p-group, metabelian, nilpotent (class 3), monomial
Aliases: C42.389C23, C4⋊C4.249D4, (C4×Q16)⋊41C2, C4⋊Q16⋊20C2, C8.12(C4○D4), C2.25(Q8○D8), C22⋊C4.89D4, C8.5Q8⋊19C2, C8.D4.4C2, C23.86(C2×D4), Q16⋊C4⋊14C2, C4⋊C4.116C23, (C2×C4).375C24, (C2×C8).568C23, (C4×C8).225C22, C4⋊Q8.117C22, (C4×Q8).92C22, C4.Q8.27C22, C8.18D4.11C2, (C2×Q16).65C22, (C2×Q8).117C23, C8⋊C4.132C22, C2.D8.186C22, C22⋊Q8.36C22, C8○2M4(2).11C2, (C22×C8).304C22, C22.635(C22×D4), C42.C2.21C22, (C22×C4).1055C23, Q8⋊C4.208C22, C42.30C22⋊21C2, C42⋊C2.332C22, (C2×M4(2)).285C22, C22.35C24.2C2, C2.72(C22.26C24), C4.60(C2×C4○D4), (C2×C4).147(C2×D4), SmallGroup(128,1909)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Subgroups: 268 in 169 conjugacy classes, 88 normal (26 characteristic)
C1, C2, C2 [×2], C2, C4 [×2], C4 [×13], C22, C22 [×3], C8 [×4], C8 [×2], C2×C4 [×2], C2×C4 [×4], C2×C4 [×10], Q8 [×8], C23, C42 [×2], C42 [×4], C22⋊C4 [×2], C22⋊C4 [×4], C4⋊C4 [×6], C4⋊C4 [×16], C2×C8 [×4], C2×C8 [×2], M4(2) [×2], Q16 [×8], C22×C4, C2×Q8 [×4], C4×C8 [×2], C8⋊C4 [×2], Q8⋊C4 [×8], C4.Q8 [×2], C2.D8 [×2], C42⋊C2, C4×Q8 [×4], C22⋊Q8 [×4], C42.C2 [×2], C42.C2 [×4], C42⋊2C2 [×4], C4⋊Q8 [×2], C22×C8, C2×M4(2), C2×Q16 [×4], C8○2M4(2), C4×Q16 [×2], Q16⋊C4 [×2], C8.18D4 [×2], C8.D4 [×2], C42.30C22 [×2], C4⋊Q16, C8.5Q8, C22.35C24 [×2], C42.389C23
Quotients:
C1, C2 [×15], C22 [×35], D4 [×4], C23 [×15], C2×D4 [×6], C4○D4 [×4], C24, C22×D4, C2×C4○D4 [×2], C22.26C24, Q8○D8 [×2], C42.389C23
Generators and relations
G = < a,b,c,d,e | a4=b4=1, c2=d2=b2, e2=cbc-1=b-1, ab=ba, ac=ca, dad-1=ab2, ae=ea, bd=db, be=eb, dcd-1=a2c, ece-1=b-1c, de=ed >
►On 64 points
(1 33 27 10)(2 34 28 11)(3 35 29 12)(4 36 30 13)(5 37 31 14)(6 38 32 15)(7 39 25 16)(8 40 26 9)(17 62 49 46)(18 63 50 47)(19 64 51 48)(20 57 52 41)(21 58 53 42)(22 59 54 43)(23 60 55 44)(24 61 56 45)
(1 7 5 3)(2 8 6 4)(9 15 13 11)(10 16 14 12)(17 23 21 19)(18 24 22 20)(25 31 29 27)(26 32 30 28)(33 39 37 35)(34 40 38 36)(41 47 45 43)(42 48 46 44)(49 55 53 51)(50 56 54 52)(57 63 61 59)(58 64 62 60)
(1 41 5 45)(2 48 6 44)(3 47 7 43)(4 46 8 42)(9 53 13 49)(10 52 14 56)(11 51 15 55)(12 50 16 54)(17 40 21 36)(18 39 22 35)(19 38 23 34)(20 37 24 33)(25 59 29 63)(26 58 30 62)(27 57 31 61)(28 64 32 60)
(1 3 5 7)(2 4 6 8)(9 15 13 11)(10 16 14 12)(17 51 21 55)(18 52 22 56)(19 53 23 49)(20 54 24 50)(25 27 29 31)(26 28 30 32)(33 39 37 35)(34 40 38 36)(41 63 45 59)(42 64 46 60)(43 57 47 61)(44 58 48 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)
G:=sub<Sym(64)| (1,33,27,10)(2,34,28,11)(3,35,29,12)(4,36,30,13)(5,37,31,14)(6,38,32,15)(7,39,25,16)(8,40,26,9)(17,62,49,46)(18,63,50,47)(19,64,51,48)(20,57,52,41)(21,58,53,42)(22,59,54,43)(23,60,55,44)(24,61,56,45), (1,7,5,3)(2,8,6,4)(9,15,13,11)(10,16,14,12)(17,23,21,19)(18,24,22,20)(25,31,29,27)(26,32,30,28)(33,39,37,35)(34,40,38,36)(41,47,45,43)(42,48,46,44)(49,55,53,51)(50,56,54,52)(57,63,61,59)(58,64,62,60), (1,41,5,45)(2,48,6,44)(3,47,7,43)(4,46,8,42)(9,53,13,49)(10,52,14,56)(11,51,15,55)(12,50,16,54)(17,40,21,36)(18,39,22,35)(19,38,23,34)(20,37,24,33)(25,59,29,63)(26,58,30,62)(27,57,31,61)(28,64,32,60), (1,3,5,7)(2,4,6,8)(9,15,13,11)(10,16,14,12)(17,51,21,55)(18,52,22,56)(19,53,23,49)(20,54,24,50)(25,27,29,31)(26,28,30,32)(33,39,37,35)(34,40,38,36)(41,63,45,59)(42,64,46,60)(43,57,47,61)(44,58,48,62), (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)>;
G:=Group( (1,33,27,10)(2,34,28,11)(3,35,29,12)(4,36,30,13)(5,37,31,14)(6,38,32,15)(7,39,25,16)(8,40,26,9)(17,62,49,46)(18,63,50,47)(19,64,51,48)(20,57,52,41)(21,58,53,42)(22,59,54,43)(23,60,55,44)(24,61,56,45), (1,7,5,3)(2,8,6,4)(9,15,13,11)(10,16,14,12)(17,23,21,19)(18,24,22,20)(25,31,29,27)(26,32,30,28)(33,39,37,35)(34,40,38,36)(41,47,45,43)(42,48,46,44)(49,55,53,51)(50,56,54,52)(57,63,61,59)(58,64,62,60), (1,41,5,45)(2,48,6,44)(3,47,7,43)(4,46,8,42)(9,53,13,49)(10,52,14,56)(11,51,15,55)(12,50,16,54)(17,40,21,36)(18,39,22,35)(19,38,23,34)(20,37,24,33)(25,59,29,63)(26,58,30,62)(27,57,31,61)(28,64,32,60), (1,3,5,7)(2,4,6,8)(9,15,13,11)(10,16,14,12)(17,51,21,55)(18,52,22,56)(19,53,23,49)(20,54,24,50)(25,27,29,31)(26,28,30,32)(33,39,37,35)(34,40,38,36)(41,63,45,59)(42,64,46,60)(43,57,47,61)(44,58,48,62), (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) );
G=PermutationGroup([(1,33,27,10),(2,34,28,11),(3,35,29,12),(4,36,30,13),(5,37,31,14),(6,38,32,15),(7,39,25,16),(8,40,26,9),(17,62,49,46),(18,63,50,47),(19,64,51,48),(20,57,52,41),(21,58,53,42),(22,59,54,43),(23,60,55,44),(24,61,56,45)], [(1,7,5,3),(2,8,6,4),(9,15,13,11),(10,16,14,12),(17,23,21,19),(18,24,22,20),(25,31,29,27),(26,32,30,28),(33,39,37,35),(34,40,38,36),(41,47,45,43),(42,48,46,44),(49,55,53,51),(50,56,54,52),(57,63,61,59),(58,64,62,60)], [(1,41,5,45),(2,48,6,44),(3,47,7,43),(4,46,8,42),(9,53,13,49),(10,52,14,56),(11,51,15,55),(12,50,16,54),(17,40,21,36),(18,39,22,35),(19,38,23,34),(20,37,24,33),(25,59,29,63),(26,58,30,62),(27,57,31,61),(28,64,32,60)], [(1,3,5,7),(2,4,6,8),(9,15,13,11),(10,16,14,12),(17,51,21,55),(18,52,22,56),(19,53,23,49),(20,54,24,50),(25,27,29,31),(26,28,30,32),(33,39,37,35),(34,40,38,36),(41,63,45,59),(42,64,46,60),(43,57,47,61),(44,58,48,62)], [(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)])
Matrix representation ►G ⊆ GL6(𝔽17)
| 4 | 0 | 0 | 0 | 0 | 0 |
| 0 | 4 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 16 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 16 | 0 |
| 13 | 15 | 0 | 0 | 0 | 0 |
| 16 | 4 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 12 | 12 |
| 0 | 0 | 0 | 0 | 12 | 5 |
| 0 | 0 | 12 | 12 | 0 | 0 |
| 0 | 0 | 12 | 5 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 13 | 16 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 16 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 16 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 14 | 3 | 0 | 0 |
| 0 | 0 | 14 | 14 | 0 | 0 |
| 0 | 0 | 0 | 0 | 14 | 3 |
| 0 | 0 | 0 | 0 | 14 | 14 |
G:=sub<GL(6,GF(17))| [4,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,0,1,0,0],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,16,0,0,0,0,1,0,0,0,0,0,0,0,0,16,0,0,0,0,1,0],[13,16,0,0,0,0,15,4,0,0,0,0,0,0,0,0,12,12,0,0,0,0,12,5,0,0,12,12,0,0,0,0,12,5,0,0],[1,13,0,0,0,0,0,16,0,0,0,0,0,0,0,1,0,0,0,0,16,0,0,0,0,0,0,0,0,16,0,0,0,0,1,0],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,14,14,0,0,0,0,3,14,0,0,0,0,0,0,14,14,0,0,0,0,3,14] >;
32 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 4A | ··· | 4F | 4G | 4H | 4I | 4J | ··· | 4Q | 8A | 8B | 8C | 8D | 8E | ··· | 8J |
| order | 1 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 8 | 8 | 8 | 8 | 8 | ··· | 8 |
| size | 1 | 1 | 1 | 1 | 4 | 2 | ··· | 2 | 4 | 4 | 4 | 8 | ··· | 8 | 2 | 2 | 2 | 2 | 4 | ··· | 4 |
32 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | - | |
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | D4 | D4 | C4○D4 | Q8○D8 |
| kernel | C42.389C23 | C8○2M4(2) | C4×Q16 | Q16⋊C4 | C8.18D4 | C8.D4 | C42.30C22 | C4⋊Q16 | C8.5Q8 | C22.35C24 | C22⋊C4 | C4⋊C4 | C8 | C2 |
| # reps | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 1 | 1 | 2 | 2 | 2 | 8 | 4 |
In GAP, Magma, Sage, TeX
C_4^2._{389}C_2^3 % in TeX
G:=Group("C4^2.389C2^3"); // GroupNames label
G:=SmallGroup(128,1909);
// by ID
G=gap.SmallGroup(128,1909);
# by ID
G:=PCGroup([7,-2,2,2,2,-2,2,-2,448,253,456,758,520,1018,80,4037,124]);
// Polycyclic
G:=Group<a,b,c,d,e|a^4=b^4=1,c^2=d^2=b^2,e^2=c*b*c^-1=b^-1,a*b=b*a,a*c=c*a,d*a*d^-1=a*b^2,a*e=e*a,b*d=d*b,b*e=e*b,d*c*d^-1=a^2*c,e*c*e^-1=b^-1*c,d*e=e*d>;
// generators/relations