p-group, metabelian, nilpotent (class 2), monomial
Aliases: C23.241C24, C24.214C23, C22.552- 1+4, C22.752+ 1+4, C22.14(C4xD4), C2.5(D4:6D4), C2.7(D4:5D4), C22:C4.184D4, C23.418(C2xD4), C22.D4:9C4, (C2xC42).22C22, C23.8Q8:16C2, C23.293(C4oD4), C22.132(C23xC4), (C23xC4).308C22, (C22xC4).763C23, C23.132(C22xC4), C23.23D4.9C2, C22.112(C22xD4), C24.C22:18C2, (C22xD4).486C22, C23.65C23:24C2, C23.63C23:17C2, C2.5(C22.45C24), C2.C42.62C22, C2.6(C22.46C24), C2.32(C23.33C23), (C4xC4:C4):41C2, C2.35(C2xC4xD4), C4:C4:29(C2xC4), (C2xC4xD4).39C2, C2.33(C4xC4oD4), (C4xC22:C4):40C2, C22:C4:15(C2xC4), (C22xC4):34(C2xC4), (C2xC4).887(C2xD4), (C2xD4).169(C2xC4), (C2xC4).40(C22xC4), (C2xC42:C2):14C2, (C2xC4).722(C4oD4), (C2xC4:C4).189C22, C22.126(C2xC4oD4), (C2xC2.C42):21C2, (C2xC22.D4).7C2, (C2xC22:C4).555C22, C22:C4o2(C2.C42), SmallGroup(128,1091)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C23.241C24
G = < a,b,c,d,e,f,g | a2=b2=c2=d2=1, e2=d, f2=g2=b, gag-1=ab=ba, faf-1=ac=ca, ad=da, ae=ea, bc=cb, bd=db, geg-1=be=eb, bf=fb, bg=gb, cd=dc, ce=ec, gfg-1=cf=fc, cg=gc, de=ed, df=fd, dg=gd, ef=fe >
Subgroups: 556 in 326 conjugacy classes, 152 normal (82 characteristic)
C1, C2, C2, C4, C22, C22, C22, C2xC4, C2xC4, D4, C23, C23, C23, C42, C22:C4, C22:C4, C4:C4, C4:C4, C22xC4, C22xC4, C22xC4, C2xD4, C2xD4, C24, C2.C42, C2xC42, C2xC22:C4, C2xC4:C4, C42:C2, C4xD4, C22.D4, C23xC4, C22xD4, C2xC2.C42, C4xC22:C4, C4xC4:C4, C23.8Q8, C23.23D4, C23.63C23, C24.C22, C23.65C23, C2xC42:C2, C2xC4xD4, C2xC22.D4, C23.241C24
Quotients: C1, C2, C4, C22, C2xC4, D4, C23, C22xC4, C2xD4, C4oD4, C24, C4xD4, C23xC4, C22xD4, C2xC4oD4, 2+ 1+4, 2- 1+4, C2xC4xD4, C4xC4oD4, C23.33C23, D4:5D4, D4:6D4, C22.45C24, C22.46C24, C23.241C24
►On 64 points
(9 34)(10 35)(11 36)(12 33)(13 62)(14 63)(15 64)(16 61)(21 25)(22 26)(23 27)(24 28)(29 54)(30 55)(31 56)(32 53)(37 48)(38 45)(39 46)(40 47)(41 57)(42 58)(43 59)(44 60)
(1 5)(2 6)(3 7)(4 8)(9 34)(10 35)(11 36)(12 33)(13 46)(14 47)(15 48)(16 45)(17 50)(18 51)(19 52)(20 49)(21 42)(22 43)(23 44)(24 41)(25 58)(26 59)(27 60)(28 57)(29 54)(30 55)(31 56)(32 53)(37 64)(38 61)(39 62)(40 63)
(1 20)(2 17)(3 18)(4 19)(5 49)(6 50)(7 51)(8 52)(9 29)(10 30)(11 31)(12 32)(13 39)(14 40)(15 37)(16 38)(21 25)(22 26)(23 27)(24 28)(33 53)(34 54)(35 55)(36 56)(41 57)(42 58)(43 59)(44 60)(45 61)(46 62)(47 63)(48 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 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 41 5 24)(2 42 6 21)(3 43 7 22)(4 44 8 23)(9 46 34 13)(10 47 35 14)(11 48 36 15)(12 45 33 16)(17 58 50 25)(18 59 51 26)(19 60 52 27)(20 57 49 28)(29 62 54 39)(30 63 55 40)(31 64 56 37)(32 61 53 38)
(1 33 5 12)(2 9 6 34)(3 35 7 10)(4 11 8 36)(13 58 46 25)(14 26 47 59)(15 60 48 27)(16 28 45 57)(17 29 50 54)(18 55 51 30)(19 31 52 56)(20 53 49 32)(21 39 42 62)(22 63 43 40)(23 37 44 64)(24 61 41 38)
G:=sub<Sym(64)| (9,34)(10,35)(11,36)(12,33)(13,62)(14,63)(15,64)(16,61)(21,25)(22,26)(23,27)(24,28)(29,54)(30,55)(31,56)(32,53)(37,48)(38,45)(39,46)(40,47)(41,57)(42,58)(43,59)(44,60), (1,5)(2,6)(3,7)(4,8)(9,34)(10,35)(11,36)(12,33)(13,46)(14,47)(15,48)(16,45)(17,50)(18,51)(19,52)(20,49)(21,42)(22,43)(23,44)(24,41)(25,58)(26,59)(27,60)(28,57)(29,54)(30,55)(31,56)(32,53)(37,64)(38,61)(39,62)(40,63), (1,20)(2,17)(3,18)(4,19)(5,49)(6,50)(7,51)(8,52)(9,29)(10,30)(11,31)(12,32)(13,39)(14,40)(15,37)(16,38)(21,25)(22,26)(23,27)(24,28)(33,53)(34,54)(35,55)(36,56)(41,57)(42,58)(43,59)(44,60)(45,61)(46,62)(47,63)(48,64), (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), (1,2,3,4)(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), (1,41,5,24)(2,42,6,21)(3,43,7,22)(4,44,8,23)(9,46,34,13)(10,47,35,14)(11,48,36,15)(12,45,33,16)(17,58,50,25)(18,59,51,26)(19,60,52,27)(20,57,49,28)(29,62,54,39)(30,63,55,40)(31,64,56,37)(32,61,53,38), (1,33,5,12)(2,9,6,34)(3,35,7,10)(4,11,8,36)(13,58,46,25)(14,26,47,59)(15,60,48,27)(16,28,45,57)(17,29,50,54)(18,55,51,30)(19,31,52,56)(20,53,49,32)(21,39,42,62)(22,63,43,40)(23,37,44,64)(24,61,41,38)>;
G:=Group( (9,34)(10,35)(11,36)(12,33)(13,62)(14,63)(15,64)(16,61)(21,25)(22,26)(23,27)(24,28)(29,54)(30,55)(31,56)(32,53)(37,48)(38,45)(39,46)(40,47)(41,57)(42,58)(43,59)(44,60), (1,5)(2,6)(3,7)(4,8)(9,34)(10,35)(11,36)(12,33)(13,46)(14,47)(15,48)(16,45)(17,50)(18,51)(19,52)(20,49)(21,42)(22,43)(23,44)(24,41)(25,58)(26,59)(27,60)(28,57)(29,54)(30,55)(31,56)(32,53)(37,64)(38,61)(39,62)(40,63), (1,20)(2,17)(3,18)(4,19)(5,49)(6,50)(7,51)(8,52)(9,29)(10,30)(11,31)(12,32)(13,39)(14,40)(15,37)(16,38)(21,25)(22,26)(23,27)(24,28)(33,53)(34,54)(35,55)(36,56)(41,57)(42,58)(43,59)(44,60)(45,61)(46,62)(47,63)(48,64), (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), (1,2,3,4)(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), (1,41,5,24)(2,42,6,21)(3,43,7,22)(4,44,8,23)(9,46,34,13)(10,47,35,14)(11,48,36,15)(12,45,33,16)(17,58,50,25)(18,59,51,26)(19,60,52,27)(20,57,49,28)(29,62,54,39)(30,63,55,40)(31,64,56,37)(32,61,53,38), (1,33,5,12)(2,9,6,34)(3,35,7,10)(4,11,8,36)(13,58,46,25)(14,26,47,59)(15,60,48,27)(16,28,45,57)(17,29,50,54)(18,55,51,30)(19,31,52,56)(20,53,49,32)(21,39,42,62)(22,63,43,40)(23,37,44,64)(24,61,41,38) );
G=PermutationGroup([[(9,34),(10,35),(11,36),(12,33),(13,62),(14,63),(15,64),(16,61),(21,25),(22,26),(23,27),(24,28),(29,54),(30,55),(31,56),(32,53),(37,48),(38,45),(39,46),(40,47),(41,57),(42,58),(43,59),(44,60)], [(1,5),(2,6),(3,7),(4,8),(9,34),(10,35),(11,36),(12,33),(13,46),(14,47),(15,48),(16,45),(17,50),(18,51),(19,52),(20,49),(21,42),(22,43),(23,44),(24,41),(25,58),(26,59),(27,60),(28,57),(29,54),(30,55),(31,56),(32,53),(37,64),(38,61),(39,62),(40,63)], [(1,20),(2,17),(3,18),(4,19),(5,49),(6,50),(7,51),(8,52),(9,29),(10,30),(11,31),(12,32),(13,39),(14,40),(15,37),(16,38),(21,25),(22,26),(23,27),(24,28),(33,53),(34,54),(35,55),(36,56),(41,57),(42,58),(43,59),(44,60),(45,61),(46,62),(47,63),(48,64)], [(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)], [(1,2,3,4),(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)], [(1,41,5,24),(2,42,6,21),(3,43,7,22),(4,44,8,23),(9,46,34,13),(10,47,35,14),(11,48,36,15),(12,45,33,16),(17,58,50,25),(18,59,51,26),(19,60,52,27),(20,57,49,28),(29,62,54,39),(30,63,55,40),(31,64,56,37),(32,61,53,38)], [(1,33,5,12),(2,9,6,34),(3,35,7,10),(4,11,8,36),(13,58,46,25),(14,26,47,59),(15,60,48,27),(16,28,45,57),(17,29,50,54),(18,55,51,30),(19,31,52,56),(20,53,49,32),(21,39,42,62),(22,63,43,40),(23,37,44,64),(24,61,41,38)]])
50 conjugacy classes
| class | 1 | 2A | ··· | 2G | 2H | 2I | 2J | 2K | 2L | 2M | 4A | ··· | 4T | 4U | ··· | 4AJ |
| order | 1 | 2 | ··· | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 4 | ··· | 4 |
| size | 1 | 1 | ··· | 1 | 2 | 2 | 2 | 2 | 4 | 4 | 2 | ··· | 2 | 4 | ··· | 4 |
50 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | - | |||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C4 | D4 | C4oD4 | C4oD4 | 2+ 1+4 | 2- 1+4 |
| kernel | C23.241C24 | C2xC2.C42 | C4xC22:C4 | C4xC4:C4 | C23.8Q8 | C23.23D4 | C23.63C23 | C24.C22 | C23.65C23 | C2xC42:C2 | C2xC4xD4 | C2xC22.D4 | C22.D4 | C22:C4 | C2xC4 | C23 | C22 | C22 |
| # reps | 1 | 1 | 2 | 1 | 2 | 1 | 2 | 2 | 1 | 1 | 1 | 1 | 16 | 4 | 8 | 4 | 1 | 1 |
Matrix representation of C23.241C24 ►in GL5(F5)
| 4 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 1 | 4 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 2 | 4 |
| 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 |
| 2 | 0 | 0 | 0 | 0 |
| 0 | 3 | 0 | 0 | 0 |
| 0 | 0 | 3 | 0 | 0 |
| 0 | 0 | 0 | 4 | 0 |
| 0 | 0 | 0 | 3 | 1 |
| 4 | 0 | 0 | 0 | 0 |
| 0 | 1 | 3 | 0 | 0 |
| 0 | 0 | 4 | 0 | 0 |
| 0 | 0 | 0 | 2 | 0 |
| 0 | 0 | 0 | 0 | 2 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 1 | 4 | 0 | 0 |
| 0 | 0 | 0 | 1 | 4 |
| 0 | 0 | 0 | 2 | 4 |
G:=sub<GL(5,GF(5))| [4,0,0,0,0,0,1,1,0,0,0,0,4,0,0,0,0,0,1,2,0,0,0,0,4],[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],[2,0,0,0,0,0,3,0,0,0,0,0,3,0,0,0,0,0,4,3,0,0,0,0,1],[4,0,0,0,0,0,1,0,0,0,0,3,4,0,0,0,0,0,2,0,0,0,0,0,2],[1,0,0,0,0,0,1,1,0,0,0,0,4,0,0,0,0,0,1,2,0,0,0,4,4] >;
C23.241C24 in GAP, Magma, Sage, TeX
C_2^3._{241}C_2^4 % in TeX
G:=Group("C2^3.241C2^4"); // GroupNames label
G:=SmallGroup(128,1091);
// by ID
G=gap.SmallGroup(128,1091);
# by ID
G:=PCGroup([7,-2,2,2,2,-2,2,2,448,253,456,758,100,346]);
// Polycyclic
G:=Group<a,b,c,d,e,f,g|a^2=b^2=c^2=d^2=1,e^2=d,f^2=g^2=b,g*a*g^-1=a*b=b*a,f*a*f^-1=a*c=c*a,a*d=d*a,a*e=e*a,b*c=c*b,b*d=d*b,g*e*g^-1=b*e=e*b,b*f=f*b,b*g=g*b,c*d=d*c,c*e=e*c,g*f*g^-1=c*f=f*c,c*g=g*c,d*e=e*d,d*f=f*d,d*g=g*d,e*f=f*e>;
// generators/relations