p-group, metabelian, nilpotent (class 2), monomial
Aliases: C24.19C23, C23.416C24, C22.2102+ 1+4, (C22×C4).84C23, C23.4Q8⋊17C2, C23.147(C4○D4), C23.11D4⋊35C2, C23.34D4⋊33C2, (C2×C42).531C22, (C23×C4).105C22, C24.C22⋊71C2, C23.10D4.13C2, C23.23D4.29C2, (C22×D4).155C22, C23.83C23⋊30C2, C23.63C23⋊70C2, C24.3C22.40C2, C2.33(C22.45C24), C2.C42.164C22, C2.44(C22.47C24), C2.20(C22.34C24), C2.14(C22.53C24), C2.59(C23.36C23), (C4×C4⋊C4)⋊78C2, (C4×C22⋊C4)⋊78C2, (C2×C4).137(C4○D4), (C2×C4⋊C4).863C22, C22.293(C2×C4○D4), (C2×C22⋊C4).164C22, SmallGroup(128,1248)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C23.416C24
G = < a,b,c,d,e,f,g | a2=b2=c2=f2=1, d2=ca=ac, e2=a, 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: 452 in 227 conjugacy classes, 92 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, C4×C22⋊C4, C4×C4⋊C4, C23.34D4, C23.23D4, C23.63C23, C24.C22, C24.3C22, C23.10D4, C23.11D4, C23.4Q8, C23.83C23, C23.416C24
Quotients: C1, C2, C22, C23, C4○D4, C24, C2×C4○D4, 2+ 1+4, C23.36C23, C22.34C24, C22.45C24, C22.47C24, C22.53C24, C23.416C24
►On 64 points
Generators in S64
(1 36)(2 33)(3 34)(4 35)(5 55)(6 56)(7 53)(8 54)(9 16)(10 13)(11 14)(12 15)(17 21)(18 22)(19 23)(20 24)(25 30)(26 31)(27 32)(28 29)(37 44)(38 41)(39 42)(40 43)(45 52)(46 49)(47 50)(48 51)(57 62)(58 63)(59 64)(60 61)
(1 58)(2 59)(3 60)(4 57)(5 27)(6 28)(7 25)(8 26)(9 41)(10 42)(11 43)(12 44)(13 39)(14 40)(15 37)(16 38)(17 52)(18 49)(19 50)(20 51)(21 45)(22 46)(23 47)(24 48)(29 56)(30 53)(31 54)(32 55)(33 64)(34 61)(35 62)(36 63)
(1 34)(2 35)(3 36)(4 33)(5 53)(6 54)(7 55)(8 56)(9 14)(10 15)(11 16)(12 13)(17 23)(18 24)(19 21)(20 22)(25 32)(26 29)(27 30)(28 31)(37 42)(38 43)(39 44)(40 41)(45 50)(46 51)(47 52)(48 49)(57 64)(58 61)(59 62)(60 63)
(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 22 36 18)(2 19 33 23)(3 24 34 20)(4 17 35 21)(5 39 55 42)(6 43 56 40)(7 37 53 44)(8 41 54 38)(9 31 16 26)(10 27 13 32)(11 29 14 28)(12 25 15 30)(45 57 52 62)(46 63 49 58)(47 59 50 64)(48 61 51 60)
(1 9)(2 42)(3 11)(4 44)(5 21)(6 46)(7 23)(8 48)(10 59)(12 57)(13 64)(14 34)(15 62)(16 36)(17 55)(18 29)(19 53)(20 31)(22 28)(24 26)(25 47)(27 45)(30 50)(32 52)(33 39)(35 37)(38 63)(40 61)(41 58)(43 60)(49 56)(51 54)
(1 30 58 53)(2 26 59 8)(3 32 60 55)(4 28 57 6)(5 34 27 61)(7 36 25 63)(9 50 41 19)(10 48 42 24)(11 52 43 17)(12 46 44 22)(13 51 39 20)(14 45 40 21)(15 49 37 18)(16 47 38 23)(29 62 56 35)(31 64 54 33)
G:=sub<Sym(64)| (1,36)(2,33)(3,34)(4,35)(5,55)(6,56)(7,53)(8,54)(9,16)(10,13)(11,14)(12,15)(17,21)(18,22)(19,23)(20,24)(25,30)(26,31)(27,32)(28,29)(37,44)(38,41)(39,42)(40,43)(45,52)(46,49)(47,50)(48,51)(57,62)(58,63)(59,64)(60,61), (1,58)(2,59)(3,60)(4,57)(5,27)(6,28)(7,25)(8,26)(9,41)(10,42)(11,43)(12,44)(13,39)(14,40)(15,37)(16,38)(17,52)(18,49)(19,50)(20,51)(21,45)(22,46)(23,47)(24,48)(29,56)(30,53)(31,54)(32,55)(33,64)(34,61)(35,62)(36,63), (1,34)(2,35)(3,36)(4,33)(5,53)(6,54)(7,55)(8,56)(9,14)(10,15)(11,16)(12,13)(17,23)(18,24)(19,21)(20,22)(25,32)(26,29)(27,30)(28,31)(37,42)(38,43)(39,44)(40,41)(45,50)(46,51)(47,52)(48,49)(57,64)(58,61)(59,62)(60,63), (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), (1,22,36,18)(2,19,33,23)(3,24,34,20)(4,17,35,21)(5,39,55,42)(6,43,56,40)(7,37,53,44)(8,41,54,38)(9,31,16,26)(10,27,13,32)(11,29,14,28)(12,25,15,30)(45,57,52,62)(46,63,49,58)(47,59,50,64)(48,61,51,60), (1,9)(2,42)(3,11)(4,44)(5,21)(6,46)(7,23)(8,48)(10,59)(12,57)(13,64)(14,34)(15,62)(16,36)(17,55)(18,29)(19,53)(20,31)(22,28)(24,26)(25,47)(27,45)(30,50)(32,52)(33,39)(35,37)(38,63)(40,61)(41,58)(43,60)(49,56)(51,54), (1,30,58,53)(2,26,59,8)(3,32,60,55)(4,28,57,6)(5,34,27,61)(7,36,25,63)(9,50,41,19)(10,48,42,24)(11,52,43,17)(12,46,44,22)(13,51,39,20)(14,45,40,21)(15,49,37,18)(16,47,38,23)(29,62,56,35)(31,64,54,33)>;
G:=Group( (1,36)(2,33)(3,34)(4,35)(5,55)(6,56)(7,53)(8,54)(9,16)(10,13)(11,14)(12,15)(17,21)(18,22)(19,23)(20,24)(25,30)(26,31)(27,32)(28,29)(37,44)(38,41)(39,42)(40,43)(45,52)(46,49)(47,50)(48,51)(57,62)(58,63)(59,64)(60,61), (1,58)(2,59)(3,60)(4,57)(5,27)(6,28)(7,25)(8,26)(9,41)(10,42)(11,43)(12,44)(13,39)(14,40)(15,37)(16,38)(17,52)(18,49)(19,50)(20,51)(21,45)(22,46)(23,47)(24,48)(29,56)(30,53)(31,54)(32,55)(33,64)(34,61)(35,62)(36,63), (1,34)(2,35)(3,36)(4,33)(5,53)(6,54)(7,55)(8,56)(9,14)(10,15)(11,16)(12,13)(17,23)(18,24)(19,21)(20,22)(25,32)(26,29)(27,30)(28,31)(37,42)(38,43)(39,44)(40,41)(45,50)(46,51)(47,52)(48,49)(57,64)(58,61)(59,62)(60,63), (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), (1,22,36,18)(2,19,33,23)(3,24,34,20)(4,17,35,21)(5,39,55,42)(6,43,56,40)(7,37,53,44)(8,41,54,38)(9,31,16,26)(10,27,13,32)(11,29,14,28)(12,25,15,30)(45,57,52,62)(46,63,49,58)(47,59,50,64)(48,61,51,60), (1,9)(2,42)(3,11)(4,44)(5,21)(6,46)(7,23)(8,48)(10,59)(12,57)(13,64)(14,34)(15,62)(16,36)(17,55)(18,29)(19,53)(20,31)(22,28)(24,26)(25,47)(27,45)(30,50)(32,52)(33,39)(35,37)(38,63)(40,61)(41,58)(43,60)(49,56)(51,54), (1,30,58,53)(2,26,59,8)(3,32,60,55)(4,28,57,6)(5,34,27,61)(7,36,25,63)(9,50,41,19)(10,48,42,24)(11,52,43,17)(12,46,44,22)(13,51,39,20)(14,45,40,21)(15,49,37,18)(16,47,38,23)(29,62,56,35)(31,64,54,33) );
G=PermutationGroup([[(1,36),(2,33),(3,34),(4,35),(5,55),(6,56),(7,53),(8,54),(9,16),(10,13),(11,14),(12,15),(17,21),(18,22),(19,23),(20,24),(25,30),(26,31),(27,32),(28,29),(37,44),(38,41),(39,42),(40,43),(45,52),(46,49),(47,50),(48,51),(57,62),(58,63),(59,64),(60,61)], [(1,58),(2,59),(3,60),(4,57),(5,27),(6,28),(7,25),(8,26),(9,41),(10,42),(11,43),(12,44),(13,39),(14,40),(15,37),(16,38),(17,52),(18,49),(19,50),(20,51),(21,45),(22,46),(23,47),(24,48),(29,56),(30,53),(31,54),(32,55),(33,64),(34,61),(35,62),(36,63)], [(1,34),(2,35),(3,36),(4,33),(5,53),(6,54),(7,55),(8,56),(9,14),(10,15),(11,16),(12,13),(17,23),(18,24),(19,21),(20,22),(25,32),(26,29),(27,30),(28,31),(37,42),(38,43),(39,44),(40,41),(45,50),(46,51),(47,52),(48,49),(57,64),(58,61),(59,62),(60,63)], [(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)], [(1,22,36,18),(2,19,33,23),(3,24,34,20),(4,17,35,21),(5,39,55,42),(6,43,56,40),(7,37,53,44),(8,41,54,38),(9,31,16,26),(10,27,13,32),(11,29,14,28),(12,25,15,30),(45,57,52,62),(46,63,49,58),(47,59,50,64),(48,61,51,60)], [(1,9),(2,42),(3,11),(4,44),(5,21),(6,46),(7,23),(8,48),(10,59),(12,57),(13,64),(14,34),(15,62),(16,36),(17,55),(18,29),(19,53),(20,31),(22,28),(24,26),(25,47),(27,45),(30,50),(32,52),(33,39),(35,37),(38,63),(40,61),(41,58),(43,60),(49,56),(51,54)], [(1,30,58,53),(2,26,59,8),(3,32,60,55),(4,28,57,6),(5,34,27,61),(7,36,25,63),(9,50,41,19),(10,48,42,24),(11,52,43,17),(12,46,44,22),(13,51,39,20),(14,45,40,21),(15,49,37,18),(16,47,38,23),(29,62,56,35),(31,64,54,33)]])
38 conjugacy classes
| class | 1 | 2A | ··· | 2G | 2H | 2I | 2J | 4A | ··· | 4H | 4I | ··· | 4X | 4Y | 4Z | 4AA |
| order | 1 | 2 | ··· | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 4 | ··· | 4 | 4 | 4 | 4 |
| size | 1 | 1 | ··· | 1 | 4 | 4 | 8 | 2 | ··· | 2 | 4 | ··· | 4 | 8 | 8 | 8 |
38 irreducible representations
| dim | 1 | 1 | 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 | C2 | C2 | C4○D4 | C4○D4 | 2+ 1+4 |
| kernel | C23.416C24 | C4×C22⋊C4 | C4×C4⋊C4 | C23.34D4 | C23.23D4 | C23.63C23 | C24.C22 | C24.3C22 | C23.10D4 | C23.11D4 | C23.4Q8 | C23.83C23 | C2×C4 | C23 | C22 |
| # reps | 1 | 1 | 1 | 1 | 1 | 2 | 4 | 1 | 1 | 1 | 1 | 1 | 16 | 4 | 2 |
Matrix representation of C23.416C24 ►in GL6(𝔽5)
| 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 |
| 0 | 4 | 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 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 3 | 0 | 0 | 0 | 0 |
| 3 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 3 | 1 | 0 | 0 |
| 0 | 0 | 2 | 2 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 3 |
| 0 | 0 | 0 | 0 | 1 | 4 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 3 | 4 |
| 0 | 0 | 0 | 0 | 0 | 2 |
| 1 | 0 | 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 | 3 |
| 0 | 0 | 0 | 0 | 0 | 4 |
| 2 | 0 | 0 | 0 | 0 | 0 |
| 0 | 2 | 0 | 0 | 0 | 0 |
| 0 | 0 | 3 | 0 | 0 | 0 |
| 0 | 0 | 0 | 3 | 0 | 0 |
| 0 | 0 | 0 | 0 | 3 | 4 |
| 0 | 0 | 0 | 0 | 0 | 2 |
G:=sub<GL(6,GF(5))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,4,0,0,0,0,0,0,4],[4,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,4],[4,0,0,0,0,0,0,4,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[0,3,0,0,0,0,3,0,0,0,0,0,0,0,3,2,0,0,0,0,1,2,0,0,0,0,0,0,1,1,0,0,0,0,3,4],[0,1,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,3,0,0,0,0,0,4,2],[1,0,0,0,0,0,0,4,0,0,0,0,0,0,1,0,0,0,0,0,2,4,0,0,0,0,0,0,1,0,0,0,0,0,3,4],[2,0,0,0,0,0,0,2,0,0,0,0,0,0,3,0,0,0,0,0,0,3,0,0,0,0,0,0,3,0,0,0,0,0,4,2] >;
C23.416C24 in GAP, Magma, Sage, TeX
C_2^3._{416}C_2^4 % in TeX
G:=Group("C2^3.416C2^4"); // GroupNames label
G:=SmallGroup(128,1248);
// by ID
G=gap.SmallGroup(128,1248);
# by ID
G:=PCGroup([7,-2,2,2,2,-2,2,2,560,253,120,758,723,675,136]);
// Polycyclic
G:=Group<a,b,c,d,e,f,g|a^2=b^2=c^2=f^2=1,d^2=c*a=a*c,e^2=a,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