p-group, metabelian, nilpotent (class 3), monomial
Aliases: C42.386C23, C8⋊D4⋊7C2, C8⋊5D4⋊6C2, C8⋊8D4⋊46C2, C4⋊C4.246D4, C8.9(C4○D4), (C4×SD16)⋊54C2, C22⋊C4.86D4, C8.5Q8⋊17C2, C23.83(C2×D4), C4⋊C4.113C23, (C4×C8).291C22, (C2×C4).372C24, (C2×C8).601C23, (C4×D4).93C22, C4⋊Q8.115C22, SD16⋊C4⋊19C2, (C4×Q8).90C22, C8○2M4(2)⋊16C2, C2.D8.96C22, C2.38(D4○SD16), (C2×D4).127C23, C4⋊1D4.64C22, C4⋊D4.34C22, (C2×Q8).115C23, C8⋊C4.129C22, C4.Q8.164C22, C22⋊Q8.34C22, (C22×C8).360C22, (C2×SD16).22C22, C22.632(C22×D4), C42.C2.19C22, D4⋊C4.205C22, C22.35C24⋊3C2, (C22×C4).1052C23, Q8⋊C4.128C22, C42.30C22⋊20C2, C42.29C22⋊20C2, C42⋊C2.329C22, (C2×M4(2)).282C22, C22.34C24.2C2, C2.69(C22.26C24), C4.57(C2×C4○D4), (C2×C4).144(C2×D4), SmallGroup(128,1906)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Subgroups: 348 in 181 conjugacy classes, 88 normal (34 characteristic)
C1, C2, C2 [×2], C2 [×3], C4 [×2], C4 [×11], C22, C22 [×9], C8 [×4], C8 [×2], C2×C4 [×2], C2×C4 [×4], C2×C4 [×10], D4 [×8], Q8 [×4], C23, C23 [×2], C42 [×2], C42 [×2], C22⋊C4 [×2], C22⋊C4 [×6], C4⋊C4 [×6], C4⋊C4 [×10], C2×C8 [×4], C2×C8 [×2], M4(2) [×2], SD16 [×8], C22×C4, C22×C4 [×2], C2×D4 [×2], C2×D4 [×4], C2×Q8 [×2], C4×C8 [×2], C8⋊C4 [×2], D4⋊C4 [×4], Q8⋊C4 [×4], C4.Q8 [×2], C2.D8 [×2], C42⋊C2, C4×D4 [×2], C4×Q8 [×2], C4⋊D4 [×2], C4⋊D4 [×2], C22⋊Q8 [×2], C22.D4 [×2], C42.C2 [×2], C42.C2 [×2], C42⋊2C2 [×2], C4⋊1D4, C4⋊Q8, C22×C8, C2×M4(2), C2×SD16 [×4], C8○2M4(2), C4×SD16 [×2], SD16⋊C4 [×2], C8⋊8D4 [×2], C8⋊D4 [×2], C42.29C22, C42.30C22, C8⋊5D4, C8.5Q8, C22.34C24, C22.35C24, C42.386C23
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, D4○SD16 [×2], C42.386C23
Generators and relations
G = < a,b,c,d,e | a4=b4=c2=1, d2=b2, e2=b, ab=ba, ac=ca, dad-1=ab2, ae=ea, cbc=b-1, bd=db, be=eb, dcd-1=a2c, ece-1=b-1c, de=ed >
►On 64 points
(1 39 31 14)(2 40 32 15)(3 33 25 16)(4 34 26 9)(5 35 27 10)(6 36 28 11)(7 37 29 12)(8 38 30 13)(17 57 44 53)(18 58 45 54)(19 59 46 55)(20 60 47 56)(21 61 48 49)(22 62 41 50)(23 63 42 51)(24 64 43 52)
(1 3 5 7)(2 4 6 8)(9 11 13 15)(10 12 14 16)(17 19 21 23)(18 20 22 24)(25 27 29 31)(26 28 30 32)(33 35 37 39)(34 36 38 40)(41 43 45 47)(42 44 46 48)(49 51 53 55)(50 52 54 56)(57 59 61 63)(58 60 62 64)
(1 46)(2 41)(3 44)(4 47)(5 42)(6 45)(7 48)(8 43)(9 60)(10 63)(11 58)(12 61)(13 64)(14 59)(15 62)(16 57)(17 25)(18 28)(19 31)(20 26)(21 29)(22 32)(23 27)(24 30)(33 53)(34 56)(35 51)(36 54)(37 49)(38 52)(39 55)(40 50)
(1 55 5 51)(2 56 6 52)(3 49 7 53)(4 50 8 54)(9 45 13 41)(10 46 14 42)(11 47 15 43)(12 48 16 44)(17 37 21 33)(18 38 22 34)(19 39 23 35)(20 40 24 36)(25 61 29 57)(26 62 30 58)(27 63 31 59)(28 64 32 60)
(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)| (1,39,31,14)(2,40,32,15)(3,33,25,16)(4,34,26,9)(5,35,27,10)(6,36,28,11)(7,37,29,12)(8,38,30,13)(17,57,44,53)(18,58,45,54)(19,59,46,55)(20,60,47,56)(21,61,48,49)(22,62,41,50)(23,63,42,51)(24,64,43,52), (1,3,5,7)(2,4,6,8)(9,11,13,15)(10,12,14,16)(17,19,21,23)(18,20,22,24)(25,27,29,31)(26,28,30,32)(33,35,37,39)(34,36,38,40)(41,43,45,47)(42,44,46,48)(49,51,53,55)(50,52,54,56)(57,59,61,63)(58,60,62,64), (1,46)(2,41)(3,44)(4,47)(5,42)(6,45)(7,48)(8,43)(9,60)(10,63)(11,58)(12,61)(13,64)(14,59)(15,62)(16,57)(17,25)(18,28)(19,31)(20,26)(21,29)(22,32)(23,27)(24,30)(33,53)(34,56)(35,51)(36,54)(37,49)(38,52)(39,55)(40,50), (1,55,5,51)(2,56,6,52)(3,49,7,53)(4,50,8,54)(9,45,13,41)(10,46,14,42)(11,47,15,43)(12,48,16,44)(17,37,21,33)(18,38,22,34)(19,39,23,35)(20,40,24,36)(25,61,29,57)(26,62,30,58)(27,63,31,59)(28,64,32,60), (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( (1,39,31,14)(2,40,32,15)(3,33,25,16)(4,34,26,9)(5,35,27,10)(6,36,28,11)(7,37,29,12)(8,38,30,13)(17,57,44,53)(18,58,45,54)(19,59,46,55)(20,60,47,56)(21,61,48,49)(22,62,41,50)(23,63,42,51)(24,64,43,52), (1,3,5,7)(2,4,6,8)(9,11,13,15)(10,12,14,16)(17,19,21,23)(18,20,22,24)(25,27,29,31)(26,28,30,32)(33,35,37,39)(34,36,38,40)(41,43,45,47)(42,44,46,48)(49,51,53,55)(50,52,54,56)(57,59,61,63)(58,60,62,64), (1,46)(2,41)(3,44)(4,47)(5,42)(6,45)(7,48)(8,43)(9,60)(10,63)(11,58)(12,61)(13,64)(14,59)(15,62)(16,57)(17,25)(18,28)(19,31)(20,26)(21,29)(22,32)(23,27)(24,30)(33,53)(34,56)(35,51)(36,54)(37,49)(38,52)(39,55)(40,50), (1,55,5,51)(2,56,6,52)(3,49,7,53)(4,50,8,54)(9,45,13,41)(10,46,14,42)(11,47,15,43)(12,48,16,44)(17,37,21,33)(18,38,22,34)(19,39,23,35)(20,40,24,36)(25,61,29,57)(26,62,30,58)(27,63,31,59)(28,64,32,60), (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([(1,39,31,14),(2,40,32,15),(3,33,25,16),(4,34,26,9),(5,35,27,10),(6,36,28,11),(7,37,29,12),(8,38,30,13),(17,57,44,53),(18,58,45,54),(19,59,46,55),(20,60,47,56),(21,61,48,49),(22,62,41,50),(23,63,42,51),(24,64,43,52)], [(1,3,5,7),(2,4,6,8),(9,11,13,15),(10,12,14,16),(17,19,21,23),(18,20,22,24),(25,27,29,31),(26,28,30,32),(33,35,37,39),(34,36,38,40),(41,43,45,47),(42,44,46,48),(49,51,53,55),(50,52,54,56),(57,59,61,63),(58,60,62,64)], [(1,46),(2,41),(3,44),(4,47),(5,42),(6,45),(7,48),(8,43),(9,60),(10,63),(11,58),(12,61),(13,64),(14,59),(15,62),(16,57),(17,25),(18,28),(19,31),(20,26),(21,29),(22,32),(23,27),(24,30),(33,53),(34,56),(35,51),(36,54),(37,49),(38,52),(39,55),(40,50)], [(1,55,5,51),(2,56,6,52),(3,49,7,53),(4,50,8,54),(9,45,13,41),(10,46,14,42),(11,47,15,43),(12,48,16,44),(17,37,21,33),(18,38,22,34),(19,39,23,35),(20,40,24,36),(25,61,29,57),(26,62,30,58),(27,63,31,59),(28,64,32,60)], [(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 | 16 | 0 | 0 | 0 |
| 0 | 0 | 0 | 16 | 0 | 0 |
| 16 | 0 | 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 | 1 |
| 0 | 0 | 0 | 0 | 16 | 0 |
| 1 | 15 | 0 | 0 | 0 | 0 |
| 0 | 16 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 14 | 5 | 0 |
| 0 | 0 | 14 | 0 | 0 | 12 |
| 0 | 0 | 12 | 0 | 0 | 14 |
| 0 | 0 | 0 | 5 | 14 | 0 |
| 4 | 9 | 0 | 0 | 0 | 0 |
| 4 | 13 | 0 | 0 | 0 | 0 |
| 0 | 0 | 12 | 0 | 0 | 14 |
| 0 | 0 | 0 | 12 | 3 | 0 |
| 0 | 0 | 0 | 14 | 5 | 0 |
| 0 | 0 | 3 | 0 | 0 | 5 |
| 1 | 15 | 0 | 0 | 0 | 0 |
| 1 | 16 | 0 | 0 | 0 | 0 |
| 0 | 0 | 5 | 12 | 0 | 0 |
| 0 | 0 | 5 | 5 | 0 | 0 |
| 0 | 0 | 0 | 0 | 5 | 12 |
| 0 | 0 | 0 | 0 | 5 | 5 |
G:=sub<GL(6,GF(17))| [4,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,1,0,0,0,0,0,0,1,0,0],[16,0,0,0,0,0,0,16,0,0,0,0,0,0,0,16,0,0,0,0,1,0,0,0,0,0,0,0,0,16,0,0,0,0,1,0],[1,0,0,0,0,0,15,16,0,0,0,0,0,0,0,14,12,0,0,0,14,0,0,5,0,0,5,0,0,14,0,0,0,12,14,0],[4,4,0,0,0,0,9,13,0,0,0,0,0,0,12,0,0,3,0,0,0,12,14,0,0,0,0,3,5,0,0,0,14,0,0,5],[1,1,0,0,0,0,15,16,0,0,0,0,0,0,5,5,0,0,0,0,12,5,0,0,0,0,0,0,5,5,0,0,0,0,12,5] >;
32 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 4A | ··· | 4F | 4G | 4H | 4I | 4J | ··· | 4O | 8A | 8B | 8C | 8D | 8E | ··· | 8J |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 8 | 8 | 8 | 8 | 8 | ··· | 8 |
| size | 1 | 1 | 1 | 1 | 4 | 8 | 8 | 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 | 1 | 1 | 2 | 2 | 2 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | ||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | D4 | D4 | C4○D4 | D4○SD16 |
| kernel | C42.386C23 | C8○2M4(2) | C4×SD16 | SD16⋊C4 | C8⋊8D4 | C8⋊D4 | C42.29C22 | C42.30C22 | C8⋊5D4 | C8.5Q8 | C22.34C24 | C22.35C24 | C22⋊C4 | C4⋊C4 | C8 | C2 |
| # reps | 1 | 1 | 2 | 2 | 2 | 2 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 8 | 4 |
In GAP, Magma, Sage, TeX
C_4^2._{386}C_2^3 % in TeX
G:=Group("C4^2.386C2^3"); // GroupNames label
G:=SmallGroup(128,1906);
// by ID
G=gap.SmallGroup(128,1906);
# by ID
G:=PCGroup([7,-2,2,2,2,-2,2,-2,253,456,758,184,1018,80,4037,124]);
// Polycyclic
G:=Group<a,b,c,d,e|a^4=b^4=c^2=1,d^2=b^2,e^2=b,a*b=b*a,a*c=c*a,d*a*d^-1=a*b^2,a*e=e*a,c*b*c=b^-1,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