p-group, metabelian, nilpotent (class 2), monomial
Aliases: C24.93C23, C23.695C24, C22.4682+ 1+4, C22.3582- 1+4, C42⋊9C4⋊38C2, C23.Q8⋊88C2, (C2×C42).111C22, (C22×C4).605C23, C23.10D4.66C2, (C22×D4).283C22, C24.C22⋊171C2, C23.83C23⋊125C2, C23.63C23⋊190C2, C2.36(C22.54C24), C2.C42.399C22, C2.47(C22.49C24), C2.117(C22.47C24), C2.113(C22.33C24), C2.118(C22.36C24), (C2×C4).236(C4○D4), (C2×C4⋊C4).505C22, C22.556(C2×C4○D4), (C2×C22⋊C4).326C22, SmallGroup(128,1527)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C23.695C24
G = < a,b,c,d,e,f,g | a2=b2=c2=1, d2=abc, e2=f2=b, g2=ba=ab, ac=ca, ede-1=ad=da, geg-1=ae=ea, af=fa, ag=ga, bc=cb, fdf-1=bd=db, be=eb, bf=fb, bg=gb, cd=dc, fef-1=ce=ec, cf=fc, cg=gc, gdg-1=abd, fg=gf >
Subgroups: 436 in 210 conjugacy classes, 88 normal (22 characteristic)
C1, C2, C2, C2, C4, C22, C22, C22, C2×C4, C2×C4, D4, C23, C23, C42, C22⋊C4, C4⋊C4, C22×C4, C22×C4, C2×D4, C24, C2.C42, C2.C42, C2×C42, C2×C42, C2×C22⋊C4, C2×C4⋊C4, C2×C4⋊C4, C22×D4, C42⋊9C4, C23.63C23, C24.C22, C23.10D4, C23.10D4, C23.Q8, C23.83C23, C23.695C24
Quotients: C1, C2, C22, C23, C4○D4, C24, C2×C4○D4, 2+ 1+4, 2- 1+4, C22.33C24, C22.36C24, C22.47C24, C22.49C24, C22.54C24, C23.695C24
►On 64 points
Generators in S64
(1 61)(2 62)(3 63)(4 64)(5 50)(6 51)(7 52)(8 49)(9 59)(10 60)(11 57)(12 58)(13 47)(14 48)(15 45)(16 46)(17 36)(18 33)(19 34)(20 35)(21 55)(22 56)(23 53)(24 54)(25 38)(26 39)(27 40)(28 37)(29 44)(30 41)(31 42)(32 43)
(1 53)(2 54)(3 55)(4 56)(5 35)(6 36)(7 33)(8 34)(9 41)(10 42)(11 43)(12 44)(13 38)(14 39)(15 40)(16 37)(17 51)(18 52)(19 49)(20 50)(21 63)(22 64)(23 61)(24 62)(25 47)(26 48)(27 45)(28 46)(29 58)(30 59)(31 60)(32 57)
(1 21)(2 22)(3 23)(4 24)(5 18)(6 19)(7 20)(8 17)(9 32)(10 29)(11 30)(12 31)(13 27)(14 28)(15 25)(16 26)(33 50)(34 51)(35 52)(36 49)(37 48)(38 45)(39 46)(40 47)(41 57)(42 58)(43 59)(44 60)(53 63)(54 64)(55 61)(56 62)
(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 37 53 16)(2 25 54 47)(3 39 55 14)(4 27 56 45)(5 59 35 30)(6 10 36 42)(7 57 33 32)(8 12 34 44)(9 20 41 50)(11 18 43 52)(13 62 38 24)(15 64 40 22)(17 31 51 60)(19 29 49 58)(21 48 63 26)(23 46 61 28)
(1 47 53 25)(2 26 54 48)(3 45 55 27)(4 28 56 46)(5 10 35 42)(6 43 36 11)(7 12 33 44)(8 41 34 9)(13 23 38 61)(14 62 39 24)(15 21 40 63)(16 64 37 22)(17 57 51 32)(18 29 52 58)(19 59 49 30)(20 31 50 60)
(1 5 23 20)(2 17 24 6)(3 7 21 18)(4 19 22 8)(9 46 30 37)(10 38 31 47)(11 48 32 39)(12 40 29 45)(13 60 25 42)(14 43 26 57)(15 58 27 44)(16 41 28 59)(33 63 52 55)(34 56 49 64)(35 61 50 53)(36 54 51 62)
G:=sub<Sym(64)| (1,61)(2,62)(3,63)(4,64)(5,50)(6,51)(7,52)(8,49)(9,59)(10,60)(11,57)(12,58)(13,47)(14,48)(15,45)(16,46)(17,36)(18,33)(19,34)(20,35)(21,55)(22,56)(23,53)(24,54)(25,38)(26,39)(27,40)(28,37)(29,44)(30,41)(31,42)(32,43), (1,53)(2,54)(3,55)(4,56)(5,35)(6,36)(7,33)(8,34)(9,41)(10,42)(11,43)(12,44)(13,38)(14,39)(15,40)(16,37)(17,51)(18,52)(19,49)(20,50)(21,63)(22,64)(23,61)(24,62)(25,47)(26,48)(27,45)(28,46)(29,58)(30,59)(31,60)(32,57), (1,21)(2,22)(3,23)(4,24)(5,18)(6,19)(7,20)(8,17)(9,32)(10,29)(11,30)(12,31)(13,27)(14,28)(15,25)(16,26)(33,50)(34,51)(35,52)(36,49)(37,48)(38,45)(39,46)(40,47)(41,57)(42,58)(43,59)(44,60)(53,63)(54,64)(55,61)(56,62), (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), (1,37,53,16)(2,25,54,47)(3,39,55,14)(4,27,56,45)(5,59,35,30)(6,10,36,42)(7,57,33,32)(8,12,34,44)(9,20,41,50)(11,18,43,52)(13,62,38,24)(15,64,40,22)(17,31,51,60)(19,29,49,58)(21,48,63,26)(23,46,61,28), (1,47,53,25)(2,26,54,48)(3,45,55,27)(4,28,56,46)(5,10,35,42)(6,43,36,11)(7,12,33,44)(8,41,34,9)(13,23,38,61)(14,62,39,24)(15,21,40,63)(16,64,37,22)(17,57,51,32)(18,29,52,58)(19,59,49,30)(20,31,50,60), (1,5,23,20)(2,17,24,6)(3,7,21,18)(4,19,22,8)(9,46,30,37)(10,38,31,47)(11,48,32,39)(12,40,29,45)(13,60,25,42)(14,43,26,57)(15,58,27,44)(16,41,28,59)(33,63,52,55)(34,56,49,64)(35,61,50,53)(36,54,51,62)>;
G:=Group( (1,61)(2,62)(3,63)(4,64)(5,50)(6,51)(7,52)(8,49)(9,59)(10,60)(11,57)(12,58)(13,47)(14,48)(15,45)(16,46)(17,36)(18,33)(19,34)(20,35)(21,55)(22,56)(23,53)(24,54)(25,38)(26,39)(27,40)(28,37)(29,44)(30,41)(31,42)(32,43), (1,53)(2,54)(3,55)(4,56)(5,35)(6,36)(7,33)(8,34)(9,41)(10,42)(11,43)(12,44)(13,38)(14,39)(15,40)(16,37)(17,51)(18,52)(19,49)(20,50)(21,63)(22,64)(23,61)(24,62)(25,47)(26,48)(27,45)(28,46)(29,58)(30,59)(31,60)(32,57), (1,21)(2,22)(3,23)(4,24)(5,18)(6,19)(7,20)(8,17)(9,32)(10,29)(11,30)(12,31)(13,27)(14,28)(15,25)(16,26)(33,50)(34,51)(35,52)(36,49)(37,48)(38,45)(39,46)(40,47)(41,57)(42,58)(43,59)(44,60)(53,63)(54,64)(55,61)(56,62), (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), (1,37,53,16)(2,25,54,47)(3,39,55,14)(4,27,56,45)(5,59,35,30)(6,10,36,42)(7,57,33,32)(8,12,34,44)(9,20,41,50)(11,18,43,52)(13,62,38,24)(15,64,40,22)(17,31,51,60)(19,29,49,58)(21,48,63,26)(23,46,61,28), (1,47,53,25)(2,26,54,48)(3,45,55,27)(4,28,56,46)(5,10,35,42)(6,43,36,11)(7,12,33,44)(8,41,34,9)(13,23,38,61)(14,62,39,24)(15,21,40,63)(16,64,37,22)(17,57,51,32)(18,29,52,58)(19,59,49,30)(20,31,50,60), (1,5,23,20)(2,17,24,6)(3,7,21,18)(4,19,22,8)(9,46,30,37)(10,38,31,47)(11,48,32,39)(12,40,29,45)(13,60,25,42)(14,43,26,57)(15,58,27,44)(16,41,28,59)(33,63,52,55)(34,56,49,64)(35,61,50,53)(36,54,51,62) );
G=PermutationGroup([[(1,61),(2,62),(3,63),(4,64),(5,50),(6,51),(7,52),(8,49),(9,59),(10,60),(11,57),(12,58),(13,47),(14,48),(15,45),(16,46),(17,36),(18,33),(19,34),(20,35),(21,55),(22,56),(23,53),(24,54),(25,38),(26,39),(27,40),(28,37),(29,44),(30,41),(31,42),(32,43)], [(1,53),(2,54),(3,55),(4,56),(5,35),(6,36),(7,33),(8,34),(9,41),(10,42),(11,43),(12,44),(13,38),(14,39),(15,40),(16,37),(17,51),(18,52),(19,49),(20,50),(21,63),(22,64),(23,61),(24,62),(25,47),(26,48),(27,45),(28,46),(29,58),(30,59),(31,60),(32,57)], [(1,21),(2,22),(3,23),(4,24),(5,18),(6,19),(7,20),(8,17),(9,32),(10,29),(11,30),(12,31),(13,27),(14,28),(15,25),(16,26),(33,50),(34,51),(35,52),(36,49),(37,48),(38,45),(39,46),(40,47),(41,57),(42,58),(43,59),(44,60),(53,63),(54,64),(55,61),(56,62)], [(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)], [(1,37,53,16),(2,25,54,47),(3,39,55,14),(4,27,56,45),(5,59,35,30),(6,10,36,42),(7,57,33,32),(8,12,34,44),(9,20,41,50),(11,18,43,52),(13,62,38,24),(15,64,40,22),(17,31,51,60),(19,29,49,58),(21,48,63,26),(23,46,61,28)], [(1,47,53,25),(2,26,54,48),(3,45,55,27),(4,28,56,46),(5,10,35,42),(6,43,36,11),(7,12,33,44),(8,41,34,9),(13,23,38,61),(14,62,39,24),(15,21,40,63),(16,64,37,22),(17,57,51,32),(18,29,52,58),(19,59,49,30),(20,31,50,60)], [(1,5,23,20),(2,17,24,6),(3,7,21,18),(4,19,22,8),(9,46,30,37),(10,38,31,47),(11,48,32,39),(12,40,29,45),(13,60,25,42),(14,43,26,57),(15,58,27,44),(16,41,28,59),(33,63,52,55),(34,56,49,64),(35,61,50,53),(36,54,51,62)]])
32 conjugacy classes
| class | 1 | 2A | ··· | 2G | 2H | 2I | 4A | ··· | 4R | 4S | 4T | 4U | 4V |
| order | 1 | 2 | ··· | 2 | 2 | 2 | 4 | ··· | 4 | 4 | 4 | 4 | 4 |
| size | 1 | 1 | ··· | 1 | 8 | 8 | 4 | ··· | 4 | 8 | 8 | 8 | 8 |
32 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | - | |
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C4○D4 | 2+ 1+4 | 2- 1+4 |
| kernel | C23.695C24 | C42⋊9C4 | C23.63C23 | C24.C22 | C23.10D4 | C23.Q8 | C23.83C23 | C2×C4 | C22 | C22 |
| # reps | 1 | 1 | 2 | 6 | 3 | 2 | 1 | 12 | 3 | 1 |
Matrix representation of C23.695C24 ►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 |
| 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 |
| 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 |
| 2 | 0 | 0 | 0 | 0 | 0 |
| 0 | 2 | 0 | 0 | 0 | 0 |
| 0 | 0 | 3 | 3 | 0 | 0 |
| 0 | 0 | 0 | 2 | 0 | 0 |
| 0 | 0 | 0 | 0 | 3 | 3 |
| 0 | 0 | 0 | 0 | 0 | 2 |
| 0 | 2 | 0 | 0 | 0 | 0 |
| 3 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 2 | 0 | 0 | 0 |
| 0 | 0 | 0 | 2 | 0 | 0 |
| 0 | 0 | 0 | 0 | 2 | 2 |
| 0 | 0 | 0 | 0 | 1 | 3 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 4 | 4 | 0 | 0 |
| 0 | 0 | 2 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 4 | 0 | 0 | 0 | 0 | 0 |
| 0 | 4 | 0 | 0 | 0 | 0 |
| 0 | 0 | 4 | 4 | 0 | 0 |
| 0 | 0 | 2 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 2 | 0 |
| 0 | 0 | 0 | 0 | 1 | 3 |
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],[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],[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],[2,0,0,0,0,0,0,2,0,0,0,0,0,0,3,0,0,0,0,0,3,2,0,0,0,0,0,0,3,0,0,0,0,0,3,2],[0,3,0,0,0,0,2,0,0,0,0,0,0,0,2,0,0,0,0,0,0,2,0,0,0,0,0,0,2,1,0,0,0,0,2,3],[0,1,0,0,0,0,1,0,0,0,0,0,0,0,4,2,0,0,0,0,4,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[4,0,0,0,0,0,0,4,0,0,0,0,0,0,4,2,0,0,0,0,4,1,0,0,0,0,0,0,2,1,0,0,0,0,0,3] >;
C23.695C24 in GAP, Magma, Sage, TeX
C_2^3._{695}C_2^4 % in TeX
G:=Group("C2^3.695C2^4"); // GroupNames label
G:=SmallGroup(128,1527);
// by ID
G=gap.SmallGroup(128,1527);
# by ID
G:=PCGroup([7,-2,2,2,2,-2,2,2,784,253,232,758,723,184,1571,346,192]);
// Polycyclic
G:=Group<a,b,c,d,e,f,g|a^2=b^2=c^2=1,d^2=a*b*c,e^2=f^2=b,g^2=b*a=a*b,a*c=c*a,e*d*e^-1=a*d=d*a,g*e*g^-1=a*e=e*a,a*f=f*a,a*g=g*a,b*c=c*b,f*d*f^-1=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,g*d*g^-1=a*b*d,f*g=g*f>;
// generators/relations