(5,14)(6,58)(7,16)(8,60)(9,20)(10,33)(11,18)(12,35)(13,45)(15,47)(17,53)(19,55)(21,39)(22,32)(23,37)(24,30)(25,63)(26,49)(27,61)(28,51)(34,54)(36,56)(41,50)(42,62)(43,52)(44,64)(46,57)(48,59), (1,47,14,31)(2,40,15,7)(3,45,16,29)(4,38,13,5)(6,57,39,24)(8,59,37,22)(9,35,25,51)(10,63,26,20)(11,33,27,49)(12,61,28,18)(17,41,64,54)(19,43,62,56)(21,32,58,48)(23,30,60,46)(34,55,50,42)(36,53,52,44), (1,50,24,61)(2,35,21,19)(3,52,22,63)(4,33,23,17)(5,11,46,54)(6,28,47,42)(7,9,48,56)(8,26,45,44)(10,29,53,37)(12,31,55,39)(13,49,60,64)(14,34,57,18)(15,51,58,62)(16,36,59,20)(25,32,43,40)(27,30,41,38)>;
G:=Group( (1,24)(2,21)(3,22)(4,23)(5,46)(6,47)(7,48)(8,45)(9,56)(10,53)(11,54)(12,55)(13,60)(14,57)(15,58)(16,59)(17,33)(18,34)(19,35)(20,36)(25,43)(26,44)(27,41)(28,42)(29,37)(30,38)(31,39)(32,40)(49,64)(50,61)(51,62)(52,63), (1,57)(2,58)(3,59)(4,60)(5,30)(6,31)(7,32)(8,29)(9,43)(10,44)(11,41)(12,42)(13,23)(14,24)(15,21)(16,22)(17,49)(18,50)(19,51)(20,52)(25,56)(26,53)(27,54)(28,55)(33,64)(34,61)(35,62)(36,63)(37,45)(38,46)(39,47)(40,48), (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), (1,38)(2,31)(3,40)(4,29)(5,14)(6,58)(7,16)(8,60)(9,20)(10,33)(11,18)(12,35)(13,45)(15,47)(17,53)(19,55)(21,39)(22,32)(23,37)(24,30)(25,63)(26,49)(27,61)(28,51)(34,54)(36,56)(41,50)(42,62)(43,52)(44,64)(46,57)(48,59), (1,47,14,31)(2,40,15,7)(3,45,16,29)(4,38,13,5)(6,57,39,24)(8,59,37,22)(9,35,25,51)(10,63,26,20)(11,33,27,49)(12,61,28,18)(17,41,64,54)(19,43,62,56)(21,32,58,48)(23,30,60,46)(34,55,50,42)(36,53,52,44), (1,50,24,61)(2,35,21,19)(3,52,22,63)(4,33,23,17)(5,11,46,54)(6,28,47,42)(7,9,48,56)(8,26,45,44)(10,29,53,37)(12,31,55,39)(13,49,60,64)(14,34,57,18)(15,51,58,62)(16,36,59,20)(25,32,43,40)(27,30,41,38) );
G=PermutationGroup([[(1,24),(2,21),(3,22),(4,23),(5,46),(6,47),(7,48),(8,45),(9,56),(10,53),(11,54),(12,55),(13,60),(14,57),(15,58),(16,59),(17,33),(18,34),(19,35),(20,36),(25,43),(26,44),(27,41),(28,42),(29,37),(30,38),(31,39),(32,40),(49,64),(50,61),(51,62),(52,63)], [(1,57),(2,58),(3,59),(4,60),(5,30),(6,31),(7,32),(8,29),(9,43),(10,44),(11,41),(12,42),(13,23),(14,24),(15,21),(16,22),(17,49),(18,50),(19,51),(20,52),(25,56),(26,53),(27,54),(28,55),(33,64),(34,61),(35,62),(36,63),(37,45),(38,46),(39,47),(40,48)], [(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)], [(1,38),(2,31),(3,40),(4,29),(5,14),(6,58),(7,16),(8,60),(9,20),(10,33),(11,18),(12,35),(13,45),(15,47),(17,53),(19,55),(21,39),(22,32),(23,37),(24,30),(25,63),(26,49),(27,61),(28,51),(34,54),(36,56),(41,50),(42,62),(43,52),(44,64),(46,57),(48,59)], [(1,47,14,31),(2,40,15,7),(3,45,16,29),(4,38,13,5),(6,57,39,24),(8,59,37,22),(9,35,25,51),(10,63,26,20),(11,33,27,49),(12,61,28,18),(17,41,64,54),(19,43,62,56),(21,32,58,48),(23,30,60,46),(34,55,50,42),(36,53,52,44)], [(1,50,24,61),(2,35,21,19),(3,52,22,63),(4,33,23,17),(5,11,46,54),(6,28,47,42),(7,9,48,56),(8,26,45,44),(10,29,53,37),(12,31,55,39),(13,49,60,64),(14,34,57,18),(15,51,58,62),(16,36,59,20),(25,32,43,40),(27,30,41,38)]])
32 conjugacy classes
| class | 1 | 2A | ··· | 2G | 2H | 2I | 2J | 2K | 4A | ··· | 4P | 4Q | 4R | 4S | 4T |
| order | 1 | 2 | ··· | 2 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 4 | 4 | 4 | 4 |
| size | 1 | 1 | ··· | 1 | 4 | 4 | 8 | 8 | 4 | ··· | 4 | 8 | 8 | 8 | 8 |
32 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | ||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C4○D4 | C4○D4 | 2+ 1+4 |
| kernel | C23.656C24 | C23.7Q8 | C42⋊5C4 | C23.23D4 | C24.C22 | C24.3C22 | C23⋊2D4 | C23.10D4 | C23.11D4 | C23.81C23 | C2×C4 | C23 | C22 |
| # reps | 1 | 1 | 1 | 2 | 4 | 1 | 1 | 3 | 1 | 1 | 8 | 4 | 4 |
Matrix representation of C23.656C24 ►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 | 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 | 1 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 2 | 0 | 0 |
| 0 | 0 | 3 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 3 | 0 |
| 0 | 0 | 0 | 0 | 0 | 3 |
| 3 | 2 | 0 | 0 | 0 | 0 |
| 1 | 2 | 0 | 0 | 0 | 0 |
| 0 | 0 | 4 | 0 | 0 | 0 |
| 0 | 0 | 0 | 4 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 3 |
| 0 | 0 | 0 | 0 | 2 | 0 |
| 3 | 0 | 0 | 0 | 0 | 0 |
| 0 | 3 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 3 | 0 | 0 |
| 0 | 0 | 3 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 3 | 0 | 0 | 0 | 0 | 0 |
| 1 | 2 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 4 | 0 |
| 0 | 0 | 0 | 0 | 0 | 4 |
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],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[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],[4,0,0,0,0,0,1,1,0,0,0,0,0,0,0,3,0,0,0,0,2,0,0,0,0,0,0,0,3,0,0,0,0,0,0,3],[3,1,0,0,0,0,2,2,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,0,2,0,0,0,0,3,0],[3,0,0,0,0,0,0,3,0,0,0,0,0,0,0,3,0,0,0,0,3,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0],[3,1,0,0,0,0,0,2,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,4,0,0,0,0,0,0,4] >;
C23.656C24 in GAP, Magma, Sage, TeX
C_2^3._{656}C_2^4 % in TeX
G:=Group("C2^3.656C2^4"); // GroupNames label
G:=SmallGroup(128,1488);
// by ID
G=gap.SmallGroup(128,1488);
# by ID
G:=PCGroup([7,-2,2,2,2,-2,2,2,448,253,758,723,268,1571,346,80]);
// Polycyclic
G:=Group<a,b,c,d,e,f,g|a^2=b^2=c^2=e^2=1,d^2=c,f^2=b*a=a*b,g^2=a,a*c=c*a,e*d*e=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