p-group, metabelian, nilpotent (class 3), monomial
Aliases: C42.283D4, C42.733C23, C4.552- (1+4), Q8.Q8⋊1C2, C8⋊2Q8⋊15C2, C8⋊3Q8⋊23C2, C4.31(C4○D8), C4⋊C4.170C23, C4⋊C8.314C22, (C2×C4).429C24, (C4×C8).269C22, (C2×C8).333C23, C4.SD16⋊28C2, C23.297(C2×D4), (C22×C4).512D4, C4⋊Q8.313C22, C2.D8.38C22, C4.Q8.86C22, C4.30(C8.C22), Q8⋊C4.6C22, (C2×Q8).163C23, (C4×Q8).110C22, C22⋊C8.196C22, (C2×C42).890C22, C23.20D4.1C2, C22.689(C22×D4), C22⋊Q8.203C22, C42.12C4.38C2, (C22×C4).1094C23, C42.C2.130C22, C42⋊C2.164C22, C23.37C23.40C2, C2.77(C23.38C23), C2.46(C2×C4○D8), (C2×C4).554(C2×D4), C2.61(C2×C8.C22), SmallGroup(128,1963)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Subgroups: 268 in 163 conjugacy classes, 88 normal (34 characteristic)
C1, C2 [×3], C2, C4 [×6], C4 [×11], C22, C22 [×3], C8 [×4], C2×C4 [×6], C2×C4 [×14], Q8 [×10], C23, C42 [×4], C42 [×4], C22⋊C4 [×4], C4⋊C4 [×6], C4⋊C4 [×13], C2×C8 [×4], C22×C4 [×3], C2×Q8 [×2], C2×Q8 [×3], C4×C8 [×2], C22⋊C8 [×2], Q8⋊C4 [×8], C4⋊C8 [×2], C4.Q8 [×4], C2.D8 [×4], C2×C42, C42⋊C2 [×2], C42⋊C2, C4×Q8 [×2], C4×Q8 [×3], C22⋊Q8 [×2], C22⋊Q8 [×3], C42.C2 [×2], C42.C2, C4⋊Q8 [×4], C42.12C4, Q8.Q8 [×4], C23.20D4 [×4], C4.SD16 [×2], C8⋊3Q8, C8⋊2Q8, C23.37C23 [×2], C42.283D4
Quotients:
C1, C2 [×15], C22 [×35], D4 [×4], C23 [×15], C2×D4 [×6], C24, C4○D8 [×2], C8.C22 [×2], C22×D4, 2- (1+4) [×2], C23.38C23, C2×C4○D8, C2×C8.C22, C42.283D4
Generators and relations
G = < a,b,c,d | a4=b4=1, c4=d2=b2, ab=ba, ac=ca, dad-1=a-1, cbc-1=dbd-1=a2b, dcd-1=a2c3 >
►On 64 points
(1 47 15 38)(2 48 16 39)(3 41 9 40)(4 42 10 33)(5 43 11 34)(6 44 12 35)(7 45 13 36)(8 46 14 37)(17 53 61 27)(18 54 62 28)(19 55 63 29)(20 56 64 30)(21 49 57 31)(22 50 58 32)(23 51 59 25)(24 52 60 26)
(1 13 5 9)(2 8 6 4)(3 15 7 11)(10 16 14 12)(17 63 21 59)(18 20 22 24)(19 57 23 61)(25 53 29 49)(26 28 30 32)(27 55 31 51)(33 39 37 35)(34 41 38 45)(36 43 40 47)(42 48 46 44)(50 52 54 56)(58 60 62 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 54 5 50)(2 31 6 27)(3 52 7 56)(4 29 8 25)(9 26 13 30)(10 55 14 51)(11 32 15 28)(12 53 16 49)(17 39 21 35)(18 43 22 47)(19 37 23 33)(20 41 24 45)(34 58 38 62)(36 64 40 60)(42 63 46 59)(44 61 48 57)
G:=sub<Sym(64)| (1,47,15,38)(2,48,16,39)(3,41,9,40)(4,42,10,33)(5,43,11,34)(6,44,12,35)(7,45,13,36)(8,46,14,37)(17,53,61,27)(18,54,62,28)(19,55,63,29)(20,56,64,30)(21,49,57,31)(22,50,58,32)(23,51,59,25)(24,52,60,26), (1,13,5,9)(2,8,6,4)(3,15,7,11)(10,16,14,12)(17,63,21,59)(18,20,22,24)(19,57,23,61)(25,53,29,49)(26,28,30,32)(27,55,31,51)(33,39,37,35)(34,41,38,45)(36,43,40,47)(42,48,46,44)(50,52,54,56)(58,60,62,64), (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), (1,54,5,50)(2,31,6,27)(3,52,7,56)(4,29,8,25)(9,26,13,30)(10,55,14,51)(11,32,15,28)(12,53,16,49)(17,39,21,35)(18,43,22,47)(19,37,23,33)(20,41,24,45)(34,58,38,62)(36,64,40,60)(42,63,46,59)(44,61,48,57)>;
G:=Group( (1,47,15,38)(2,48,16,39)(3,41,9,40)(4,42,10,33)(5,43,11,34)(6,44,12,35)(7,45,13,36)(8,46,14,37)(17,53,61,27)(18,54,62,28)(19,55,63,29)(20,56,64,30)(21,49,57,31)(22,50,58,32)(23,51,59,25)(24,52,60,26), (1,13,5,9)(2,8,6,4)(3,15,7,11)(10,16,14,12)(17,63,21,59)(18,20,22,24)(19,57,23,61)(25,53,29,49)(26,28,30,32)(27,55,31,51)(33,39,37,35)(34,41,38,45)(36,43,40,47)(42,48,46,44)(50,52,54,56)(58,60,62,64), (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), (1,54,5,50)(2,31,6,27)(3,52,7,56)(4,29,8,25)(9,26,13,30)(10,55,14,51)(11,32,15,28)(12,53,16,49)(17,39,21,35)(18,43,22,47)(19,37,23,33)(20,41,24,45)(34,58,38,62)(36,64,40,60)(42,63,46,59)(44,61,48,57) );
G=PermutationGroup([(1,47,15,38),(2,48,16,39),(3,41,9,40),(4,42,10,33),(5,43,11,34),(6,44,12,35),(7,45,13,36),(8,46,14,37),(17,53,61,27),(18,54,62,28),(19,55,63,29),(20,56,64,30),(21,49,57,31),(22,50,58,32),(23,51,59,25),(24,52,60,26)], [(1,13,5,9),(2,8,6,4),(3,15,7,11),(10,16,14,12),(17,63,21,59),(18,20,22,24),(19,57,23,61),(25,53,29,49),(26,28,30,32),(27,55,31,51),(33,39,37,35),(34,41,38,45),(36,43,40,47),(42,48,46,44),(50,52,54,56),(58,60,62,64)], [(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)], [(1,54,5,50),(2,31,6,27),(3,52,7,56),(4,29,8,25),(9,26,13,30),(10,55,14,51),(11,32,15,28),(12,53,16,49),(17,39,21,35),(18,43,22,47),(19,37,23,33),(20,41,24,45),(34,58,38,62),(36,64,40,60),(42,63,46,59),(44,61,48,57)])
Matrix representation ►G ⊆ GL6(𝔽17)
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 8 | 12 | 0 | 0 |
| 0 | 0 | 13 | 9 | 0 | 0 |
| 0 | 0 | 2 | 10 | 16 | 7 |
| 0 | 0 | 11 | 7 | 7 | 1 |
| 4 | 0 | 0 | 0 | 0 | 0 |
| 0 | 4 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 13 | 0 | 16 | 0 |
| 0 | 0 | 4 | 0 | 0 | 16 |
| 9 | 0 | 0 | 0 | 0 | 0 |
| 0 | 15 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 8 |
| 0 | 0 | 13 | 0 | 16 | 1 |
| 0 | 0 | 0 | 1 | 0 | 1 |
| 0 | 0 | 4 | 0 | 0 | 16 |
| 0 | 15 | 0 | 0 | 0 | 0 |
| 9 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 16 | 0 | 8 | 0 |
| 0 | 0 | 0 | 0 | 1 | 1 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 | 16 | 0 |
G:=sub<GL(6,GF(17))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,8,13,2,11,0,0,12,9,10,7,0,0,0,0,16,7,0,0,0,0,7,1],[4,0,0,0,0,0,0,4,0,0,0,0,0,0,1,0,13,4,0,0,0,1,0,0,0,0,0,0,16,0,0,0,0,0,0,16],[9,0,0,0,0,0,0,15,0,0,0,0,0,0,1,13,0,4,0,0,0,0,1,0,0,0,0,16,0,0,0,0,8,1,1,16],[0,9,0,0,0,0,15,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,1,0,0,8,1,1,16,0,0,0,1,0,0] >;
32 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 4A | ··· | 4J | 4K | 4L | ··· | 4S | 8A | ··· | 8H |
| order | 1 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 4 | 4 | ··· | 4 | 8 | ··· | 8 |
| size | 1 | 1 | 1 | 1 | 4 | 2 | ··· | 2 | 4 | 8 | ··· | 8 | 4 | ··· | 4 |
32 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | - | - | |
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | D4 | D4 | C4○D8 | C8.C22 | 2- (1+4) |
| kernel | C42.283D4 | C42.12C4 | Q8.Q8 | C23.20D4 | C4.SD16 | C8⋊3Q8 | C8⋊2Q8 | C23.37C23 | C42 | C22×C4 | C4 | C4 | C4 |
| # reps | 1 | 1 | 4 | 4 | 2 | 1 | 1 | 2 | 2 | 2 | 8 | 2 | 2 |
In GAP, Magma, Sage, TeX
C_4^2._{283}D_4 % in TeX
G:=Group("C4^2.283D4"); // GroupNames label
G:=SmallGroup(128,1963);
// by ID
G=gap.SmallGroup(128,1963);
# by ID
G:=PCGroup([7,-2,2,2,2,-2,2,-2,448,253,120,758,219,100,675,248,4037,1027,124]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^4=1,c^4=d^2=b^2,a*b=b*a,a*c=c*a,d*a*d^-1=a^-1,c*b*c^-1=d*b*d^-1=a^2*b,d*c*d^-1=a^2*c^3>;
// generators/relations