p-group, metabelian, nilpotent (class 2), monomial
Aliases: C24.221C23, C23.254C24, C22.852+ 1+4, C42⋊29(C2×C4), C4.4D4⋊25C4, C42⋊4C4⋊16C2, (C23×C4).59C22, C23.23(C22×C4), C22.145(C23×C4), (C2×C42).444C22, (C22×C4).773C23, (C22×Q8).93C22, C24.C22⋊23C2, C23.23D4.10C2, (C22×D4).110C22, C23.67C23⋊24C2, C2.7(C22.45C24), C2.35(C22.11C24), C2.C42.481C22, C2.5(C22.49C24), (C2×Q8)⋊16(C2×C4), C2.41(C4×C4○D4), (C4×C22⋊C4)⋊45C2, C22⋊C4⋊32(C2×C4), (C2×D4).131(C2×C4), (C2×C4).726(C4○D4), (C2×C4⋊C4).193C22, (C2×C4).290(C22×C4), (C2×C4.4D4).18C2, C22.139(C2×C4○D4), (C2×C22⋊C4).445C22, SmallGroup(128,1104)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C24.221C23
G = < a,b,c,d,e,f,g | a2=b2=c2=d2=1, e2=d, f2=b, g2=c, eae-1=gag-1=ab=ba, faf-1=ac=ca, ad=da, bc=cb, bd=db, be=eb, bf=fb, bg=gb, cd=dc, fef-1=ce=ec, cf=fc, cg=gc, de=ed, df=fd, dg=gd, eg=ge, fg=gf >
Subgroups: 508 in 280 conjugacy classes, 140 normal (12 characteristic)
C1, C2, C2, C2, C4, C22, C22, C22, C2×C4, C2×C4, D4, Q8, C23, C23, C23, C42, C42, C22⋊C4, C22⋊C4, C4⋊C4, C22×C4, C22×C4, C22×C4, C2×D4, C2×D4, C2×Q8, C2×Q8, C24, C2.C42, C2×C42, C2×C42, C2×C22⋊C4, C2×C4⋊C4, C4.4D4, C23×C4, C22×D4, C22×Q8, C42⋊4C4, C4×C22⋊C4, C23.23D4, C24.C22, C23.67C23, C2×C4.4D4, C24.221C23
Quotients: C1, C2, C4, C22, C2×C4, C23, C22×C4, C4○D4, C24, C23×C4, C2×C4○D4, 2+ 1+4, C4×C4○D4, C22.11C24, C22.45C24, C22.49C24, C24.221C23
►On 64 points
Generators in S64
(1 20)(2 41)(3 18)(4 43)(5 39)(6 21)(7 37)(8 23)(9 33)(10 29)(11 35)(12 31)(13 64)(14 58)(15 62)(16 60)(17 27)(19 25)(22 56)(24 54)(26 44)(28 42)(30 50)(32 52)(34 49)(36 51)(38 53)(40 55)(45 61)(46 59)(47 63)(48 57)
(1 26)(2 27)(3 28)(4 25)(5 54)(6 55)(7 56)(8 53)(9 52)(10 49)(11 50)(12 51)(13 48)(14 45)(15 46)(16 47)(17 41)(18 42)(19 43)(20 44)(21 40)(22 37)(23 38)(24 39)(29 34)(30 35)(31 36)(32 33)(57 64)(58 61)(59 62)(60 63)
(1 61)(2 62)(3 63)(4 64)(5 35)(6 36)(7 33)(8 34)(9 37)(10 38)(11 39)(12 40)(13 43)(14 44)(15 41)(16 42)(17 46)(18 47)(19 48)(20 45)(21 51)(22 52)(23 49)(24 50)(25 57)(26 58)(27 59)(28 60)(29 53)(30 54)(31 55)(32 56)
(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 49 26 10)(2 24 27 39)(3 51 28 12)(4 22 25 37)(5 15 54 46)(6 42 55 18)(7 13 56 48)(8 44 53 20)(9 64 52 57)(11 62 50 59)(14 29 45 34)(16 31 47 36)(17 35 41 30)(19 33 43 32)(21 60 40 63)(23 58 38 61)
(1 34 61 8)(2 35 62 5)(3 36 63 6)(4 33 64 7)(9 48 37 19)(10 45 38 20)(11 46 39 17)(12 47 40 18)(13 22 43 52)(14 23 44 49)(15 24 41 50)(16 21 42 51)(25 32 57 56)(26 29 58 53)(27 30 59 54)(28 31 60 55)
G:=sub<Sym(64)| (1,20)(2,41)(3,18)(4,43)(5,39)(6,21)(7,37)(8,23)(9,33)(10,29)(11,35)(12,31)(13,64)(14,58)(15,62)(16,60)(17,27)(19,25)(22,56)(24,54)(26,44)(28,42)(30,50)(32,52)(34,49)(36,51)(38,53)(40,55)(45,61)(46,59)(47,63)(48,57), (1,26)(2,27)(3,28)(4,25)(5,54)(6,55)(7,56)(8,53)(9,52)(10,49)(11,50)(12,51)(13,48)(14,45)(15,46)(16,47)(17,41)(18,42)(19,43)(20,44)(21,40)(22,37)(23,38)(24,39)(29,34)(30,35)(31,36)(32,33)(57,64)(58,61)(59,62)(60,63), (1,61)(2,62)(3,63)(4,64)(5,35)(6,36)(7,33)(8,34)(9,37)(10,38)(11,39)(12,40)(13,43)(14,44)(15,41)(16,42)(17,46)(18,47)(19,48)(20,45)(21,51)(22,52)(23,49)(24,50)(25,57)(26,58)(27,59)(28,60)(29,53)(30,54)(31,55)(32,56), (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), (1,49,26,10)(2,24,27,39)(3,51,28,12)(4,22,25,37)(5,15,54,46)(6,42,55,18)(7,13,56,48)(8,44,53,20)(9,64,52,57)(11,62,50,59)(14,29,45,34)(16,31,47,36)(17,35,41,30)(19,33,43,32)(21,60,40,63)(23,58,38,61), (1,34,61,8)(2,35,62,5)(3,36,63,6)(4,33,64,7)(9,48,37,19)(10,45,38,20)(11,46,39,17)(12,47,40,18)(13,22,43,52)(14,23,44,49)(15,24,41,50)(16,21,42,51)(25,32,57,56)(26,29,58,53)(27,30,59,54)(28,31,60,55)>;
G:=Group( (1,20)(2,41)(3,18)(4,43)(5,39)(6,21)(7,37)(8,23)(9,33)(10,29)(11,35)(12,31)(13,64)(14,58)(15,62)(16,60)(17,27)(19,25)(22,56)(24,54)(26,44)(28,42)(30,50)(32,52)(34,49)(36,51)(38,53)(40,55)(45,61)(46,59)(47,63)(48,57), (1,26)(2,27)(3,28)(4,25)(5,54)(6,55)(7,56)(8,53)(9,52)(10,49)(11,50)(12,51)(13,48)(14,45)(15,46)(16,47)(17,41)(18,42)(19,43)(20,44)(21,40)(22,37)(23,38)(24,39)(29,34)(30,35)(31,36)(32,33)(57,64)(58,61)(59,62)(60,63), (1,61)(2,62)(3,63)(4,64)(5,35)(6,36)(7,33)(8,34)(9,37)(10,38)(11,39)(12,40)(13,43)(14,44)(15,41)(16,42)(17,46)(18,47)(19,48)(20,45)(21,51)(22,52)(23,49)(24,50)(25,57)(26,58)(27,59)(28,60)(29,53)(30,54)(31,55)(32,56), (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), (1,49,26,10)(2,24,27,39)(3,51,28,12)(4,22,25,37)(5,15,54,46)(6,42,55,18)(7,13,56,48)(8,44,53,20)(9,64,52,57)(11,62,50,59)(14,29,45,34)(16,31,47,36)(17,35,41,30)(19,33,43,32)(21,60,40,63)(23,58,38,61), (1,34,61,8)(2,35,62,5)(3,36,63,6)(4,33,64,7)(9,48,37,19)(10,45,38,20)(11,46,39,17)(12,47,40,18)(13,22,43,52)(14,23,44,49)(15,24,41,50)(16,21,42,51)(25,32,57,56)(26,29,58,53)(27,30,59,54)(28,31,60,55) );
G=PermutationGroup([[(1,20),(2,41),(3,18),(4,43),(5,39),(6,21),(7,37),(8,23),(9,33),(10,29),(11,35),(12,31),(13,64),(14,58),(15,62),(16,60),(17,27),(19,25),(22,56),(24,54),(26,44),(28,42),(30,50),(32,52),(34,49),(36,51),(38,53),(40,55),(45,61),(46,59),(47,63),(48,57)], [(1,26),(2,27),(3,28),(4,25),(5,54),(6,55),(7,56),(8,53),(9,52),(10,49),(11,50),(12,51),(13,48),(14,45),(15,46),(16,47),(17,41),(18,42),(19,43),(20,44),(21,40),(22,37),(23,38),(24,39),(29,34),(30,35),(31,36),(32,33),(57,64),(58,61),(59,62),(60,63)], [(1,61),(2,62),(3,63),(4,64),(5,35),(6,36),(7,33),(8,34),(9,37),(10,38),(11,39),(12,40),(13,43),(14,44),(15,41),(16,42),(17,46),(18,47),(19,48),(20,45),(21,51),(22,52),(23,49),(24,50),(25,57),(26,58),(27,59),(28,60),(29,53),(30,54),(31,55),(32,56)], [(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)], [(1,49,26,10),(2,24,27,39),(3,51,28,12),(4,22,25,37),(5,15,54,46),(6,42,55,18),(7,13,56,48),(8,44,53,20),(9,64,52,57),(11,62,50,59),(14,29,45,34),(16,31,47,36),(17,35,41,30),(19,33,43,32),(21,60,40,63),(23,58,38,61)], [(1,34,61,8),(2,35,62,5),(3,36,63,6),(4,33,64,7),(9,48,37,19),(10,45,38,20),(11,46,39,17),(12,47,40,18),(13,22,43,52),(14,23,44,49),(15,24,41,50),(16,21,42,51),(25,32,57,56),(26,29,58,53),(27,30,59,54),(28,31,60,55)]])
50 conjugacy classes
| class | 1 | 2A | ··· | 2G | 2H | 2I | 2J | 2K | 4A | ··· | 4X | 4Y | ··· | 4AL |
| order | 1 | 2 | ··· | 2 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 4 | ··· | 4 |
| size | 1 | 1 | ··· | 1 | 4 | 4 | 4 | 4 | 2 | ··· | 2 | 4 | ··· | 4 |
50 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 4 |
| type | + | + | + | + | + | + | + | + | ||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C4 | C4○D4 | 2+ 1+4 |
| kernel | C24.221C23 | C42⋊4C4 | C4×C22⋊C4 | C23.23D4 | C24.C22 | C23.67C23 | C2×C4.4D4 | C4.4D4 | C2×C4 | C22 |
| # reps | 1 | 2 | 4 | 2 | 4 | 2 | 1 | 16 | 16 | 2 |
Matrix representation of C24.221C23 ►in GL5(𝔽5)
| 1 | 0 | 0 | 0 | 0 |
| 0 | 2 | 4 | 0 | 0 |
| 0 | 3 | 3 | 0 | 0 |
| 0 | 0 | 0 | 1 | 2 |
| 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 |
| 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 |
| 4 | 0 | 0 | 0 | 0 |
| 0 | 4 | 0 | 0 | 0 |
| 0 | 0 | 4 | 0 | 0 |
| 0 | 0 | 0 | 4 | 0 |
| 0 | 0 | 0 | 0 | 4 |
| 3 | 0 | 0 | 0 | 0 |
| 0 | 2 | 4 | 0 | 0 |
| 0 | 0 | 3 | 0 | 0 |
| 0 | 0 | 0 | 3 | 1 |
| 0 | 0 | 0 | 0 | 2 |
| 4 | 0 | 0 | 0 | 0 |
| 0 | 2 | 0 | 0 | 0 |
| 0 | 0 | 2 | 0 | 0 |
| 0 | 0 | 0 | 3 | 1 |
| 0 | 0 | 0 | 2 | 2 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 1 | 2 | 0 | 0 |
| 0 | 0 | 4 | 0 | 0 |
| 0 | 0 | 0 | 2 | 0 |
| 0 | 0 | 0 | 0 | 2 |
G:=sub<GL(5,GF(5))| [1,0,0,0,0,0,2,3,0,0,0,4,3,0,0,0,0,0,1,0,0,0,0,2,4],[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],[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],[4,0,0,0,0,0,4,0,0,0,0,0,4,0,0,0,0,0,4,0,0,0,0,0,4],[3,0,0,0,0,0,2,0,0,0,0,4,3,0,0,0,0,0,3,0,0,0,0,1,2],[4,0,0,0,0,0,2,0,0,0,0,0,2,0,0,0,0,0,3,2,0,0,0,1,2],[1,0,0,0,0,0,1,0,0,0,0,2,4,0,0,0,0,0,2,0,0,0,0,0,2] >;
C24.221C23 in GAP, Magma, Sage, TeX
C_2^4._{221}C_2^3 % in TeX
G:=Group("C2^4.221C2^3"); // GroupNames label
G:=SmallGroup(128,1104);
// by ID
G=gap.SmallGroup(128,1104);
# by ID
G:=PCGroup([7,-2,2,2,2,-2,2,2,448,253,568,758,100,346,136]);
// Polycyclic
G:=Group<a,b,c,d,e,f,g|a^2=b^2=c^2=d^2=1,e^2=d,f^2=b,g^2=c,e*a*e^-1=g*a*g^-1=a*b=b*a,f*a*f^-1=a*c=c*a,a*d=d*a,b*c=c*b,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,d*e=e*d,d*f=f*d,d*g=g*d,e*g=g*e,f*g=g*f>;
// generators/relations