p-group, metabelian, nilpotent (class 2), monomial
Aliases: C23.255C24, C24.222C23, C22.862+ 1+4, C22.632- 1+4, C42⋊30(C2×C4), C42⋊2C2⋊2C4, C42⋊4C4⋊17C2, (C23×C4).60C22, C23.24(C22×C4), C23.8Q8.9C2, (C2×C42).445C22, (C22×C4).774C23, C22.146(C23×C4), C24.C22.8C2, C23.65C23⋊29C2, C23.63C23⋊22C2, C2.8(C22.45C24), C2.C42.482C22, C2.8(C22.50C24), C2.11(C22.46C24), C2.37(C23.33C23), C2.10(C22.47C24), (C4×C4⋊C4)⋊49C2, C4⋊C4⋊18(C2×C4), C2.42(C4×C4○D4), (C4×C22⋊C4).32C2, C22⋊C4.12(C2×C4), (C2×C4).52(C22×C4), (C2×C4).727(C4○D4), (C2×C4⋊C4).831C22, (C2×C42⋊2C2).3C2, C22.140(C2×C4○D4), (C2×C22⋊C4).446C22, SmallGroup(128,1105)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C23.255C24
G = < a,b,c,d,e,f,g | a2=b2=c2=1, d2=e2=c, f2=a, g2=ba=ab, 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, ce=ec, cf=fc, cg=gc, dg=gd, ef=fe, fg=gf >
Subgroups: 396 in 248 conjugacy classes, 140 normal (82 characteristic)
C1, C2, C2, C4, C22, C22, C2×C4, C2×C4, C23, C23, C23, C42, C42, C22⋊C4, C22⋊C4, C4⋊C4, C4⋊C4, C22×C4, C22×C4, C24, C2.C42, C2×C42, C2×C22⋊C4, C2×C4⋊C4, C42⋊2C2, C23×C4, C42⋊4C4, C4×C22⋊C4, C4×C4⋊C4, C23.8Q8, C23.63C23, C24.C22, C23.65C23, C2×C42⋊2C2, C23.255C24
Quotients: C1, C2, C4, C22, C2×C4, C23, C22×C4, C4○D4, C24, C23×C4, C2×C4○D4, 2+ 1+4, 2- 1+4, C4×C4○D4, C23.33C23, C22.45C24, C22.46C24, C22.47C24, C22.50C24, C23.255C24
►On 64 points
Generators in S64
(1 32)(2 29)(3 30)(4 31)(5 54)(6 55)(7 56)(8 53)(9 48)(10 45)(11 46)(12 47)(13 52)(14 49)(15 50)(16 51)(17 42)(18 43)(19 44)(20 41)(21 40)(22 37)(23 38)(24 39)(25 34)(26 35)(27 36)(28 33)(57 64)(58 61)(59 62)(60 63)
(1 61)(2 62)(3 63)(4 64)(5 26)(6 27)(7 28)(8 25)(9 43)(10 44)(11 41)(12 42)(13 37)(14 38)(15 39)(16 40)(17 47)(18 48)(19 45)(20 46)(21 51)(22 52)(23 49)(24 50)(29 59)(30 60)(31 57)(32 58)(33 56)(34 53)(35 54)(36 55)
(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 18 3 20)(2 44 4 42)(5 23 7 21)(6 39 8 37)(9 60 11 58)(10 64 12 62)(13 27 15 25)(14 33 16 35)(17 29 19 31)(22 55 24 53)(26 49 28 51)(30 41 32 43)(34 52 36 50)(38 56 40 54)(45 57 47 59)(46 61 48 63)
(1 49 32 14)(2 24 29 39)(3 51 30 16)(4 22 31 37)(5 11 54 46)(6 42 55 17)(7 9 56 48)(8 44 53 19)(10 34 45 25)(12 36 47 27)(13 64 52 57)(15 62 50 59)(18 28 43 33)(20 26 41 35)(21 60 40 63)(23 58 38 61)
(1 25 58 53)(2 26 59 54)(3 27 60 55)(4 28 57 56)(5 29 35 62)(6 30 36 63)(7 31 33 64)(8 32 34 61)(9 37 18 52)(10 38 19 49)(11 39 20 50)(12 40 17 51)(13 48 22 43)(14 45 23 44)(15 46 24 41)(16 47 21 42)
G:=sub<Sym(64)| (1,32)(2,29)(3,30)(4,31)(5,54)(6,55)(7,56)(8,53)(9,48)(10,45)(11,46)(12,47)(13,52)(14,49)(15,50)(16,51)(17,42)(18,43)(19,44)(20,41)(21,40)(22,37)(23,38)(24,39)(25,34)(26,35)(27,36)(28,33)(57,64)(58,61)(59,62)(60,63), (1,61)(2,62)(3,63)(4,64)(5,26)(6,27)(7,28)(8,25)(9,43)(10,44)(11,41)(12,42)(13,37)(14,38)(15,39)(16,40)(17,47)(18,48)(19,45)(20,46)(21,51)(22,52)(23,49)(24,50)(29,59)(30,60)(31,57)(32,58)(33,56)(34,53)(35,54)(36,55), (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), (1,18,3,20)(2,44,4,42)(5,23,7,21)(6,39,8,37)(9,60,11,58)(10,64,12,62)(13,27,15,25)(14,33,16,35)(17,29,19,31)(22,55,24,53)(26,49,28,51)(30,41,32,43)(34,52,36,50)(38,56,40,54)(45,57,47,59)(46,61,48,63), (1,49,32,14)(2,24,29,39)(3,51,30,16)(4,22,31,37)(5,11,54,46)(6,42,55,17)(7,9,56,48)(8,44,53,19)(10,34,45,25)(12,36,47,27)(13,64,52,57)(15,62,50,59)(18,28,43,33)(20,26,41,35)(21,60,40,63)(23,58,38,61), (1,25,58,53)(2,26,59,54)(3,27,60,55)(4,28,57,56)(5,29,35,62)(6,30,36,63)(7,31,33,64)(8,32,34,61)(9,37,18,52)(10,38,19,49)(11,39,20,50)(12,40,17,51)(13,48,22,43)(14,45,23,44)(15,46,24,41)(16,47,21,42)>;
G:=Group( (1,32)(2,29)(3,30)(4,31)(5,54)(6,55)(7,56)(8,53)(9,48)(10,45)(11,46)(12,47)(13,52)(14,49)(15,50)(16,51)(17,42)(18,43)(19,44)(20,41)(21,40)(22,37)(23,38)(24,39)(25,34)(26,35)(27,36)(28,33)(57,64)(58,61)(59,62)(60,63), (1,61)(2,62)(3,63)(4,64)(5,26)(6,27)(7,28)(8,25)(9,43)(10,44)(11,41)(12,42)(13,37)(14,38)(15,39)(16,40)(17,47)(18,48)(19,45)(20,46)(21,51)(22,52)(23,49)(24,50)(29,59)(30,60)(31,57)(32,58)(33,56)(34,53)(35,54)(36,55), (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), (1,18,3,20)(2,44,4,42)(5,23,7,21)(6,39,8,37)(9,60,11,58)(10,64,12,62)(13,27,15,25)(14,33,16,35)(17,29,19,31)(22,55,24,53)(26,49,28,51)(30,41,32,43)(34,52,36,50)(38,56,40,54)(45,57,47,59)(46,61,48,63), (1,49,32,14)(2,24,29,39)(3,51,30,16)(4,22,31,37)(5,11,54,46)(6,42,55,17)(7,9,56,48)(8,44,53,19)(10,34,45,25)(12,36,47,27)(13,64,52,57)(15,62,50,59)(18,28,43,33)(20,26,41,35)(21,60,40,63)(23,58,38,61), (1,25,58,53)(2,26,59,54)(3,27,60,55)(4,28,57,56)(5,29,35,62)(6,30,36,63)(7,31,33,64)(8,32,34,61)(9,37,18,52)(10,38,19,49)(11,39,20,50)(12,40,17,51)(13,48,22,43)(14,45,23,44)(15,46,24,41)(16,47,21,42) );
G=PermutationGroup([[(1,32),(2,29),(3,30),(4,31),(5,54),(6,55),(7,56),(8,53),(9,48),(10,45),(11,46),(12,47),(13,52),(14,49),(15,50),(16,51),(17,42),(18,43),(19,44),(20,41),(21,40),(22,37),(23,38),(24,39),(25,34),(26,35),(27,36),(28,33),(57,64),(58,61),(59,62),(60,63)], [(1,61),(2,62),(3,63),(4,64),(5,26),(6,27),(7,28),(8,25),(9,43),(10,44),(11,41),(12,42),(13,37),(14,38),(15,39),(16,40),(17,47),(18,48),(19,45),(20,46),(21,51),(22,52),(23,49),(24,50),(29,59),(30,60),(31,57),(32,58),(33,56),(34,53),(35,54),(36,55)], [(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)], [(1,18,3,20),(2,44,4,42),(5,23,7,21),(6,39,8,37),(9,60,11,58),(10,64,12,62),(13,27,15,25),(14,33,16,35),(17,29,19,31),(22,55,24,53),(26,49,28,51),(30,41,32,43),(34,52,36,50),(38,56,40,54),(45,57,47,59),(46,61,48,63)], [(1,49,32,14),(2,24,29,39),(3,51,30,16),(4,22,31,37),(5,11,54,46),(6,42,55,17),(7,9,56,48),(8,44,53,19),(10,34,45,25),(12,36,47,27),(13,64,52,57),(15,62,50,59),(18,28,43,33),(20,26,41,35),(21,60,40,63),(23,58,38,61)], [(1,25,58,53),(2,26,59,54),(3,27,60,55),(4,28,57,56),(5,29,35,62),(6,30,36,63),(7,31,33,64),(8,32,34,61),(9,37,18,52),(10,38,19,49),(11,39,20,50),(12,40,17,51),(13,48,22,43),(14,45,23,44),(15,46,24,41),(16,47,21,42)]])
50 conjugacy classes
| class | 1 | 2A | ··· | 2G | 2H | 2I | 4A | ··· | 4X | 4Y | ··· | 4AN |
| order | 1 | 2 | ··· | 2 | 2 | 2 | 4 | ··· | 4 | 4 | ··· | 4 |
| size | 1 | 1 | ··· | 1 | 4 | 4 | 2 | ··· | 2 | 4 | ··· | 4 |
50 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 | C4 | C4○D4 | 2+ 1+4 | 2- 1+4 |
| kernel | C23.255C24 | C42⋊4C4 | C4×C22⋊C4 | C4×C4⋊C4 | C23.8Q8 | C23.63C23 | C24.C22 | C23.65C23 | C2×C42⋊2C2 | C42⋊2C2 | C2×C4 | C22 | C22 |
| # reps | 1 | 1 | 2 | 3 | 1 | 3 | 3 | 1 | 1 | 16 | 16 | 1 | 1 |
Matrix representation of C23.255C24 ►in GL5(𝔽5)
| 1 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 4 | 0 |
| 0 | 0 | 0 | 0 | 4 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 4 | 0 | 0 | 0 |
| 0 | 0 | 4 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 1 |
| 4 | 0 | 0 | 0 | 0 |
| 0 | 4 | 0 | 0 | 0 |
| 0 | 0 | 4 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 1 |
| 3 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 4 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 2 |
| 0 | 0 | 0 | 3 | 0 |
| 3 | 0 | 0 | 0 | 0 |
| 0 | 3 | 0 | 0 | 0 |
| 0 | 0 | 3 | 0 | 0 |
| 0 | 0 | 0 | 4 | 0 |
| 0 | 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 2 | 0 |
| 0 | 0 | 0 | 0 | 2 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 3 | 0 | 0 | 0 |
| 0 | 0 | 3 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 4 | 0 |
G:=sub<GL(5,GF(5))| [1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,4,0,0,0,0,0,4],[1,0,0,0,0,0,4,0,0,0,0,0,4,0,0,0,0,0,1,0,0,0,0,0,1],[4,0,0,0,0,0,4,0,0,0,0,0,4,0,0,0,0,0,1,0,0,0,0,0,1],[3,0,0,0,0,0,0,4,0,0,0,1,0,0,0,0,0,0,0,3,0,0,0,2,0],[3,0,0,0,0,0,3,0,0,0,0,0,3,0,0,0,0,0,4,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,1,0,0,0,0,0,0,2,0,0,0,0,0,2],[1,0,0,0,0,0,3,0,0,0,0,0,3,0,0,0,0,0,0,4,0,0,0,1,0] >;
C23.255C24 in GAP, Magma, Sage, TeX
C_2^3._{255}C_2^4 % in TeX
G:=Group("C2^3.255C2^4"); // GroupNames label
G:=SmallGroup(128,1105);
// by ID
G=gap.SmallGroup(128,1105);
# by ID
G:=PCGroup([7,-2,2,2,2,-2,2,2,448,253,456,758,100,346,192]);
// Polycyclic
G:=Group<a,b,c,d,e,f,g|a^2=b^2=c^2=1,d^2=e^2=c,f^2=a,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,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,c*e=e*c,c*f=f*c,c*g=g*c,d*g=g*d,e*f=f*e,f*g=g*f>;
// generators/relations