p-group, metabelian, nilpotent (class 2), monomial
Aliases: C23.508C24, C24.356C23, C22.2882+ (1+4), (C22×C4).15Q8, C23.65(C2×Q8), C2.7(C23⋊2Q8), (C23×C4).135C22, (C2×C42).595C22, (C22×C4).554C23, C23.Q8.18C2, C22.129(C22×Q8), C23.34D4.22C2, C23.65C23⋊100C2, C23.63C23⋊109C2, C2.C42.237C22, C2.81(C22.47C24), C2.40(C23.37C23), (C2×C4).167(C2×Q8), (C4×C22⋊C4).72C2, (C2×C4).166(C4○D4), (C2×C4⋊C4).347C22, C22.384(C2×C4○D4), (C2×C22⋊C4).517C22, SmallGroup(128,1340)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Subgroups: 388 in 214 conjugacy classes, 100 normal (8 characteristic)
C1, C2, C2 [×6], C2 [×2], C4 [×20], C22, C22 [×6], C22 [×10], C2×C4 [×12], C2×C4 [×40], C23, C23 [×2], C23 [×6], C42 [×4], C22⋊C4 [×10], C4⋊C4 [×20], C22×C4 [×18], C22×C4 [×2], C24, C2.C42 [×12], C2×C42 [×4], C2×C22⋊C4 [×6], C2×C4⋊C4 [×12], C23×C4, C4×C22⋊C4 [×2], C23.34D4, C23.63C23 [×4], C23.65C23 [×4], C23.Q8 [×4], C23.508C24
Quotients:
C1, C2 [×15], C22 [×35], Q8 [×4], C23 [×15], C2×Q8 [×6], C4○D4 [×8], C24, C22×Q8, C2×C4○D4 [×4], 2+ (1+4) [×2], C23.37C23 [×2], C23⋊2Q8, C22.47C24 [×4], C23.508C24
Generators and relations
G = < a,b,c,d,e,f,g | a2=b2=c2=e2=1, d2=ba=ab, f2=b, g2=ca=ac, ede=gdg-1=ad=da, 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, eg=ge, fg=gf >
(1 9)(2 10)(3 11)(4 12)(5 37)(6 38)(7 39)(8 40)(13 45)(14 46)(15 47)(16 48)(17 49)(18 50)(19 51)(20 52)(21 43)(22 44)(23 41)(24 42)(25 55)(26 56)(27 53)(28 54)(29 59)(30 60)(31 57)(32 58)(33 64)(34 61)(35 62)(36 63)
(1 11)(2 12)(3 9)(4 10)(5 39)(6 40)(7 37)(8 38)(13 47)(14 48)(15 45)(16 46)(17 51)(18 52)(19 49)(20 50)(21 41)(22 42)(23 43)(24 44)(25 53)(26 54)(27 55)(28 56)(29 57)(30 58)(31 59)(32 60)(33 62)(34 63)(35 64)(36 61)
(1 25)(2 26)(3 27)(4 28)(5 22)(6 23)(7 24)(8 21)(9 55)(10 56)(11 53)(12 54)(13 59)(14 60)(15 57)(16 58)(17 63)(18 64)(19 61)(20 62)(29 45)(30 46)(31 47)(32 48)(33 50)(34 51)(35 52)(36 49)(37 44)(38 41)(39 42)(40 43)
(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)
(2 10)(4 12)(5 44)(6 23)(7 42)(8 21)(14 46)(16 48)(17 63)(18 33)(19 61)(20 35)(22 37)(24 39)(26 56)(28 54)(30 60)(32 58)(34 51)(36 49)(38 41)(40 43)(50 64)(52 62)
(1 19 11 49)(2 50 12 20)(3 17 9 51)(4 52 10 18)(5 32 39 60)(6 57 40 29)(7 30 37 58)(8 59 38 31)(13 41 47 21)(14 22 48 42)(15 43 45 23)(16 24 46 44)(25 61 53 36)(26 33 54 62)(27 63 55 34)(28 35 56 64)
(1 57 55 47)(2 32 56 16)(3 59 53 45)(4 30 54 14)(5 35 44 20)(6 63 41 49)(7 33 42 18)(8 61 43 51)(9 31 25 15)(10 58 26 48)(11 29 27 13)(12 60 28 46)(17 38 36 23)(19 40 34 21)(22 52 37 62)(24 50 39 64)
G:=sub<Sym(64)| (1,9)(2,10)(3,11)(4,12)(5,37)(6,38)(7,39)(8,40)(13,45)(14,46)(15,47)(16,48)(17,49)(18,50)(19,51)(20,52)(21,43)(22,44)(23,41)(24,42)(25,55)(26,56)(27,53)(28,54)(29,59)(30,60)(31,57)(32,58)(33,64)(34,61)(35,62)(36,63), (1,11)(2,12)(3,9)(4,10)(5,39)(6,40)(7,37)(8,38)(13,47)(14,48)(15,45)(16,46)(17,51)(18,52)(19,49)(20,50)(21,41)(22,42)(23,43)(24,44)(25,53)(26,54)(27,55)(28,56)(29,57)(30,58)(31,59)(32,60)(33,62)(34,63)(35,64)(36,61), (1,25)(2,26)(3,27)(4,28)(5,22)(6,23)(7,24)(8,21)(9,55)(10,56)(11,53)(12,54)(13,59)(14,60)(15,57)(16,58)(17,63)(18,64)(19,61)(20,62)(29,45)(30,46)(31,47)(32,48)(33,50)(34,51)(35,52)(36,49)(37,44)(38,41)(39,42)(40,43), (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), (2,10)(4,12)(5,44)(6,23)(7,42)(8,21)(14,46)(16,48)(17,63)(18,33)(19,61)(20,35)(22,37)(24,39)(26,56)(28,54)(30,60)(32,58)(34,51)(36,49)(38,41)(40,43)(50,64)(52,62), (1,19,11,49)(2,50,12,20)(3,17,9,51)(4,52,10,18)(5,32,39,60)(6,57,40,29)(7,30,37,58)(8,59,38,31)(13,41,47,21)(14,22,48,42)(15,43,45,23)(16,24,46,44)(25,61,53,36)(26,33,54,62)(27,63,55,34)(28,35,56,64), (1,57,55,47)(2,32,56,16)(3,59,53,45)(4,30,54,14)(5,35,44,20)(6,63,41,49)(7,33,42,18)(8,61,43,51)(9,31,25,15)(10,58,26,48)(11,29,27,13)(12,60,28,46)(17,38,36,23)(19,40,34,21)(22,52,37,62)(24,50,39,64)>;
G:=Group( (1,9)(2,10)(3,11)(4,12)(5,37)(6,38)(7,39)(8,40)(13,45)(14,46)(15,47)(16,48)(17,49)(18,50)(19,51)(20,52)(21,43)(22,44)(23,41)(24,42)(25,55)(26,56)(27,53)(28,54)(29,59)(30,60)(31,57)(32,58)(33,64)(34,61)(35,62)(36,63), (1,11)(2,12)(3,9)(4,10)(5,39)(6,40)(7,37)(8,38)(13,47)(14,48)(15,45)(16,46)(17,51)(18,52)(19,49)(20,50)(21,41)(22,42)(23,43)(24,44)(25,53)(26,54)(27,55)(28,56)(29,57)(30,58)(31,59)(32,60)(33,62)(34,63)(35,64)(36,61), (1,25)(2,26)(3,27)(4,28)(5,22)(6,23)(7,24)(8,21)(9,55)(10,56)(11,53)(12,54)(13,59)(14,60)(15,57)(16,58)(17,63)(18,64)(19,61)(20,62)(29,45)(30,46)(31,47)(32,48)(33,50)(34,51)(35,52)(36,49)(37,44)(38,41)(39,42)(40,43), (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), (2,10)(4,12)(5,44)(6,23)(7,42)(8,21)(14,46)(16,48)(17,63)(18,33)(19,61)(20,35)(22,37)(24,39)(26,56)(28,54)(30,60)(32,58)(34,51)(36,49)(38,41)(40,43)(50,64)(52,62), (1,19,11,49)(2,50,12,20)(3,17,9,51)(4,52,10,18)(5,32,39,60)(6,57,40,29)(7,30,37,58)(8,59,38,31)(13,41,47,21)(14,22,48,42)(15,43,45,23)(16,24,46,44)(25,61,53,36)(26,33,54,62)(27,63,55,34)(28,35,56,64), (1,57,55,47)(2,32,56,16)(3,59,53,45)(4,30,54,14)(5,35,44,20)(6,63,41,49)(7,33,42,18)(8,61,43,51)(9,31,25,15)(10,58,26,48)(11,29,27,13)(12,60,28,46)(17,38,36,23)(19,40,34,21)(22,52,37,62)(24,50,39,64) );
G=PermutationGroup([(1,9),(2,10),(3,11),(4,12),(5,37),(6,38),(7,39),(8,40),(13,45),(14,46),(15,47),(16,48),(17,49),(18,50),(19,51),(20,52),(21,43),(22,44),(23,41),(24,42),(25,55),(26,56),(27,53),(28,54),(29,59),(30,60),(31,57),(32,58),(33,64),(34,61),(35,62),(36,63)], [(1,11),(2,12),(3,9),(4,10),(5,39),(6,40),(7,37),(8,38),(13,47),(14,48),(15,45),(16,46),(17,51),(18,52),(19,49),(20,50),(21,41),(22,42),(23,43),(24,44),(25,53),(26,54),(27,55),(28,56),(29,57),(30,58),(31,59),(32,60),(33,62),(34,63),(35,64),(36,61)], [(1,25),(2,26),(3,27),(4,28),(5,22),(6,23),(7,24),(8,21),(9,55),(10,56),(11,53),(12,54),(13,59),(14,60),(15,57),(16,58),(17,63),(18,64),(19,61),(20,62),(29,45),(30,46),(31,47),(32,48),(33,50),(34,51),(35,52),(36,49),(37,44),(38,41),(39,42),(40,43)], [(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)], [(2,10),(4,12),(5,44),(6,23),(7,42),(8,21),(14,46),(16,48),(17,63),(18,33),(19,61),(20,35),(22,37),(24,39),(26,56),(28,54),(30,60),(32,58),(34,51),(36,49),(38,41),(40,43),(50,64),(52,62)], [(1,19,11,49),(2,50,12,20),(3,17,9,51),(4,52,10,18),(5,32,39,60),(6,57,40,29),(7,30,37,58),(8,59,38,31),(13,41,47,21),(14,22,48,42),(15,43,45,23),(16,24,46,44),(25,61,53,36),(26,33,54,62),(27,63,55,34),(28,35,56,64)], [(1,57,55,47),(2,32,56,16),(3,59,53,45),(4,30,54,14),(5,35,44,20),(6,63,41,49),(7,33,42,18),(8,61,43,51),(9,31,25,15),(10,58,26,48),(11,29,27,13),(12,60,28,46),(17,38,36,23),(19,40,34,21),(22,52,37,62),(24,50,39,64)])
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 | 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 |
| 0 | 3 | 0 | 0 | 0 | 0 |
| 3 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 4 | 0 | 0 | 0 |
| 0 | 0 | 0 | 4 | 0 | 0 |
| 0 | 0 | 0 | 0 | 4 | 2 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 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 | 1 | 0 |
| 0 | 0 | 0 | 0 | 1 | 4 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 4 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 2 | 0 |
| 0 | 0 | 0 | 0 | 2 | 3 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 3 | 0 | 0 | 0 |
| 0 | 0 | 0 | 3 | 0 | 0 |
| 0 | 0 | 0 | 0 | 2 | 0 |
| 0 | 0 | 0 | 0 | 2 | 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],[4,0,0,0,0,0,0,4,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,4,0,0,0,0,0,0,4],[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],[0,3,0,0,0,0,3,0,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,2,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,4,0,0,0,0,0,0,1,1,0,0,0,0,0,4],[0,4,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,2,2,0,0,0,0,0,3],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,3,0,0,0,0,0,0,3,0,0,0,0,0,0,2,2,0,0,0,0,0,3] >;
38 conjugacy classes
| class | 1 | 2A | ··· | 2G | 2H | 2I | 4A | ··· | 4H | 4I | ··· | 4X | 4Y | 4Z | 4AA | 4AB |
| order | 1 | 2 | ··· | 2 | 2 | 2 | 4 | ··· | 4 | 4 | ··· | 4 | 4 | 4 | 4 | 4 |
| size | 1 | 1 | ··· | 1 | 4 | 4 | 2 | ··· | 2 | 4 | ··· | 4 | 8 | 8 | 8 | 8 |
38 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 4 |
| type | + | + | + | + | + | + | - | + | |
| image | C1 | C2 | C2 | C2 | C2 | C2 | Q8 | C4○D4 | 2+ (1+4) |
| kernel | C23.508C24 | C4×C22⋊C4 | C23.34D4 | C23.63C23 | C23.65C23 | C23.Q8 | C22×C4 | C2×C4 | C22 |
| # reps | 1 | 2 | 1 | 4 | 4 | 4 | 4 | 16 | 2 |
In GAP, Magma, Sage, TeX
C_2^3._{508}C_2^4 % in TeX
G:=Group("C2^3.508C2^4"); // GroupNames label
G:=SmallGroup(128,1340);
// by ID
G=gap.SmallGroup(128,1340);
# by ID
G:=PCGroup([7,-2,2,2,2,-2,2,2,336,253,758,723,184,675,304]);
// Polycyclic
G:=Group<a,b,c,d,e,f,g|a^2=b^2=c^2=e^2=1,d^2=b*a=a*b,f^2=b,g^2=c*a=a*c,e*d*e=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^-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,e*g=g*e,f*g=g*f>;
// generators/relations