p-group, metabelian, nilpotent (class 2), monomial
Aliases: C24.97C23, C23.701C24, C22.4742+ 1+4, C22.3632- 1+4, C23.Q8⋊90C2, C23.4Q8⋊63C2, (C22×C4).610C23, (C2×C42).722C22, C23.10D4.69C2, (C22×D4).287C22, C23.84C23⋊16C2, C24.C22⋊175C2, C23.65C23⋊160C2, C2.39(C22.54C24), C2.C42.405C22, C2.121(C22.47C24), C2.123(C22.36C24), (C2×C4).242(C4○D4), (C2×C4⋊C4).511C22, C22.562(C2×C4○D4), (C2×C22⋊C4).80C22, SmallGroup(128,1533)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C23.701C24
G = < a,b,c,d,e,f,g | a2=b2=c2=1, d2=abc, e2=a, f2=ca=ac, g2=b, ab=ba, 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 (13 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×C22⋊C4, C2×C4⋊C4, C22×D4, C24.C22, C23.65C23, C23.10D4, C23.Q8, C23.4Q8, C23.84C23, C23.701C24
Quotients: C1, C2, C22, C23, C4○D4, C24, C2×C4○D4, 2+ 1+4, 2- 1+4, C22.36C24, C22.47C24, C22.54C24, C23.701C24
►On 64 points
Generators in S64
(1 29)(2 30)(3 31)(4 32)(5 18)(6 19)(7 20)(8 17)(9 13)(10 14)(11 15)(12 16)(21 61)(22 62)(23 63)(24 64)(25 33)(26 34)(27 35)(28 36)(37 49)(38 50)(39 51)(40 52)(41 59)(42 60)(43 57)(44 58)(45 55)(46 56)(47 53)(48 54)
(1 14)(2 15)(3 16)(4 13)(5 59)(6 60)(7 57)(8 58)(9 32)(10 29)(11 30)(12 31)(17 44)(18 41)(19 42)(20 43)(21 54)(22 55)(23 56)(24 53)(25 40)(26 37)(27 38)(28 39)(33 52)(34 49)(35 50)(36 51)(45 62)(46 63)(47 64)(48 61)
(1 12)(2 9)(3 10)(4 11)(5 43)(6 44)(7 41)(8 42)(13 30)(14 31)(15 32)(16 29)(17 60)(18 57)(19 58)(20 59)(21 46)(22 47)(23 48)(24 45)(25 50)(26 51)(27 52)(28 49)(33 38)(34 39)(35 40)(36 37)(53 62)(54 63)(55 64)(56 61)
(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 21 29 61)(2 62 30 22)(3 23 31 63)(4 64 32 24)(5 38 18 50)(6 51 19 39)(7 40 20 52)(8 49 17 37)(9 53 13 47)(10 48 14 54)(11 55 15 45)(12 46 16 56)(25 43 33 57)(26 58 34 44)(27 41 35 59)(28 60 36 42)
(1 13 16 2)(3 15 14 4)(5 42 57 17)(6 20 58 41)(7 44 59 19)(8 18 60 43)(9 12 30 29)(10 32 31 11)(21 22 56 53)(23 24 54 55)(25 36 38 49)(26 52 39 35)(27 34 40 51)(28 50 37 33)(45 63 64 48)(46 47 61 62)
(1 50 14 35)(2 28 15 39)(3 52 16 33)(4 26 13 37)(5 48 59 61)(6 22 60 55)(7 46 57 63)(8 24 58 53)(9 49 32 34)(10 27 29 38)(11 51 30 36)(12 25 31 40)(17 64 44 47)(18 54 41 21)(19 62 42 45)(20 56 43 23)
G:=sub<Sym(64)| (1,29)(2,30)(3,31)(4,32)(5,18)(6,19)(7,20)(8,17)(9,13)(10,14)(11,15)(12,16)(21,61)(22,62)(23,63)(24,64)(25,33)(26,34)(27,35)(28,36)(37,49)(38,50)(39,51)(40,52)(41,59)(42,60)(43,57)(44,58)(45,55)(46,56)(47,53)(48,54), (1,14)(2,15)(3,16)(4,13)(5,59)(6,60)(7,57)(8,58)(9,32)(10,29)(11,30)(12,31)(17,44)(18,41)(19,42)(20,43)(21,54)(22,55)(23,56)(24,53)(25,40)(26,37)(27,38)(28,39)(33,52)(34,49)(35,50)(36,51)(45,62)(46,63)(47,64)(48,61), (1,12)(2,9)(3,10)(4,11)(5,43)(6,44)(7,41)(8,42)(13,30)(14,31)(15,32)(16,29)(17,60)(18,57)(19,58)(20,59)(21,46)(22,47)(23,48)(24,45)(25,50)(26,51)(27,52)(28,49)(33,38)(34,39)(35,40)(36,37)(53,62)(54,63)(55,64)(56,61), (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), (1,21,29,61)(2,62,30,22)(3,23,31,63)(4,64,32,24)(5,38,18,50)(6,51,19,39)(7,40,20,52)(8,49,17,37)(9,53,13,47)(10,48,14,54)(11,55,15,45)(12,46,16,56)(25,43,33,57)(26,58,34,44)(27,41,35,59)(28,60,36,42), (1,13,16,2)(3,15,14,4)(5,42,57,17)(6,20,58,41)(7,44,59,19)(8,18,60,43)(9,12,30,29)(10,32,31,11)(21,22,56,53)(23,24,54,55)(25,36,38,49)(26,52,39,35)(27,34,40,51)(28,50,37,33)(45,63,64,48)(46,47,61,62), (1,50,14,35)(2,28,15,39)(3,52,16,33)(4,26,13,37)(5,48,59,61)(6,22,60,55)(7,46,57,63)(8,24,58,53)(9,49,32,34)(10,27,29,38)(11,51,30,36)(12,25,31,40)(17,64,44,47)(18,54,41,21)(19,62,42,45)(20,56,43,23)>;
G:=Group( (1,29)(2,30)(3,31)(4,32)(5,18)(6,19)(7,20)(8,17)(9,13)(10,14)(11,15)(12,16)(21,61)(22,62)(23,63)(24,64)(25,33)(26,34)(27,35)(28,36)(37,49)(38,50)(39,51)(40,52)(41,59)(42,60)(43,57)(44,58)(45,55)(46,56)(47,53)(48,54), (1,14)(2,15)(3,16)(4,13)(5,59)(6,60)(7,57)(8,58)(9,32)(10,29)(11,30)(12,31)(17,44)(18,41)(19,42)(20,43)(21,54)(22,55)(23,56)(24,53)(25,40)(26,37)(27,38)(28,39)(33,52)(34,49)(35,50)(36,51)(45,62)(46,63)(47,64)(48,61), (1,12)(2,9)(3,10)(4,11)(5,43)(6,44)(7,41)(8,42)(13,30)(14,31)(15,32)(16,29)(17,60)(18,57)(19,58)(20,59)(21,46)(22,47)(23,48)(24,45)(25,50)(26,51)(27,52)(28,49)(33,38)(34,39)(35,40)(36,37)(53,62)(54,63)(55,64)(56,61), (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), (1,21,29,61)(2,62,30,22)(3,23,31,63)(4,64,32,24)(5,38,18,50)(6,51,19,39)(7,40,20,52)(8,49,17,37)(9,53,13,47)(10,48,14,54)(11,55,15,45)(12,46,16,56)(25,43,33,57)(26,58,34,44)(27,41,35,59)(28,60,36,42), (1,13,16,2)(3,15,14,4)(5,42,57,17)(6,20,58,41)(7,44,59,19)(8,18,60,43)(9,12,30,29)(10,32,31,11)(21,22,56,53)(23,24,54,55)(25,36,38,49)(26,52,39,35)(27,34,40,51)(28,50,37,33)(45,63,64,48)(46,47,61,62), (1,50,14,35)(2,28,15,39)(3,52,16,33)(4,26,13,37)(5,48,59,61)(6,22,60,55)(7,46,57,63)(8,24,58,53)(9,49,32,34)(10,27,29,38)(11,51,30,36)(12,25,31,40)(17,64,44,47)(18,54,41,21)(19,62,42,45)(20,56,43,23) );
G=PermutationGroup([[(1,29),(2,30),(3,31),(4,32),(5,18),(6,19),(7,20),(8,17),(9,13),(10,14),(11,15),(12,16),(21,61),(22,62),(23,63),(24,64),(25,33),(26,34),(27,35),(28,36),(37,49),(38,50),(39,51),(40,52),(41,59),(42,60),(43,57),(44,58),(45,55),(46,56),(47,53),(48,54)], [(1,14),(2,15),(3,16),(4,13),(5,59),(6,60),(7,57),(8,58),(9,32),(10,29),(11,30),(12,31),(17,44),(18,41),(19,42),(20,43),(21,54),(22,55),(23,56),(24,53),(25,40),(26,37),(27,38),(28,39),(33,52),(34,49),(35,50),(36,51),(45,62),(46,63),(47,64),(48,61)], [(1,12),(2,9),(3,10),(4,11),(5,43),(6,44),(7,41),(8,42),(13,30),(14,31),(15,32),(16,29),(17,60),(18,57),(19,58),(20,59),(21,46),(22,47),(23,48),(24,45),(25,50),(26,51),(27,52),(28,49),(33,38),(34,39),(35,40),(36,37),(53,62),(54,63),(55,64),(56,61)], [(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)], [(1,21,29,61),(2,62,30,22),(3,23,31,63),(4,64,32,24),(5,38,18,50),(6,51,19,39),(7,40,20,52),(8,49,17,37),(9,53,13,47),(10,48,14,54),(11,55,15,45),(12,46,16,56),(25,43,33,57),(26,58,34,44),(27,41,35,59),(28,60,36,42)], [(1,13,16,2),(3,15,14,4),(5,42,57,17),(6,20,58,41),(7,44,59,19),(8,18,60,43),(9,12,30,29),(10,32,31,11),(21,22,56,53),(23,24,54,55),(25,36,38,49),(26,52,39,35),(27,34,40,51),(28,50,37,33),(45,63,64,48),(46,47,61,62)], [(1,50,14,35),(2,28,15,39),(3,52,16,33),(4,26,13,37),(5,48,59,61),(6,22,60,55),(7,46,57,63),(8,24,58,53),(9,49,32,34),(10,27,29,38),(11,51,30,36),(12,25,31,40),(17,64,44,47),(18,54,41,21),(19,62,42,45),(20,56,43,23)]])
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 | C23.701C24 | C24.C22 | C23.65C23 | C23.10D4 | C23.Q8 | C23.4Q8 | C23.84C23 | C2×C4 | C22 | C22 |
| # reps | 1 | 6 | 3 | 3 | 1 | 1 | 1 | 12 | 3 | 1 |
Matrix representation of C23.701C24 ►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 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 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 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 2 | 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 | 3 | 0 |
| 0 | 0 | 0 | 0 | 1 | 2 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 3 | 2 | 0 | 0 |
| 0 | 0 | 1 | 2 | 0 | 0 |
| 0 | 0 | 0 | 0 | 4 | 1 |
| 0 | 0 | 0 | 0 | 3 | 1 |
| 0 | 3 | 0 | 0 | 0 | 0 |
| 2 | 0 | 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 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 4 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 4 | 0 | 0 | 0 |
| 0 | 0 | 0 | 4 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 4 |
| 0 | 0 | 0 | 0 | 0 | 4 |
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],[4,0,0,0,0,0,0,4,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[0,2,0,0,0,0,2,0,0,0,0,0,0,0,2,0,0,0,0,0,0,2,0,0,0,0,0,0,3,1,0,0,0,0,0,2],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,3,1,0,0,0,0,2,2,0,0,0,0,0,0,4,3,0,0,0,0,1,1],[0,2,0,0,0,0,3,0,0,0,0,0,0,0,3,1,0,0,0,0,0,2,0,0,0,0,0,0,2,0,0,0,0,0,0,2],[0,4,0,0,0,0,1,0,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,1,0,0,0,0,0,4,4] >;
C23.701C24 in GAP, Magma, Sage, TeX
C_2^3._{701}C_2^4 % in TeX
G:=Group("C2^3.701C2^4"); // GroupNames label
G:=SmallGroup(128,1533);
// by ID
G=gap.SmallGroup(128,1533);
# by ID
G:=PCGroup([7,-2,2,2,2,-2,2,2,784,253,120,758,723,436,1571,346,136]);
// Polycyclic
G:=Group<a,b,c,d,e,f,g|a^2=b^2=c^2=1,d^2=a*b*c,e^2=a,f^2=c*a=a*c,g^2=b,a*b=b*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