p-group, metabelian, nilpotent (class 3), monomial
Aliases: C42.367D4, C42.724C23, (C2×C4)⋊8Q16, (C4×Q16)⋊8C2, C4○3(C8⋊2Q8), C8⋊2Q8⋊37C2, C4.60(C2×Q16), C4○2(C4⋊Q16), C4⋊Q16⋊26C2, C4.27(C4○D8), C8.57(C4○D4), C4○3(C8.18D4), C22.3(C2×Q16), C4⋊C4.109C23, (C2×C8).565C23, (C4×C8).412C22, (C2×C4).368C24, C4○3(C4.SD16), C4.SD16⋊48C2, (C22×C4).620D4, C23.396(C2×D4), C4⋊Q8.291C22, (C4×Q8).86C22, C2.13(C22×Q16), C8.18D4.12C2, (C2×Q8).111C23, C2.D8.183C22, (C22×C8).540C22, (C2×Q16).129C22, C22.628(C22×D4), C22⋊Q8.177C22, (C22×C4).1573C23, (C2×C42).1137C22, Q8⋊C4.139C22, C23.37C23.35C2, C2.65(C22.26C24), (C2×C4×C8).40C2, C2.37(C2×C4○D8), C4.53(C2×C4○D4), (C2×C4).701(C2×D4), SmallGroup(128,1902)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Subgroups: 292 in 184 conjugacy classes, 100 normal (26 characteristic)
C1, C2 [×3], C2 [×2], C4 [×8], C4 [×10], C22, C22 [×2], C22 [×2], C8 [×4], C8 [×2], C2×C4 [×6], C2×C4 [×4], C2×C4 [×12], Q8 [×12], C23, C42 [×4], C42 [×4], C22⋊C4 [×4], C4⋊C4 [×4], C4⋊C4 [×14], C2×C8 [×4], C2×C8 [×4], Q16 [×8], C22×C4 [×3], C2×Q8 [×4], C2×Q8 [×2], C4×C8 [×2], C4×C8 [×2], Q8⋊C4 [×8], C2.D8 [×4], C2×C42, C42⋊C2 [×2], C4×Q8 [×4], C4×Q8 [×2], C22⋊Q8 [×4], C22⋊Q8 [×2], C42.C2 [×2], C4⋊Q8 [×4], C22×C8 [×2], C2×Q16 [×4], C2×C4×C8, C4×Q16 [×4], C8.18D4 [×4], C4.SD16 [×2], C4⋊Q16, C8⋊2Q8, C23.37C23 [×2], C42.367D4
Quotients:
C1, C2 [×15], C22 [×35], D4 [×4], C23 [×15], Q16 [×4], C2×D4 [×6], C4○D4 [×4], C24, C2×Q16 [×6], C4○D8 [×2], C22×D4, C2×C4○D4 [×2], C22.26C24, C22×Q16, C2×C4○D8, C42.367D4
Generators and relations
G = < a,b,c,d | a4=b4=1, c4=d2=a2, ab=ba, ac=ca, dad-1=ab2, bc=cb, dbd-1=a2b, dcd-1=a2c3 >
►On 64 points
(1 7 5 3)(2 8 6 4)(9 15 13 11)(10 16 14 12)(17 55 21 51)(18 56 22 52)(19 49 23 53)(20 50 24 54)(25 31 29 27)(26 32 30 28)(33 39 37 35)(34 40 38 36)(41 59 45 63)(42 60 46 64)(43 61 47 57)(44 62 48 58)
(1 39 31 10)(2 40 32 11)(3 33 25 12)(4 34 26 13)(5 35 27 14)(6 36 28 15)(7 37 29 16)(8 38 30 9)(17 60 53 44)(18 61 54 45)(19 62 55 46)(20 63 56 47)(21 64 49 48)(22 57 50 41)(23 58 51 42)(24 59 52 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)
(1 44 5 48)(2 43 6 47)(3 42 7 46)(4 41 8 45)(9 50 13 54)(10 49 14 53)(11 56 15 52)(12 55 16 51)(17 39 21 35)(18 38 22 34)(19 37 23 33)(20 36 24 40)(25 58 29 62)(26 57 30 61)(27 64 31 60)(28 63 32 59)
G:=sub<Sym(64)| (1,7,5,3)(2,8,6,4)(9,15,13,11)(10,16,14,12)(17,55,21,51)(18,56,22,52)(19,49,23,53)(20,50,24,54)(25,31,29,27)(26,32,30,28)(33,39,37,35)(34,40,38,36)(41,59,45,63)(42,60,46,64)(43,61,47,57)(44,62,48,58), (1,39,31,10)(2,40,32,11)(3,33,25,12)(4,34,26,13)(5,35,27,14)(6,36,28,15)(7,37,29,16)(8,38,30,9)(17,60,53,44)(18,61,54,45)(19,62,55,46)(20,63,56,47)(21,64,49,48)(22,57,50,41)(23,58,51,42)(24,59,52,43), (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), (1,44,5,48)(2,43,6,47)(3,42,7,46)(4,41,8,45)(9,50,13,54)(10,49,14,53)(11,56,15,52)(12,55,16,51)(17,39,21,35)(18,38,22,34)(19,37,23,33)(20,36,24,40)(25,58,29,62)(26,57,30,61)(27,64,31,60)(28,63,32,59)>;
G:=Group( (1,7,5,3)(2,8,6,4)(9,15,13,11)(10,16,14,12)(17,55,21,51)(18,56,22,52)(19,49,23,53)(20,50,24,54)(25,31,29,27)(26,32,30,28)(33,39,37,35)(34,40,38,36)(41,59,45,63)(42,60,46,64)(43,61,47,57)(44,62,48,58), (1,39,31,10)(2,40,32,11)(3,33,25,12)(4,34,26,13)(5,35,27,14)(6,36,28,15)(7,37,29,16)(8,38,30,9)(17,60,53,44)(18,61,54,45)(19,62,55,46)(20,63,56,47)(21,64,49,48)(22,57,50,41)(23,58,51,42)(24,59,52,43), (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), (1,44,5,48)(2,43,6,47)(3,42,7,46)(4,41,8,45)(9,50,13,54)(10,49,14,53)(11,56,15,52)(12,55,16,51)(17,39,21,35)(18,38,22,34)(19,37,23,33)(20,36,24,40)(25,58,29,62)(26,57,30,61)(27,64,31,60)(28,63,32,59) );
G=PermutationGroup([(1,7,5,3),(2,8,6,4),(9,15,13,11),(10,16,14,12),(17,55,21,51),(18,56,22,52),(19,49,23,53),(20,50,24,54),(25,31,29,27),(26,32,30,28),(33,39,37,35),(34,40,38,36),(41,59,45,63),(42,60,46,64),(43,61,47,57),(44,62,48,58)], [(1,39,31,10),(2,40,32,11),(3,33,25,12),(4,34,26,13),(5,35,27,14),(6,36,28,15),(7,37,29,16),(8,38,30,9),(17,60,53,44),(18,61,54,45),(19,62,55,46),(20,63,56,47),(21,64,49,48),(22,57,50,41),(23,58,51,42),(24,59,52,43)], [(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)], [(1,44,5,48),(2,43,6,47),(3,42,7,46),(4,41,8,45),(9,50,13,54),(10,49,14,53),(11,56,15,52),(12,55,16,51),(17,39,21,35),(18,38,22,34),(19,37,23,33),(20,36,24,40),(25,58,29,62),(26,57,30,61),(27,64,31,60),(28,63,32,59)])
Matrix representation ►G ⊆ GL4(𝔽17) generated by
| 1 | 0 | 0 | 0 |
| 0 | 16 | 0 | 0 |
| 0 | 0 | 1 | 15 |
| 0 | 0 | 1 | 16 |
| 4 | 0 | 0 | 0 |
| 0 | 4 | 0 | 0 |
| 0 | 0 | 1 | 15 |
| 0 | 0 | 1 | 16 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 11 |
| 0 | 0 | 3 | 11 |
| 0 | 1 | 0 | 0 |
| 1 | 0 | 0 | 0 |
| 0 | 0 | 11 | 14 |
| 0 | 0 | 1 | 6 |
G:=sub<GL(4,GF(17))| [1,0,0,0,0,16,0,0,0,0,1,1,0,0,15,16],[4,0,0,0,0,4,0,0,0,0,1,1,0,0,15,16],[1,0,0,0,0,1,0,0,0,0,0,3,0,0,11,11],[0,1,0,0,1,0,0,0,0,0,11,1,0,0,14,6] >;
44 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 4A | 4B | 4C | 4D | 4E | ··· | 4N | 4O | ··· | 4V | 8A | ··· | 8P |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 4 | ··· | 4 | 8 | ··· | 8 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 1 | 1 | 1 | 1 | 2 | ··· | 2 | 8 | ··· | 8 | 2 | ··· | 2 |
44 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | + | + | + | - | ||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | D4 | D4 | C4○D4 | Q16 | C4○D8 |
| kernel | C42.367D4 | C2×C4×C8 | C4×Q16 | C8.18D4 | C4.SD16 | C4⋊Q16 | C8⋊2Q8 | C23.37C23 | C42 | C22×C4 | C8 | C2×C4 | C4 |
| # reps | 1 | 1 | 4 | 4 | 2 | 1 | 1 | 2 | 2 | 2 | 8 | 8 | 8 |
In GAP, Magma, Sage, TeX
C_4^2._{367}D_4 % in TeX
G:=Group("C4^2.367D4"); // GroupNames label
G:=SmallGroup(128,1902);
// by ID
G=gap.SmallGroup(128,1902);
# by ID
G:=PCGroup([7,-2,2,2,2,-2,2,-2,448,253,758,520,80,4037,124]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^4=1,c^4=d^2=a^2,a*b=b*a,a*c=c*a,d*a*d^-1=a*b^2,b*c=c*b,d*b*d^-1=a^2*b,d*c*d^-1=a^2*c^3>;
// generators/relations