p-group, metabelian, nilpotent (class 2), monomial, rational
Aliases: C24.360C23, C23.515C24, C22.2142- 1+4, C22.2932+ 1+4, (C22×C4)⋊32D4, C4.103C22≀C2, C23⋊2D4⋊23C2, C23⋊Q8⋊27C2, C23.188(C2×D4), C23.7Q8⋊74C2, (C23×C4).418C22, (C22×C4).853C23, C22.340(C22×D4), (C22×D4).189C22, (C22×Q8).447C22, C2.32(C22.29C24), C2.C42.243C22, C2.21(C22.31C24), (C2×C4⋊D4)⋊21C2, (C2×C4).375(C2×D4), (C22×C4○D4)⋊6C2, C2.27(C2×C22≀C2), (C2×C4⋊C4).353C22, (C2×C22⋊C4).208C22, SmallGroup(128,1347)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C24.360C23
G = < a,b,c,d,e,f,g | a2=b2=c2=d2=e2=f2=1, g2=a, ab=ba, ac=ca, ede=ad=da, ae=ea, gfg-1=af=fa, ag=ga, bc=cb, fdf=bd=db, be=eb, bf=fb, bg=gb, cd=dc, fef=ce=ec, cf=fc, cg=gc, dg=gd, eg=ge >
Subgroups: 996 in 466 conjugacy classes, 116 normal (9 characteristic)
C1, C2, C2, C2, C4, C4, C22, C22, C22, C2×C4, C2×C4, D4, Q8, C23, C23, C23, C22⋊C4, C4⋊C4, C22×C4, C22×C4, C22×C4, C2×D4, C2×Q8, C4○D4, C24, C2.C42, C2×C22⋊C4, C2×C4⋊C4, C4⋊D4, C23×C4, C22×D4, C22×D4, C22×Q8, C2×C4○D4, C23.7Q8, C23⋊2D4, C23⋊Q8, C2×C4⋊D4, C22×C4○D4, C24.360C23
Quotients: C1, C2, C22, D4, C23, C2×D4, C24, C22≀C2, C22×D4, 2+ 1+4, 2- 1+4, C2×C22≀C2, C22.29C24, C22.31C24, C24.360C23
►On 64 points
Generators in S64
(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 17)(2 18)(3 19)(4 20)(5 15)(6 16)(7 13)(8 14)(9 64)(10 61)(11 62)(12 63)(21 27)(22 28)(23 25)(24 26)(29 33)(30 34)(31 35)(32 36)(37 43)(38 44)(39 41)(40 42)(45 49)(46 50)(47 51)(48 52)(53 59)(54 60)(55 57)(56 58)
(1 25)(2 26)(3 27)(4 28)(5 64)(6 61)(7 62)(8 63)(9 15)(10 16)(11 13)(12 14)(17 23)(18 24)(19 21)(20 22)(29 41)(30 42)(31 43)(32 44)(33 39)(34 40)(35 37)(36 38)(45 57)(46 58)(47 59)(48 60)(49 55)(50 56)(51 53)(52 54)
(1 45)(2 46)(3 47)(4 48)(5 44)(6 41)(7 42)(8 43)(9 36)(10 33)(11 34)(12 35)(13 40)(14 37)(15 38)(16 39)(17 49)(18 50)(19 51)(20 52)(21 53)(22 54)(23 55)(24 56)(25 57)(26 58)(27 59)(28 60)(29 61)(30 62)(31 63)(32 64)
(1 29)(2 30)(3 31)(4 32)(5 58)(6 59)(7 60)(8 57)(9 50)(10 51)(11 52)(12 49)(13 54)(14 55)(15 56)(16 53)(17 33)(18 34)(19 35)(20 36)(21 37)(22 38)(23 39)(24 40)(25 41)(26 42)(27 43)(28 44)(45 63)(46 64)(47 61)(48 62)
(2 4)(5 11)(6 10)(7 9)(8 12)(13 64)(14 63)(15 62)(16 61)(18 20)(22 24)(26 28)(29 41)(30 44)(31 43)(32 42)(33 39)(34 38)(35 37)(36 40)(45 49)(46 52)(47 51)(48 50)(53 59)(54 58)(55 57)(56 60)
(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)
G:=sub<Sym(64)| (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), (1,17)(2,18)(3,19)(4,20)(5,15)(6,16)(7,13)(8,14)(9,64)(10,61)(11,62)(12,63)(21,27)(22,28)(23,25)(24,26)(29,33)(30,34)(31,35)(32,36)(37,43)(38,44)(39,41)(40,42)(45,49)(46,50)(47,51)(48,52)(53,59)(54,60)(55,57)(56,58), (1,25)(2,26)(3,27)(4,28)(5,64)(6,61)(7,62)(8,63)(9,15)(10,16)(11,13)(12,14)(17,23)(18,24)(19,21)(20,22)(29,41)(30,42)(31,43)(32,44)(33,39)(34,40)(35,37)(36,38)(45,57)(46,58)(47,59)(48,60)(49,55)(50,56)(51,53)(52,54), (1,45)(2,46)(3,47)(4,48)(5,44)(6,41)(7,42)(8,43)(9,36)(10,33)(11,34)(12,35)(13,40)(14,37)(15,38)(16,39)(17,49)(18,50)(19,51)(20,52)(21,53)(22,54)(23,55)(24,56)(25,57)(26,58)(27,59)(28,60)(29,61)(30,62)(31,63)(32,64), (1,29)(2,30)(3,31)(4,32)(5,58)(6,59)(7,60)(8,57)(9,50)(10,51)(11,52)(12,49)(13,54)(14,55)(15,56)(16,53)(17,33)(18,34)(19,35)(20,36)(21,37)(22,38)(23,39)(24,40)(25,41)(26,42)(27,43)(28,44)(45,63)(46,64)(47,61)(48,62), (2,4)(5,11)(6,10)(7,9)(8,12)(13,64)(14,63)(15,62)(16,61)(18,20)(22,24)(26,28)(29,41)(30,44)(31,43)(32,42)(33,39)(34,38)(35,37)(36,40)(45,49)(46,52)(47,51)(48,50)(53,59)(54,58)(55,57)(56,60), (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)>;
G:=Group( (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), (1,17)(2,18)(3,19)(4,20)(5,15)(6,16)(7,13)(8,14)(9,64)(10,61)(11,62)(12,63)(21,27)(22,28)(23,25)(24,26)(29,33)(30,34)(31,35)(32,36)(37,43)(38,44)(39,41)(40,42)(45,49)(46,50)(47,51)(48,52)(53,59)(54,60)(55,57)(56,58), (1,25)(2,26)(3,27)(4,28)(5,64)(6,61)(7,62)(8,63)(9,15)(10,16)(11,13)(12,14)(17,23)(18,24)(19,21)(20,22)(29,41)(30,42)(31,43)(32,44)(33,39)(34,40)(35,37)(36,38)(45,57)(46,58)(47,59)(48,60)(49,55)(50,56)(51,53)(52,54), (1,45)(2,46)(3,47)(4,48)(5,44)(6,41)(7,42)(8,43)(9,36)(10,33)(11,34)(12,35)(13,40)(14,37)(15,38)(16,39)(17,49)(18,50)(19,51)(20,52)(21,53)(22,54)(23,55)(24,56)(25,57)(26,58)(27,59)(28,60)(29,61)(30,62)(31,63)(32,64), (1,29)(2,30)(3,31)(4,32)(5,58)(6,59)(7,60)(8,57)(9,50)(10,51)(11,52)(12,49)(13,54)(14,55)(15,56)(16,53)(17,33)(18,34)(19,35)(20,36)(21,37)(22,38)(23,39)(24,40)(25,41)(26,42)(27,43)(28,44)(45,63)(46,64)(47,61)(48,62), (2,4)(5,11)(6,10)(7,9)(8,12)(13,64)(14,63)(15,62)(16,61)(18,20)(22,24)(26,28)(29,41)(30,44)(31,43)(32,42)(33,39)(34,38)(35,37)(36,40)(45,49)(46,52)(47,51)(48,50)(53,59)(54,58)(55,57)(56,60), (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) );
G=PermutationGroup([[(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)], [(1,17),(2,18),(3,19),(4,20),(5,15),(6,16),(7,13),(8,14),(9,64),(10,61),(11,62),(12,63),(21,27),(22,28),(23,25),(24,26),(29,33),(30,34),(31,35),(32,36),(37,43),(38,44),(39,41),(40,42),(45,49),(46,50),(47,51),(48,52),(53,59),(54,60),(55,57),(56,58)], [(1,25),(2,26),(3,27),(4,28),(5,64),(6,61),(7,62),(8,63),(9,15),(10,16),(11,13),(12,14),(17,23),(18,24),(19,21),(20,22),(29,41),(30,42),(31,43),(32,44),(33,39),(34,40),(35,37),(36,38),(45,57),(46,58),(47,59),(48,60),(49,55),(50,56),(51,53),(52,54)], [(1,45),(2,46),(3,47),(4,48),(5,44),(6,41),(7,42),(8,43),(9,36),(10,33),(11,34),(12,35),(13,40),(14,37),(15,38),(16,39),(17,49),(18,50),(19,51),(20,52),(21,53),(22,54),(23,55),(24,56),(25,57),(26,58),(27,59),(28,60),(29,61),(30,62),(31,63),(32,64)], [(1,29),(2,30),(3,31),(4,32),(5,58),(6,59),(7,60),(8,57),(9,50),(10,51),(11,52),(12,49),(13,54),(14,55),(15,56),(16,53),(17,33),(18,34),(19,35),(20,36),(21,37),(22,38),(23,39),(24,40),(25,41),(26,42),(27,43),(28,44),(45,63),(46,64),(47,61),(48,62)], [(2,4),(5,11),(6,10),(7,9),(8,12),(13,64),(14,63),(15,62),(16,61),(18,20),(22,24),(26,28),(29,41),(30,44),(31,43),(32,42),(33,39),(34,38),(35,37),(36,40),(45,49),(46,52),(47,51),(48,50),(53,59),(54,58),(55,57),(56,60)], [(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)]])
32 conjugacy classes
| class | 1 | 2A | ··· | 2G | 2H | ··· | 2M | 2N | 2O | 4A | 4B | 4C | 4D | 4E | ··· | 4J | 4K | ··· | 4P |
| order | 1 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 4 | ··· | 4 |
| size | 1 | 1 | ··· | 1 | 4 | ··· | 4 | 8 | 8 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 8 | ··· | 8 |
32 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | - |
| image | C1 | C2 | C2 | C2 | C2 | C2 | D4 | 2+ 1+4 | 2- 1+4 |
| kernel | C24.360C23 | C23.7Q8 | C23⋊2D4 | C23⋊Q8 | C2×C4⋊D4 | C22×C4○D4 | C22×C4 | C22 | C22 |
| # reps | 1 | 3 | 6 | 2 | 3 | 1 | 12 | 3 | 1 |
Matrix representation of C24.360C23 ►in GL8(ℤ)
| 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | -1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | -1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | -1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | -1 |
| 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | -1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | -1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 |
| -1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | -1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | -1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | -1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | -2 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | -1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | -2 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | -1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | -1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | -1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | -1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | -1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | -1 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | -1 | -1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | -1 | -1 |
| 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 2 | 0 | 0 |
| 0 | 0 | 0 | 0 | -1 | -1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 1 | 2 |
| 0 | 0 | 0 | 0 | 0 | 0 | -1 | -1 |
G:=sub<GL(8,Integers())| [1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,-1],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[-1,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,-2,-1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0],[0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,-2,-1,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,-1,-1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,-1,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,1,-1,0,0,0,0,0,0,0,-1],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,-1,0,0,0,0,0,0,2,-1,0,0,0,0,0,0,0,0,1,-1,0,0,0,0,0,0,2,-1] >;
C24.360C23 in GAP, Magma, Sage, TeX
C_2^4._{360}C_2^3 % in TeX
G:=Group("C2^4.360C2^3"); // GroupNames label
G:=SmallGroup(128,1347);
// by ID
G=gap.SmallGroup(128,1347);
# by ID
G:=PCGroup([7,-2,2,2,2,-2,2,2,253,758,723,185,80]);
// Polycyclic
G:=Group<a,b,c,d,e,f,g|a^2=b^2=c^2=d^2=e^2=f^2=1,g^2=a,a*b=b*a,a*c=c*a,e*d*e=a*d=d*a,a*e=e*a,g*f*g^-1=a*f=f*a,a*g=g*a,b*c=c*b,f*d*f=b*d=d*b,b*e=e*b,b*f=f*b,b*g=g*b,c*d=d*c,f*e*f=c*e=e*c,c*f=f*c,c*g=g*c,d*g=g*d,e*g=g*e>;
// generators/relations