p-group, metabelian, nilpotent (class 3), monomial
Aliases: C42.501C23, C4.222- (1+4), (C8×Q8)⋊10C2, (C4×D8).8C2, Q8.Q8⋊9C2, C4⋊C4.274D4, D4⋊2Q8⋊43C2, D4⋊3Q8⋊10C2, (C4×SD16)⋊45C2, C4.48(C4○D8), (C2×Q8).182D4, D4.34(C4○D4), D4.D4⋊45C2, C4⋊C4.428C23, C4⋊C8.323C22, (C4×C8).279C22, (C2×C4).552C24, (C2×C8).368C23, C4.SD16⋊16C2, D4.2D4.1C2, C4⋊Q8.181C22, C2.60(Q8⋊5D4), C2.97(D4○SD16), (C2×D8).146C22, (C4×D4).192C22, (C2×D4).429C23, (C2×Q8).251C23, (C4×Q8).306C22, C4.Q8.173C22, C2.D8.199C22, C4.4D4.74C22, C22.812(C22×D4), C42.C2.59C22, D4⋊C4.151C22, Q8⋊C4.121C22, (C2×SD16).168C22, C42.78C22⋊19C2, C22.50C24⋊10C2, C2.72(C2×C4○D8), C4.253(C2×C4○D4), (C2×C4).174(C2×D4), SmallGroup(128,2092)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Subgroups: 320 in 175 conjugacy classes, 88 normal (38 characteristic)
C1, C2 [×3], C2 [×3], C4 [×2], C4 [×2], C4 [×11], C22, C22 [×7], C8 [×4], C2×C4 [×3], C2×C4 [×4], C2×C4 [×12], D4 [×2], D4 [×3], Q8 [×7], C23 [×2], C42, C42 [×2], C42 [×2], C22⋊C4 [×8], C4⋊C4 [×3], C4⋊C4 [×4], C4⋊C4 [×9], C2×C8 [×2], C2×C8 [×2], D8 [×2], SD16 [×4], C22×C4 [×4], C2×D4 [×2], C2×Q8, C2×Q8 [×2], C2×Q8, C4×C8, C4×C8 [×2], D4⋊C4 [×2], D4⋊C4 [×2], Q8⋊C4 [×6], C4⋊C8, C4⋊C8 [×2], C4.Q8 [×2], C2.D8, C2×C4⋊C4, C42⋊C2, C4×D4 [×2], C4×D4 [×2], C4×Q8, C4×Q8 [×2], C22⋊Q8 [×4], C4.4D4 [×2], C42.C2 [×2], C42⋊2C2 [×2], C4⋊Q8 [×2], C2×D8, C2×SD16 [×2], C4×D8, C4×SD16 [×2], C8×Q8, D4.D4, D4.2D4 [×2], D4⋊2Q8, Q8.Q8 [×2], C4.SD16, C42.78C22 [×2], D4⋊3Q8, C22.50C24, C42.501C23
Quotients:
C1, C2 [×15], C22 [×35], D4 [×4], C23 [×15], C2×D4 [×6], C4○D4 [×2], C24, C4○D8 [×2], C22×D4, C2×C4○D4, 2- (1+4), Q8⋊5D4, C2×C4○D8, D4○SD16, C42.501C23
Generators and relations
G = < a,b,c,d,e | a4=b4=c2=e2=1, d2=a2b2, ab=ba, ac=ca, dad-1=a-1, ae=ea, cbc=ebe=b-1, bd=db, dcd-1=a2c, ece=bc, de=ed >
►On 64 points
(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 23 25 17)(2 24 26 18)(3 21 27 19)(4 22 28 20)(5 15 9 64)(6 16 10 61)(7 13 11 62)(8 14 12 63)(29 37 41 35)(30 38 42 36)(31 39 43 33)(32 40 44 34)(45 53 51 57)(46 54 52 58)(47 55 49 59)(48 56 50 60)
(5 13)(6 14)(7 15)(8 16)(9 62)(10 63)(11 64)(12 61)(17 23)(18 24)(19 21)(20 22)(29 37)(30 38)(31 39)(32 40)(33 43)(34 44)(35 41)(36 42)(45 49)(46 50)(47 51)(48 52)(53 55)(54 56)(57 59)(58 60)
(1 57 27 55)(2 60 28 54)(3 59 25 53)(4 58 26 56)(5 44 11 30)(6 43 12 29)(7 42 9 32)(8 41 10 31)(13 36 64 40)(14 35 61 39)(15 34 62 38)(16 33 63 37)(17 51 21 47)(18 50 22 46)(19 49 23 45)(20 52 24 48)
(1 29)(2 30)(3 31)(4 32)(5 60)(6 57)(7 58)(8 59)(9 56)(10 53)(11 54)(12 55)(13 52)(14 49)(15 50)(16 51)(17 37)(18 38)(19 39)(20 40)(21 33)(22 34)(23 35)(24 36)(25 41)(26 42)(27 43)(28 44)(45 61)(46 62)(47 63)(48 64)
G:=sub<Sym(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), (1,23,25,17)(2,24,26,18)(3,21,27,19)(4,22,28,20)(5,15,9,64)(6,16,10,61)(7,13,11,62)(8,14,12,63)(29,37,41,35)(30,38,42,36)(31,39,43,33)(32,40,44,34)(45,53,51,57)(46,54,52,58)(47,55,49,59)(48,56,50,60), (5,13)(6,14)(7,15)(8,16)(9,62)(10,63)(11,64)(12,61)(17,23)(18,24)(19,21)(20,22)(29,37)(30,38)(31,39)(32,40)(33,43)(34,44)(35,41)(36,42)(45,49)(46,50)(47,51)(48,52)(53,55)(54,56)(57,59)(58,60), (1,57,27,55)(2,60,28,54)(3,59,25,53)(4,58,26,56)(5,44,11,30)(6,43,12,29)(7,42,9,32)(8,41,10,31)(13,36,64,40)(14,35,61,39)(15,34,62,38)(16,33,63,37)(17,51,21,47)(18,50,22,46)(19,49,23,45)(20,52,24,48), (1,29)(2,30)(3,31)(4,32)(5,60)(6,57)(7,58)(8,59)(9,56)(10,53)(11,54)(12,55)(13,52)(14,49)(15,50)(16,51)(17,37)(18,38)(19,39)(20,40)(21,33)(22,34)(23,35)(24,36)(25,41)(26,42)(27,43)(28,44)(45,61)(46,62)(47,63)(48,64)>;
G:=Group( (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), (1,23,25,17)(2,24,26,18)(3,21,27,19)(4,22,28,20)(5,15,9,64)(6,16,10,61)(7,13,11,62)(8,14,12,63)(29,37,41,35)(30,38,42,36)(31,39,43,33)(32,40,44,34)(45,53,51,57)(46,54,52,58)(47,55,49,59)(48,56,50,60), (5,13)(6,14)(7,15)(8,16)(9,62)(10,63)(11,64)(12,61)(17,23)(18,24)(19,21)(20,22)(29,37)(30,38)(31,39)(32,40)(33,43)(34,44)(35,41)(36,42)(45,49)(46,50)(47,51)(48,52)(53,55)(54,56)(57,59)(58,60), (1,57,27,55)(2,60,28,54)(3,59,25,53)(4,58,26,56)(5,44,11,30)(6,43,12,29)(7,42,9,32)(8,41,10,31)(13,36,64,40)(14,35,61,39)(15,34,62,38)(16,33,63,37)(17,51,21,47)(18,50,22,46)(19,49,23,45)(20,52,24,48), (1,29)(2,30)(3,31)(4,32)(5,60)(6,57)(7,58)(8,59)(9,56)(10,53)(11,54)(12,55)(13,52)(14,49)(15,50)(16,51)(17,37)(18,38)(19,39)(20,40)(21,33)(22,34)(23,35)(24,36)(25,41)(26,42)(27,43)(28,44)(45,61)(46,62)(47,63)(48,64) );
G=PermutationGroup([(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)], [(1,23,25,17),(2,24,26,18),(3,21,27,19),(4,22,28,20),(5,15,9,64),(6,16,10,61),(7,13,11,62),(8,14,12,63),(29,37,41,35),(30,38,42,36),(31,39,43,33),(32,40,44,34),(45,53,51,57),(46,54,52,58),(47,55,49,59),(48,56,50,60)], [(5,13),(6,14),(7,15),(8,16),(9,62),(10,63),(11,64),(12,61),(17,23),(18,24),(19,21),(20,22),(29,37),(30,38),(31,39),(32,40),(33,43),(34,44),(35,41),(36,42),(45,49),(46,50),(47,51),(48,52),(53,55),(54,56),(57,59),(58,60)], [(1,57,27,55),(2,60,28,54),(3,59,25,53),(4,58,26,56),(5,44,11,30),(6,43,12,29),(7,42,9,32),(8,41,10,31),(13,36,64,40),(14,35,61,39),(15,34,62,38),(16,33,63,37),(17,51,21,47),(18,50,22,46),(19,49,23,45),(20,52,24,48)], [(1,29),(2,30),(3,31),(4,32),(5,60),(6,57),(7,58),(8,59),(9,56),(10,53),(11,54),(12,55),(13,52),(14,49),(15,50),(16,51),(17,37),(18,38),(19,39),(20,40),(21,33),(22,34),(23,35),(24,36),(25,41),(26,42),(27,43),(28,44),(45,61),(46,62),(47,63),(48,64)])
Matrix representation ►G ⊆ GL4(𝔽17) generated by
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 4 | 0 |
| 0 | 0 | 16 | 13 |
| 0 | 1 | 0 | 0 |
| 16 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 |
| 0 | 16 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 4 | 16 |
| 4 | 0 | 0 | 0 |
| 0 | 4 | 0 | 0 |
| 0 | 0 | 8 | 13 |
| 0 | 0 | 12 | 9 |
| 3 | 3 | 0 | 0 |
| 3 | 14 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
G:=sub<GL(4,GF(17))| [1,0,0,0,0,1,0,0,0,0,4,16,0,0,0,13],[0,16,0,0,1,0,0,0,0,0,1,0,0,0,0,1],[1,0,0,0,0,16,0,0,0,0,1,4,0,0,0,16],[4,0,0,0,0,4,0,0,0,0,8,12,0,0,13,9],[3,3,0,0,3,14,0,0,0,0,1,0,0,0,0,1] >;
35 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 4A | ··· | 4H | 4I | ··· | 4M | 4N | ··· | 4R | 8A | 8B | 8C | 8D | 8E | ··· | 8J |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 4 | ··· | 4 | 4 | ··· | 4 | 8 | 8 | 8 | 8 | 8 | ··· | 8 |
| size | 1 | 1 | 1 | 1 | 4 | 4 | 8 | 2 | ··· | 2 | 4 | ··· | 4 | 8 | ··· | 8 | 2 | 2 | 2 | 2 | 4 | ··· | 4 |
35 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | - | |||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | D4 | D4 | C4○D4 | C4○D8 | 2- (1+4) | D4○SD16 |
| kernel | C42.501C23 | C4×D8 | C4×SD16 | C8×Q8 | D4.D4 | D4.2D4 | D4⋊2Q8 | Q8.Q8 | C4.SD16 | C42.78C22 | D4⋊3Q8 | C22.50C24 | C4⋊C4 | C2×Q8 | D4 | C4 | C4 | C2 |
| # reps | 1 | 1 | 2 | 1 | 1 | 2 | 1 | 2 | 1 | 2 | 1 | 1 | 3 | 1 | 4 | 8 | 1 | 2 |
In GAP, Magma, Sage, TeX
C_4^2._{501}C_2^3 % in TeX
G:=Group("C4^2.501C2^3"); // GroupNames label
G:=SmallGroup(128,2092);
// by ID
G=gap.SmallGroup(128,2092);
# by ID
G:=PCGroup([7,-2,2,2,2,-2,2,-2,253,568,758,346,80,4037,1027,124]);
// Polycyclic
G:=Group<a,b,c,d,e|a^4=b^4=c^2=e^2=1,d^2=a^2*b^2,a*b=b*a,a*c=c*a,d*a*d^-1=a^-1,a*e=e*a,c*b*c=e*b*e=b^-1,b*d=d*b,d*c*d^-1=a^2*c,e*c*e=b*c,d*e=e*d>;
// generators/relations