direct product, p-group, metabelian, nilpotent (class 3), monomial
Aliases: C2×C23.25D4, C24.140D4, (C22×C8)⋊16C4, C4.3(C22×Q8), C4.47(C23×C4), (C23×C8).18C2, C8.57(C22×C4), C2.D8⋊73C22, C4.Q8⋊72C22, C4⋊C4.348C23, C23.75(C4⋊C4), (C2×C4).185C24, (C2×C8).588C23, C23.378(C2×D4), (C22×C4).822D4, (C22×C4).103Q8, C22.82(C4○D8), C4○(C23.25D4), (C23×C4).696C22, (C22×C8).565C22, C22.132(C22×D4), (C22×C4).1506C23, C42⋊C2.285C22, C4○(C2×C2.D8), C4○(C2×C4.Q8), (C2×C8)⋊37(C2×C4), C4.85(C2×C4⋊C4), C2.2(C2×C4○D8), (C2×C2.D8)⋊44C2, (C2×C4.Q8)⋊38C2, (C2×C4)○2(C4.Q8), (C2×C4)○2(C2.D8), C22.35(C2×C4⋊C4), C2.24(C22×C4⋊C4), (C2×C4).239(C2×Q8), (C2×C4).151(C4⋊C4), (C22×C4)○(C2.D8), (C22×C4)○(C4.Q8), (C2×C4).1567(C2×D4), (C2×C4⋊C4).903C22, (C2×C4).573(C22×C4), (C22×C4).496(C2×C4), (C2×C42⋊C2).53C2, (C2×C4)○(C23.25D4), (C2×C4)○(C2×C2.D8), (C2×C4)○(C2×C4.Q8), (C22×C4)○(C2×C2.D8), (C22×C4)○(C2×C4.Q8), SmallGroup(128,1641)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Subgroups: 364 in 256 conjugacy classes, 180 normal (16 characteristic)
C1, C2, C2 [×6], C2 [×4], C4 [×2], C4 [×6], C4 [×8], C22, C22 [×10], C22 [×12], C8 [×8], C2×C4 [×2], C2×C4 [×26], C2×C4 [×16], C23, C23 [×6], C23 [×4], C42 [×8], C22⋊C4 [×8], C4⋊C4 [×8], C4⋊C4 [×4], C2×C8 [×28], C22×C4 [×2], C22×C4 [×12], C22×C4 [×4], C24, C4.Q8 [×8], C2.D8 [×8], C2×C42 [×2], C2×C22⋊C4 [×2], C2×C4⋊C4 [×4], C42⋊C2 [×8], C42⋊C2 [×4], C22×C8 [×2], C22×C8 [×12], C23×C4, C2×C4.Q8 [×2], C2×C2.D8 [×2], C23.25D4 [×8], C2×C42⋊C2 [×2], C23×C8, C2×C23.25D4
Quotients:
C1, C2 [×15], C4 [×8], C22 [×35], C2×C4 [×28], D4 [×4], Q8 [×4], C23 [×15], C4⋊C4 [×16], C22×C4 [×14], C2×D4 [×6], C2×Q8 [×6], C24, C2×C4⋊C4 [×12], C4○D8 [×4], C23×C4, C22×D4, C22×Q8, C23.25D4 [×4], C22×C4⋊C4, C2×C4○D8 [×2], C2×C23.25D4
Generators and relations
G = < a,b,c,d,e,f | a2=b2=c2=d2=1, e4=d, f2=dc=cd, ab=ba, ac=ca, ad=da, ae=ea, af=fa, bc=cb, fbf-1=bd=db, be=eb, ce=ec, cf=fc, de=ed, df=fd, fef-1=e3 >
►On 64 points
(1 19)(2 20)(3 21)(4 22)(5 23)(6 24)(7 17)(8 18)(9 42)(10 43)(11 44)(12 45)(13 46)(14 47)(15 48)(16 41)(25 40)(26 33)(27 34)(28 35)(29 36)(30 37)(31 38)(32 39)(49 58)(50 59)(51 60)(52 61)(53 62)(54 63)(55 64)(56 57)
(25 29)(26 30)(27 31)(28 32)(33 37)(34 38)(35 39)(36 40)(49 53)(50 54)(51 55)(52 56)(57 61)(58 62)(59 63)(60 64)
(1 43)(2 44)(3 45)(4 46)(5 47)(6 48)(7 41)(8 42)(9 18)(10 19)(11 20)(12 21)(13 22)(14 23)(15 24)(16 17)(25 54)(26 55)(27 56)(28 49)(29 50)(30 51)(31 52)(32 53)(33 64)(34 57)(35 58)(36 59)(37 60)(38 61)(39 62)(40 63)
(1 5)(2 6)(3 7)(4 8)(9 13)(10 14)(11 15)(12 16)(17 21)(18 22)(19 23)(20 24)(25 29)(26 30)(27 31)(28 32)(33 37)(34 38)(35 39)(36 40)(41 45)(42 46)(43 47)(44 48)(49 53)(50 54)(51 55)(52 56)(57 61)(58 62)(59 63)(60 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 64 47 37)(2 59 48 40)(3 62 41 35)(4 57 42 38)(5 60 43 33)(6 63 44 36)(7 58 45 39)(8 61 46 34)(9 31 22 56)(10 26 23 51)(11 29 24 54)(12 32 17 49)(13 27 18 52)(14 30 19 55)(15 25 20 50)(16 28 21 53)
G:=sub<Sym(64)| (1,19)(2,20)(3,21)(4,22)(5,23)(6,24)(7,17)(8,18)(9,42)(10,43)(11,44)(12,45)(13,46)(14,47)(15,48)(16,41)(25,40)(26,33)(27,34)(28,35)(29,36)(30,37)(31,38)(32,39)(49,58)(50,59)(51,60)(52,61)(53,62)(54,63)(55,64)(56,57), (25,29)(26,30)(27,31)(28,32)(33,37)(34,38)(35,39)(36,40)(49,53)(50,54)(51,55)(52,56)(57,61)(58,62)(59,63)(60,64), (1,43)(2,44)(3,45)(4,46)(5,47)(6,48)(7,41)(8,42)(9,18)(10,19)(11,20)(12,21)(13,22)(14,23)(15,24)(16,17)(25,54)(26,55)(27,56)(28,49)(29,50)(30,51)(31,52)(32,53)(33,64)(34,57)(35,58)(36,59)(37,60)(38,61)(39,62)(40,63), (1,5)(2,6)(3,7)(4,8)(9,13)(10,14)(11,15)(12,16)(17,21)(18,22)(19,23)(20,24)(25,29)(26,30)(27,31)(28,32)(33,37)(34,38)(35,39)(36,40)(41,45)(42,46)(43,47)(44,48)(49,53)(50,54)(51,55)(52,56)(57,61)(58,62)(59,63)(60,64), (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), (1,64,47,37)(2,59,48,40)(3,62,41,35)(4,57,42,38)(5,60,43,33)(6,63,44,36)(7,58,45,39)(8,61,46,34)(9,31,22,56)(10,26,23,51)(11,29,24,54)(12,32,17,49)(13,27,18,52)(14,30,19,55)(15,25,20,50)(16,28,21,53)>;
G:=Group( (1,19)(2,20)(3,21)(4,22)(5,23)(6,24)(7,17)(8,18)(9,42)(10,43)(11,44)(12,45)(13,46)(14,47)(15,48)(16,41)(25,40)(26,33)(27,34)(28,35)(29,36)(30,37)(31,38)(32,39)(49,58)(50,59)(51,60)(52,61)(53,62)(54,63)(55,64)(56,57), (25,29)(26,30)(27,31)(28,32)(33,37)(34,38)(35,39)(36,40)(49,53)(50,54)(51,55)(52,56)(57,61)(58,62)(59,63)(60,64), (1,43)(2,44)(3,45)(4,46)(5,47)(6,48)(7,41)(8,42)(9,18)(10,19)(11,20)(12,21)(13,22)(14,23)(15,24)(16,17)(25,54)(26,55)(27,56)(28,49)(29,50)(30,51)(31,52)(32,53)(33,64)(34,57)(35,58)(36,59)(37,60)(38,61)(39,62)(40,63), (1,5)(2,6)(3,7)(4,8)(9,13)(10,14)(11,15)(12,16)(17,21)(18,22)(19,23)(20,24)(25,29)(26,30)(27,31)(28,32)(33,37)(34,38)(35,39)(36,40)(41,45)(42,46)(43,47)(44,48)(49,53)(50,54)(51,55)(52,56)(57,61)(58,62)(59,63)(60,64), (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), (1,64,47,37)(2,59,48,40)(3,62,41,35)(4,57,42,38)(5,60,43,33)(6,63,44,36)(7,58,45,39)(8,61,46,34)(9,31,22,56)(10,26,23,51)(11,29,24,54)(12,32,17,49)(13,27,18,52)(14,30,19,55)(15,25,20,50)(16,28,21,53) );
G=PermutationGroup([(1,19),(2,20),(3,21),(4,22),(5,23),(6,24),(7,17),(8,18),(9,42),(10,43),(11,44),(12,45),(13,46),(14,47),(15,48),(16,41),(25,40),(26,33),(27,34),(28,35),(29,36),(30,37),(31,38),(32,39),(49,58),(50,59),(51,60),(52,61),(53,62),(54,63),(55,64),(56,57)], [(25,29),(26,30),(27,31),(28,32),(33,37),(34,38),(35,39),(36,40),(49,53),(50,54),(51,55),(52,56),(57,61),(58,62),(59,63),(60,64)], [(1,43),(2,44),(3,45),(4,46),(5,47),(6,48),(7,41),(8,42),(9,18),(10,19),(11,20),(12,21),(13,22),(14,23),(15,24),(16,17),(25,54),(26,55),(27,56),(28,49),(29,50),(30,51),(31,52),(32,53),(33,64),(34,57),(35,58),(36,59),(37,60),(38,61),(39,62),(40,63)], [(1,5),(2,6),(3,7),(4,8),(9,13),(10,14),(11,15),(12,16),(17,21),(18,22),(19,23),(20,24),(25,29),(26,30),(27,31),(28,32),(33,37),(34,38),(35,39),(36,40),(41,45),(42,46),(43,47),(44,48),(49,53),(50,54),(51,55),(52,56),(57,61),(58,62),(59,63),(60,64)], [(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)], [(1,64,47,37),(2,59,48,40),(3,62,41,35),(4,57,42,38),(5,60,43,33),(6,63,44,36),(7,58,45,39),(8,61,46,34),(9,31,22,56),(10,26,23,51),(11,29,24,54),(12,32,17,49),(13,27,18,52),(14,30,19,55),(15,25,20,50),(16,28,21,53)])
Matrix representation ►G ⊆ GL4(𝔽17) generated by
| 16 | 0 | 0 | 0 |
| 0 | 16 | 0 | 0 |
| 0 | 0 | 16 | 0 |
| 0 | 0 | 0 | 16 |
| 16 | 0 | 0 | 0 |
| 0 | 16 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 15 | 16 |
| 16 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 16 | 0 |
| 0 | 0 | 0 | 16 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 16 | 0 |
| 0 | 0 | 0 | 16 |
| 16 | 0 | 0 | 0 |
| 0 | 16 | 0 | 0 |
| 0 | 0 | 2 | 0 |
| 0 | 0 | 6 | 8 |
| 4 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 16 | 16 |
| 0 | 0 | 0 | 1 |
G:=sub<GL(4,GF(17))| [16,0,0,0,0,16,0,0,0,0,16,0,0,0,0,16],[16,0,0,0,0,16,0,0,0,0,1,15,0,0,0,16],[16,0,0,0,0,1,0,0,0,0,16,0,0,0,0,16],[1,0,0,0,0,1,0,0,0,0,16,0,0,0,0,16],[16,0,0,0,0,16,0,0,0,0,2,6,0,0,0,8],[4,0,0,0,0,1,0,0,0,0,16,0,0,0,16,1] >;
56 conjugacy classes
| class | 1 | 2A | ··· | 2G | 2H | 2I | 2J | 2K | 4A | ··· | 4H | 4I | 4J | 4K | 4L | 4M | ··· | 4AB | 8A | ··· | 8P |
| order | 1 | 2 | ··· | 2 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 8 | ··· | 8 |
| size | 1 | 1 | ··· | 1 | 2 | 2 | 2 | 2 | 1 | ··· | 1 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 2 | ··· | 2 |
56 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | - | + | ||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C4 | D4 | Q8 | D4 | C4○D8 |
| kernel | C2×C23.25D4 | C2×C4.Q8 | C2×C2.D8 | C23.25D4 | C2×C42⋊C2 | C23×C8 | C22×C8 | C22×C4 | C22×C4 | C24 | C22 |
| # reps | 1 | 2 | 2 | 8 | 2 | 1 | 16 | 3 | 4 | 1 | 16 |
In GAP, Magma, Sage, TeX
C_2\times C_2^3._{25}D_4 % in TeX
G:=Group("C2xC2^3.25D4"); // GroupNames label
G:=SmallGroup(128,1641);
// by ID
G=gap.SmallGroup(128,1641);
# by ID
G:=PCGroup([7,-2,2,2,2,-2,2,-2,224,253,120,352,2804,172]);
// Polycyclic
G:=Group<a,b,c,d,e,f|a^2=b^2=c^2=d^2=1,e^4=d,f^2=d*c=c*d,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,a*f=f*a,b*c=c*b,f*b*f^-1=b*d=d*b,b*e=e*b,c*e=e*c,c*f=f*c,d*e=e*d,d*f=f*d,f*e*f^-1=e^3>;
// generators/relations