p-group, metabelian, nilpotent (class 2), monomial
Aliases: C24.97C23, C23.701C24, C22.3632- (1+4), C22.4742+ (1+4), C23.4Q8⋊63C2, C23.Q8⋊90C2, (C2×C42).722C22, (C22×C4).610C23, C23.10D4.69C2, (C22×D4).287C22, C24.C22⋊175C2, C23.84C23⋊16C2, C23.65C23⋊160C2, C2.39(C22.54C24), C2.C42.405C22, C2.123(C22.36C24), C2.121(C22.47C24), (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
Subgroups: 436 in 210 conjugacy classes, 88 normal (13 characteristic)
C1, C2, C2 [×6], C2 [×2], C4 [×16], C22, C22 [×6], C22 [×14], C2×C4 [×6], C2×C4 [×36], D4 [×4], C23, C23 [×14], C42 [×3], C22⋊C4 [×18], C4⋊C4 [×12], C22×C4, C22×C4 [×12], C2×D4 [×3], C24 [×2], C2.C42, C2.C42 [×9], C2×C42 [×3], C2×C22⋊C4 [×12], C2×C4⋊C4 [×9], C22×D4, C24.C22 [×6], C23.65C23 [×3], C23.10D4 [×3], C23.Q8, C23.4Q8, C23.84C23, C23.701C24
Quotients:
C1, C2 [×15], C22 [×35], C23 [×15], C4○D4 [×6], C24, C2×C4○D4 [×3], 2+ (1+4) [×3], 2- (1+4), C22.36C24 [×3], C22.47C24 [×3], C22.54C24, C23.701C24
Generators and relations
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 >
(1 52)(2 49)(3 50)(4 51)(5 61)(6 62)(7 63)(8 64)(9 26)(10 27)(11 28)(12 25)(13 29)(14 30)(15 31)(16 32)(17 24)(18 21)(19 22)(20 23)(33 38)(34 39)(35 40)(36 37)(41 48)(42 45)(43 46)(44 47)(53 57)(54 58)(55 59)(56 60)
(1 11)(2 12)(3 9)(4 10)(5 57)(6 58)(7 59)(8 60)(13 33)(14 34)(15 35)(16 36)(17 44)(18 41)(19 42)(20 43)(21 48)(22 45)(23 46)(24 47)(25 49)(26 50)(27 51)(28 52)(29 38)(30 39)(31 40)(32 37)(53 61)(54 62)(55 63)(56 64)
(1 26)(2 27)(3 28)(4 25)(5 55)(6 56)(7 53)(8 54)(9 52)(10 49)(11 50)(12 51)(13 40)(14 37)(15 38)(16 39)(17 45)(18 46)(19 47)(20 48)(21 43)(22 44)(23 41)(24 42)(29 35)(30 36)(31 33)(32 34)(57 63)(58 64)(59 61)(60 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 5 52 61)(2 62 49 6)(3 7 50 63)(4 64 51 8)(9 59 26 55)(10 56 27 60)(11 57 28 53)(12 54 25 58)(13 48 29 41)(14 42 30 45)(15 46 31 43)(16 44 32 47)(17 37 24 36)(18 33 21 38)(19 39 22 34)(20 35 23 40)
(1 10 9 2)(3 12 11 4)(5 6 59 60)(7 8 57 58)(13 32 35 39)(14 38 36 31)(15 30 33 37)(16 40 34 29)(17 21 42 46)(18 45 43 24)(19 23 44 48)(20 47 41 22)(25 28 51 50)(26 49 52 27)(53 54 63 64)(55 56 61 62)
(1 33 11 13)(2 30 12 39)(3 35 9 15)(4 32 10 37)(5 18 57 41)(6 45 58 22)(7 20 59 43)(8 47 60 24)(14 25 34 49)(16 27 36 51)(17 64 44 56)(19 62 42 54)(21 53 48 61)(23 55 46 63)(26 31 50 40)(28 29 52 38)
G:=sub<Sym(64)| (1,52)(2,49)(3,50)(4,51)(5,61)(6,62)(7,63)(8,64)(9,26)(10,27)(11,28)(12,25)(13,29)(14,30)(15,31)(16,32)(17,24)(18,21)(19,22)(20,23)(33,38)(34,39)(35,40)(36,37)(41,48)(42,45)(43,46)(44,47)(53,57)(54,58)(55,59)(56,60), (1,11)(2,12)(3,9)(4,10)(5,57)(6,58)(7,59)(8,60)(13,33)(14,34)(15,35)(16,36)(17,44)(18,41)(19,42)(20,43)(21,48)(22,45)(23,46)(24,47)(25,49)(26,50)(27,51)(28,52)(29,38)(30,39)(31,40)(32,37)(53,61)(54,62)(55,63)(56,64), (1,26)(2,27)(3,28)(4,25)(5,55)(6,56)(7,53)(8,54)(9,52)(10,49)(11,50)(12,51)(13,40)(14,37)(15,38)(16,39)(17,45)(18,46)(19,47)(20,48)(21,43)(22,44)(23,41)(24,42)(29,35)(30,36)(31,33)(32,34)(57,63)(58,64)(59,61)(60,62), (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), (1,5,52,61)(2,62,49,6)(3,7,50,63)(4,64,51,8)(9,59,26,55)(10,56,27,60)(11,57,28,53)(12,54,25,58)(13,48,29,41)(14,42,30,45)(15,46,31,43)(16,44,32,47)(17,37,24,36)(18,33,21,38)(19,39,22,34)(20,35,23,40), (1,10,9,2)(3,12,11,4)(5,6,59,60)(7,8,57,58)(13,32,35,39)(14,38,36,31)(15,30,33,37)(16,40,34,29)(17,21,42,46)(18,45,43,24)(19,23,44,48)(20,47,41,22)(25,28,51,50)(26,49,52,27)(53,54,63,64)(55,56,61,62), (1,33,11,13)(2,30,12,39)(3,35,9,15)(4,32,10,37)(5,18,57,41)(6,45,58,22)(7,20,59,43)(8,47,60,24)(14,25,34,49)(16,27,36,51)(17,64,44,56)(19,62,42,54)(21,53,48,61)(23,55,46,63)(26,31,50,40)(28,29,52,38)>;
G:=Group( (1,52)(2,49)(3,50)(4,51)(5,61)(6,62)(7,63)(8,64)(9,26)(10,27)(11,28)(12,25)(13,29)(14,30)(15,31)(16,32)(17,24)(18,21)(19,22)(20,23)(33,38)(34,39)(35,40)(36,37)(41,48)(42,45)(43,46)(44,47)(53,57)(54,58)(55,59)(56,60), (1,11)(2,12)(3,9)(4,10)(5,57)(6,58)(7,59)(8,60)(13,33)(14,34)(15,35)(16,36)(17,44)(18,41)(19,42)(20,43)(21,48)(22,45)(23,46)(24,47)(25,49)(26,50)(27,51)(28,52)(29,38)(30,39)(31,40)(32,37)(53,61)(54,62)(55,63)(56,64), (1,26)(2,27)(3,28)(4,25)(5,55)(6,56)(7,53)(8,54)(9,52)(10,49)(11,50)(12,51)(13,40)(14,37)(15,38)(16,39)(17,45)(18,46)(19,47)(20,48)(21,43)(22,44)(23,41)(24,42)(29,35)(30,36)(31,33)(32,34)(57,63)(58,64)(59,61)(60,62), (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), (1,5,52,61)(2,62,49,6)(3,7,50,63)(4,64,51,8)(9,59,26,55)(10,56,27,60)(11,57,28,53)(12,54,25,58)(13,48,29,41)(14,42,30,45)(15,46,31,43)(16,44,32,47)(17,37,24,36)(18,33,21,38)(19,39,22,34)(20,35,23,40), (1,10,9,2)(3,12,11,4)(5,6,59,60)(7,8,57,58)(13,32,35,39)(14,38,36,31)(15,30,33,37)(16,40,34,29)(17,21,42,46)(18,45,43,24)(19,23,44,48)(20,47,41,22)(25,28,51,50)(26,49,52,27)(53,54,63,64)(55,56,61,62), (1,33,11,13)(2,30,12,39)(3,35,9,15)(4,32,10,37)(5,18,57,41)(6,45,58,22)(7,20,59,43)(8,47,60,24)(14,25,34,49)(16,27,36,51)(17,64,44,56)(19,62,42,54)(21,53,48,61)(23,55,46,63)(26,31,50,40)(28,29,52,38) );
G=PermutationGroup([(1,52),(2,49),(3,50),(4,51),(5,61),(6,62),(7,63),(8,64),(9,26),(10,27),(11,28),(12,25),(13,29),(14,30),(15,31),(16,32),(17,24),(18,21),(19,22),(20,23),(33,38),(34,39),(35,40),(36,37),(41,48),(42,45),(43,46),(44,47),(53,57),(54,58),(55,59),(56,60)], [(1,11),(2,12),(3,9),(4,10),(5,57),(6,58),(7,59),(8,60),(13,33),(14,34),(15,35),(16,36),(17,44),(18,41),(19,42),(20,43),(21,48),(22,45),(23,46),(24,47),(25,49),(26,50),(27,51),(28,52),(29,38),(30,39),(31,40),(32,37),(53,61),(54,62),(55,63),(56,64)], [(1,26),(2,27),(3,28),(4,25),(5,55),(6,56),(7,53),(8,54),(9,52),(10,49),(11,50),(12,51),(13,40),(14,37),(15,38),(16,39),(17,45),(18,46),(19,47),(20,48),(21,43),(22,44),(23,41),(24,42),(29,35),(30,36),(31,33),(32,34),(57,63),(58,64),(59,61),(60,62)], [(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)], [(1,5,52,61),(2,62,49,6),(3,7,50,63),(4,64,51,8),(9,59,26,55),(10,56,27,60),(11,57,28,53),(12,54,25,58),(13,48,29,41),(14,42,30,45),(15,46,31,43),(16,44,32,47),(17,37,24,36),(18,33,21,38),(19,39,22,34),(20,35,23,40)], [(1,10,9,2),(3,12,11,4),(5,6,59,60),(7,8,57,58),(13,32,35,39),(14,38,36,31),(15,30,33,37),(16,40,34,29),(17,21,42,46),(18,45,43,24),(19,23,44,48),(20,47,41,22),(25,28,51,50),(26,49,52,27),(53,54,63,64),(55,56,61,62)], [(1,33,11,13),(2,30,12,39),(3,35,9,15),(4,32,10,37),(5,18,57,41),(6,45,58,22),(7,20,59,43),(8,47,60,24),(14,25,34,49),(16,27,36,51),(17,64,44,56),(19,62,42,54),(21,53,48,61),(23,55,46,63),(26,31,50,40),(28,29,52,38)])
Matrix representation ►G ⊆ 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] >;
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 |
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