p-group, metabelian, nilpotent (class 2), monomial
Aliases: C23.473C24, C24.337C23, C22.1932- 1+4, C22.2562+ 1+4, C42⋊5C4⋊19C2, (C2×C42).69C22, C23.158(C4○D4), (C23×C4).120C22, (C22×C4).105C23, C23.Q8.14C2, C23.8Q8.32C2, C23.11D4.19C2, C23.34D4.19C2, C23.65C23⋊90C2, C23.83C23⋊43C2, C23.63C23⋊90C2, C24.C22.32C2, C2.54(C22.45C24), C2.C42.209C22, C2.61(C22.47C24), C2.84(C23.36C23), C2.67(C22.46C24), C2.43(C22.50C24), C2.28(C22.33C24), (C4×C4⋊C4)⋊99C2, (C4×C22⋊C4).64C2, (C2×C4).858(C4○D4), (C2×C4⋊C4).320C22, C22.349(C2×C4○D4), (C2×C22⋊C4).189C22, SmallGroup(128,1305)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C23.473C24
G = < a,b,c,d,e,f,g | a2=b2=c2=f2=1, d2=e2=ca=ac, g2=b, ab=ba, ede-1=gdg-1=ad=da, ae=ea, af=fa, ag=ga, bc=cb, fdf=bd=db, be=eb, bf=fb, bg=gb, cd=dc, fef=ce=ec, cf=fc, cg=gc, eg=ge, fg=gf >
Subgroups: 372 in 206 conjugacy classes, 92 normal (82 characteristic)
C1, C2, C2, C4, C22, C22, C2×C4, C2×C4, C23, C23, C23, C42, C22⋊C4, C4⋊C4, C22×C4, C22×C4, C24, C2.C42, C2×C42, C2×C22⋊C4, C2×C4⋊C4, C23×C4, C4×C22⋊C4, C4×C4⋊C4, C23.34D4, C42⋊5C4, C23.8Q8, C23.63C23, C24.C22, C23.65C23, C23.Q8, C23.11D4, C23.83C23, C23.473C24
Quotients: C1, C2, C22, C23, C4○D4, C24, C2×C4○D4, 2+ 1+4, 2- 1+4, C23.36C23, C22.33C24, C22.45C24, C22.46C24, C22.47C24, C22.50C24, C23.473C24
►On 64 points
Generators in S64
(1 45)(2 46)(3 47)(4 48)(5 63)(6 64)(7 61)(8 62)(9 23)(10 24)(11 21)(12 22)(13 20)(14 17)(15 18)(16 19)(25 30)(26 31)(27 32)(28 29)(33 38)(34 39)(35 40)(36 37)(41 49)(42 50)(43 51)(44 52)(53 60)(54 57)(55 58)(56 59)
(1 58)(2 59)(3 60)(4 57)(5 52)(6 49)(7 50)(8 51)(9 34)(10 35)(11 36)(12 33)(13 31)(14 32)(15 29)(16 30)(17 27)(18 28)(19 25)(20 26)(21 37)(22 38)(23 39)(24 40)(41 64)(42 61)(43 62)(44 63)(45 55)(46 56)(47 53)(48 54)
(1 47)(2 48)(3 45)(4 46)(5 61)(6 62)(7 63)(8 64)(9 21)(10 22)(11 23)(12 24)(13 18)(14 19)(15 20)(16 17)(25 32)(26 29)(27 30)(28 31)(33 40)(34 37)(35 38)(36 39)(41 51)(42 52)(43 49)(44 50)(53 58)(54 59)(55 60)(56 57)
(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 50 3 52)(2 43 4 41)(5 58 7 60)(6 56 8 54)(9 18 11 20)(10 16 12 14)(13 23 15 21)(17 24 19 22)(25 38 27 40)(26 34 28 36)(29 37 31 39)(30 33 32 35)(42 47 44 45)(46 51 48 49)(53 63 55 61)(57 64 59 62)
(2 59)(4 57)(5 61)(6 43)(7 63)(8 41)(9 21)(10 38)(11 23)(12 40)(14 32)(16 30)(17 27)(19 25)(22 35)(24 33)(34 37)(36 39)(42 52)(44 50)(46 56)(48 54)(49 62)(51 64)
(1 31 58 13)(2 27 59 17)(3 29 60 15)(4 25 57 19)(5 21 52 37)(6 12 49 33)(7 23 50 39)(8 10 51 35)(9 42 34 61)(11 44 36 63)(14 46 32 56)(16 48 30 54)(18 47 28 53)(20 45 26 55)(22 41 38 64)(24 43 40 62)
G:=sub<Sym(64)| (1,45)(2,46)(3,47)(4,48)(5,63)(6,64)(7,61)(8,62)(9,23)(10,24)(11,21)(12,22)(13,20)(14,17)(15,18)(16,19)(25,30)(26,31)(27,32)(28,29)(33,38)(34,39)(35,40)(36,37)(41,49)(42,50)(43,51)(44,52)(53,60)(54,57)(55,58)(56,59), (1,58)(2,59)(3,60)(4,57)(5,52)(6,49)(7,50)(8,51)(9,34)(10,35)(11,36)(12,33)(13,31)(14,32)(15,29)(16,30)(17,27)(18,28)(19,25)(20,26)(21,37)(22,38)(23,39)(24,40)(41,64)(42,61)(43,62)(44,63)(45,55)(46,56)(47,53)(48,54), (1,47)(2,48)(3,45)(4,46)(5,61)(6,62)(7,63)(8,64)(9,21)(10,22)(11,23)(12,24)(13,18)(14,19)(15,20)(16,17)(25,32)(26,29)(27,30)(28,31)(33,40)(34,37)(35,38)(36,39)(41,51)(42,52)(43,49)(44,50)(53,58)(54,59)(55,60)(56,57), (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), (1,50,3,52)(2,43,4,41)(5,58,7,60)(6,56,8,54)(9,18,11,20)(10,16,12,14)(13,23,15,21)(17,24,19,22)(25,38,27,40)(26,34,28,36)(29,37,31,39)(30,33,32,35)(42,47,44,45)(46,51,48,49)(53,63,55,61)(57,64,59,62), (2,59)(4,57)(5,61)(6,43)(7,63)(8,41)(9,21)(10,38)(11,23)(12,40)(14,32)(16,30)(17,27)(19,25)(22,35)(24,33)(34,37)(36,39)(42,52)(44,50)(46,56)(48,54)(49,62)(51,64), (1,31,58,13)(2,27,59,17)(3,29,60,15)(4,25,57,19)(5,21,52,37)(6,12,49,33)(7,23,50,39)(8,10,51,35)(9,42,34,61)(11,44,36,63)(14,46,32,56)(16,48,30,54)(18,47,28,53)(20,45,26,55)(22,41,38,64)(24,43,40,62)>;
G:=Group( (1,45)(2,46)(3,47)(4,48)(5,63)(6,64)(7,61)(8,62)(9,23)(10,24)(11,21)(12,22)(13,20)(14,17)(15,18)(16,19)(25,30)(26,31)(27,32)(28,29)(33,38)(34,39)(35,40)(36,37)(41,49)(42,50)(43,51)(44,52)(53,60)(54,57)(55,58)(56,59), (1,58)(2,59)(3,60)(4,57)(5,52)(6,49)(7,50)(8,51)(9,34)(10,35)(11,36)(12,33)(13,31)(14,32)(15,29)(16,30)(17,27)(18,28)(19,25)(20,26)(21,37)(22,38)(23,39)(24,40)(41,64)(42,61)(43,62)(44,63)(45,55)(46,56)(47,53)(48,54), (1,47)(2,48)(3,45)(4,46)(5,61)(6,62)(7,63)(8,64)(9,21)(10,22)(11,23)(12,24)(13,18)(14,19)(15,20)(16,17)(25,32)(26,29)(27,30)(28,31)(33,40)(34,37)(35,38)(36,39)(41,51)(42,52)(43,49)(44,50)(53,58)(54,59)(55,60)(56,57), (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), (1,50,3,52)(2,43,4,41)(5,58,7,60)(6,56,8,54)(9,18,11,20)(10,16,12,14)(13,23,15,21)(17,24,19,22)(25,38,27,40)(26,34,28,36)(29,37,31,39)(30,33,32,35)(42,47,44,45)(46,51,48,49)(53,63,55,61)(57,64,59,62), (2,59)(4,57)(5,61)(6,43)(7,63)(8,41)(9,21)(10,38)(11,23)(12,40)(14,32)(16,30)(17,27)(19,25)(22,35)(24,33)(34,37)(36,39)(42,52)(44,50)(46,56)(48,54)(49,62)(51,64), (1,31,58,13)(2,27,59,17)(3,29,60,15)(4,25,57,19)(5,21,52,37)(6,12,49,33)(7,23,50,39)(8,10,51,35)(9,42,34,61)(11,44,36,63)(14,46,32,56)(16,48,30,54)(18,47,28,53)(20,45,26,55)(22,41,38,64)(24,43,40,62) );
G=PermutationGroup([[(1,45),(2,46),(3,47),(4,48),(5,63),(6,64),(7,61),(8,62),(9,23),(10,24),(11,21),(12,22),(13,20),(14,17),(15,18),(16,19),(25,30),(26,31),(27,32),(28,29),(33,38),(34,39),(35,40),(36,37),(41,49),(42,50),(43,51),(44,52),(53,60),(54,57),(55,58),(56,59)], [(1,58),(2,59),(3,60),(4,57),(5,52),(6,49),(7,50),(8,51),(9,34),(10,35),(11,36),(12,33),(13,31),(14,32),(15,29),(16,30),(17,27),(18,28),(19,25),(20,26),(21,37),(22,38),(23,39),(24,40),(41,64),(42,61),(43,62),(44,63),(45,55),(46,56),(47,53),(48,54)], [(1,47),(2,48),(3,45),(4,46),(5,61),(6,62),(7,63),(8,64),(9,21),(10,22),(11,23),(12,24),(13,18),(14,19),(15,20),(16,17),(25,32),(26,29),(27,30),(28,31),(33,40),(34,37),(35,38),(36,39),(41,51),(42,52),(43,49),(44,50),(53,58),(54,59),(55,60),(56,57)], [(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)], [(1,50,3,52),(2,43,4,41),(5,58,7,60),(6,56,8,54),(9,18,11,20),(10,16,12,14),(13,23,15,21),(17,24,19,22),(25,38,27,40),(26,34,28,36),(29,37,31,39),(30,33,32,35),(42,47,44,45),(46,51,48,49),(53,63,55,61),(57,64,59,62)], [(2,59),(4,57),(5,61),(6,43),(7,63),(8,41),(9,21),(10,38),(11,23),(12,40),(14,32),(16,30),(17,27),(19,25),(22,35),(24,33),(34,37),(36,39),(42,52),(44,50),(46,56),(48,54),(49,62),(51,64)], [(1,31,58,13),(2,27,59,17),(3,29,60,15),(4,25,57,19),(5,21,52,37),(6,12,49,33),(7,23,50,39),(8,10,51,35),(9,42,34,61),(11,44,36,63),(14,46,32,56),(16,48,30,54),(18,47,28,53),(20,45,26,55),(22,41,38,64),(24,43,40,62)]])
38 conjugacy classes
| class | 1 | 2A | ··· | 2G | 2H | 2I | 4A | ··· | 4H | 4I | ··· | 4X | 4Y | 4Z | 4AA | 4AB |
| order | 1 | 2 | ··· | 2 | 2 | 2 | 4 | ··· | 4 | 4 | ··· | 4 | 4 | 4 | 4 | 4 |
| size | 1 | 1 | ··· | 1 | 4 | 4 | 2 | ··· | 2 | 4 | ··· | 4 | 8 | 8 | 8 | 8 |
38 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | - | ||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C4○D4 | C4○D4 | 2+ 1+4 | 2- 1+4 |
| kernel | C23.473C24 | C4×C22⋊C4 | C4×C4⋊C4 | C23.34D4 | C42⋊5C4 | C23.8Q8 | C23.63C23 | C24.C22 | C23.65C23 | C23.Q8 | C23.11D4 | C23.83C23 | C2×C4 | C23 | C22 | C22 |
| # reps | 1 | 1 | 1 | 1 | 1 | 1 | 3 | 2 | 1 | 1 | 1 | 2 | 16 | 4 | 1 | 1 |
Matrix representation of C23.473C24 ►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 |
| 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 | 4 | 0 |
| 0 | 0 | 0 | 0 | 0 | 4 |
| 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 | 4 | 0 |
| 0 | 0 | 0 | 0 | 0 | 4 |
| 3 | 0 | 0 | 0 | 0 | 0 |
| 0 | 3 | 0 | 0 | 0 | 0 |
| 0 | 0 | 3 | 2 | 0 | 0 |
| 0 | 0 | 0 | 2 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 4 | 0 |
| 3 | 3 | 0 | 0 | 0 | 0 |
| 0 | 2 | 0 | 0 | 0 | 0 |
| 0 | 0 | 2 | 0 | 0 | 0 |
| 0 | 0 | 4 | 3 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 4 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 3 | 4 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 2 | 4 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 4 |
| 4 | 0 | 0 | 0 | 0 | 0 |
| 0 | 4 | 0 | 0 | 0 | 0 |
| 0 | 0 | 3 | 0 | 0 | 0 |
| 0 | 0 | 1 | 2 | 0 | 0 |
| 0 | 0 | 0 | 0 | 2 | 0 |
| 0 | 0 | 0 | 0 | 0 | 2 |
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],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,4],[4,0,0,0,0,0,0,4,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,4,0,0,0,0,0,0,4],[3,0,0,0,0,0,0,3,0,0,0,0,0,0,3,0,0,0,0,0,2,2,0,0,0,0,0,0,0,4,0,0,0,0,1,0],[3,0,0,0,0,0,3,2,0,0,0,0,0,0,2,4,0,0,0,0,0,3,0,0,0,0,0,0,0,4,0,0,0,0,1,0],[1,3,0,0,0,0,0,4,0,0,0,0,0,0,1,2,0,0,0,0,0,4,0,0,0,0,0,0,1,0,0,0,0,0,0,4],[4,0,0,0,0,0,0,4,0,0,0,0,0,0,3,1,0,0,0,0,0,2,0,0,0,0,0,0,2,0,0,0,0,0,0,2] >;
C23.473C24 in GAP, Magma, Sage, TeX
C_2^3._{473}C_2^4 % in TeX
G:=Group("C2^3.473C2^4"); // GroupNames label
G:=SmallGroup(128,1305);
// by ID
G=gap.SmallGroup(128,1305);
# by ID
G:=PCGroup([7,-2,2,2,2,-2,2,2,560,253,568,758,723,675,136]);
// Polycyclic
G:=Group<a,b,c,d,e,f,g|a^2=b^2=c^2=f^2=1,d^2=e^2=c*a=a*c,g^2=b,a*b=b*a,e*d*e^-1=g*d*g^-1=a*d=d*a,a*e=e*a,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,e*g=g*e,f*g=g*f>;
// generators/relations