p-group, metabelian, nilpotent (class 3), monomial
Aliases: C42.285D4, C42.735C23, C4.572- (1+4), D4⋊Q8⋊9C2, C4.Q16⋊9C2, C8.5Q8⋊4C2, Q8⋊Q8⋊38C2, D4⋊2Q8⋊36C2, C4.115(C4○D8), C4⋊C4.172C23, C4⋊C8.290C22, (C2×C8).335C23, (C4×C8).116C22, (C2×C4).431C24, (C22×C4).513D4, C23.298(C2×D4), C4⋊Q8.314C22, C2.D8.40C22, C4.Q8.88C22, (C4×D4).115C22, (C2×D4).177C23, C23.20D4⋊3C2, (C4×Q8).112C22, (C2×Q8).165C23, C42.12C4⋊39C2, C4⋊D4.200C22, C22⋊C8.197C22, (C2×C42).892C22, C23.19D4.1C2, C22.691(C22×D4), C22⋊Q8.205C22, D4⋊C4.112C22, C2.62(D8⋊C22), (C22×C4).1096C23, Q8⋊C4.106C22, C4.4D4.159C22, C42.C2.132C22, C42.78C22⋊10C2, C42⋊C2.165C22, C23.37C23⋊21C2, C23.36C23.26C2, C2.79(C23.38C23), C2.48(C2×C4○D8), (C2×C4).708(C2×D4), SmallGroup(128,1965)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Subgroups: 292 in 166 conjugacy classes, 86 normal (44 characteristic)
C1, C2 [×3], C2 [×2], C4 [×2], C4 [×2], C4 [×11], C22, C22 [×6], C8 [×4], C2×C4 [×6], C2×C4 [×14], D4 [×4], Q8 [×6], C23, C23, C42 [×4], C42 [×3], C22⋊C4 [×7], C4⋊C4 [×2], C4⋊C4 [×4], C4⋊C4 [×10], C2×C8 [×4], C22×C4 [×3], C22×C4, C2×D4, C2×D4, C2×Q8, C2×Q8 [×2], C4×C8 [×2], C22⋊C8 [×2], D4⋊C4 [×4], Q8⋊C4 [×4], C4⋊C8 [×2], C4.Q8 [×4], C2.D8 [×4], C2×C42, C42⋊C2 [×2], C42⋊C2, C4×D4, C4×D4, C4×Q8, C4×Q8 [×2], C4⋊D4, C22⋊Q8, C22⋊Q8 [×2], C22.D4, C4.4D4, C42.C2, C42.C2 [×2], C42⋊2C2, C4⋊Q8 [×2], C42.12C4, D4⋊Q8, Q8⋊Q8, D4⋊2Q8, C4.Q16, C23.19D4 [×2], C23.20D4 [×2], C42.78C22 [×2], C8.5Q8 [×2], C23.36C23, C23.37C23, C42.285D4
Quotients:
C1, C2 [×15], C22 [×35], D4 [×4], C23 [×15], C2×D4 [×6], C24, C4○D8 [×2], C22×D4, 2- (1+4) [×2], C23.38C23, C2×C4○D8, D8⋊C22, C42.285D4
Generators and relations
G = < a,b,c,d | a4=b4=d2=1, c4=b2, ab=ba, ac=ca, dad=a-1b2, cbc-1=dbd=a2b-1, dcd=a2c3 >
►On 64 points
(1 57 25 50)(2 58 26 51)(3 59 27 52)(4 60 28 53)(5 61 29 54)(6 62 30 55)(7 63 31 56)(8 64 32 49)(9 21 38 41)(10 22 39 42)(11 23 40 43)(12 24 33 44)(13 17 34 45)(14 18 35 46)(15 19 36 47)(16 20 37 48)
(1 39 5 35)(2 15 6 11)(3 33 7 37)(4 9 8 13)(10 29 14 25)(12 31 16 27)(17 60 21 64)(18 50 22 54)(19 62 23 58)(20 52 24 56)(26 36 30 40)(28 38 32 34)(41 49 45 53)(42 61 46 57)(43 51 47 55)(44 63 48 59)
(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)
(2 28)(3 7)(4 26)(6 32)(8 30)(9 11)(10 35)(12 33)(13 15)(14 39)(16 37)(17 43)(19 41)(20 24)(21 47)(23 45)(27 31)(34 36)(38 40)(44 48)(49 51)(50 61)(52 59)(53 55)(54 57)(56 63)(58 64)(60 62)
G:=sub<Sym(64)| (1,57,25,50)(2,58,26,51)(3,59,27,52)(4,60,28,53)(5,61,29,54)(6,62,30,55)(7,63,31,56)(8,64,32,49)(9,21,38,41)(10,22,39,42)(11,23,40,43)(12,24,33,44)(13,17,34,45)(14,18,35,46)(15,19,36,47)(16,20,37,48), (1,39,5,35)(2,15,6,11)(3,33,7,37)(4,9,8,13)(10,29,14,25)(12,31,16,27)(17,60,21,64)(18,50,22,54)(19,62,23,58)(20,52,24,56)(26,36,30,40)(28,38,32,34)(41,49,45,53)(42,61,46,57)(43,51,47,55)(44,63,48,59), (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), (2,28)(3,7)(4,26)(6,32)(8,30)(9,11)(10,35)(12,33)(13,15)(14,39)(16,37)(17,43)(19,41)(20,24)(21,47)(23,45)(27,31)(34,36)(38,40)(44,48)(49,51)(50,61)(52,59)(53,55)(54,57)(56,63)(58,64)(60,62)>;
G:=Group( (1,57,25,50)(2,58,26,51)(3,59,27,52)(4,60,28,53)(5,61,29,54)(6,62,30,55)(7,63,31,56)(8,64,32,49)(9,21,38,41)(10,22,39,42)(11,23,40,43)(12,24,33,44)(13,17,34,45)(14,18,35,46)(15,19,36,47)(16,20,37,48), (1,39,5,35)(2,15,6,11)(3,33,7,37)(4,9,8,13)(10,29,14,25)(12,31,16,27)(17,60,21,64)(18,50,22,54)(19,62,23,58)(20,52,24,56)(26,36,30,40)(28,38,32,34)(41,49,45,53)(42,61,46,57)(43,51,47,55)(44,63,48,59), (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), (2,28)(3,7)(4,26)(6,32)(8,30)(9,11)(10,35)(12,33)(13,15)(14,39)(16,37)(17,43)(19,41)(20,24)(21,47)(23,45)(27,31)(34,36)(38,40)(44,48)(49,51)(50,61)(52,59)(53,55)(54,57)(56,63)(58,64)(60,62) );
G=PermutationGroup([(1,57,25,50),(2,58,26,51),(3,59,27,52),(4,60,28,53),(5,61,29,54),(6,62,30,55),(7,63,31,56),(8,64,32,49),(9,21,38,41),(10,22,39,42),(11,23,40,43),(12,24,33,44),(13,17,34,45),(14,18,35,46),(15,19,36,47),(16,20,37,48)], [(1,39,5,35),(2,15,6,11),(3,33,7,37),(4,9,8,13),(10,29,14,25),(12,31,16,27),(17,60,21,64),(18,50,22,54),(19,62,23,58),(20,52,24,56),(26,36,30,40),(28,38,32,34),(41,49,45,53),(42,61,46,57),(43,51,47,55),(44,63,48,59)], [(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)], [(2,28),(3,7),(4,26),(6,32),(8,30),(9,11),(10,35),(12,33),(13,15),(14,39),(16,37),(17,43),(19,41),(20,24),(21,47),(23,45),(27,31),(34,36),(38,40),(44,48),(49,51),(50,61),(52,59),(53,55),(54,57),(56,63),(58,64),(60,62)])
Matrix representation ►G ⊆ GL6(𝔽17)
| 4 | 0 | 0 | 0 | 0 | 0 |
| 0 | 4 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 4 | 0 | 0 |
| 0 | 0 | 13 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 4 |
| 0 | 0 | 0 | 0 | 13 | 0 |
| 13 | 0 | 0 | 0 | 0 | 0 |
| 0 | 13 | 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 |
| 6 | 6 | 0 | 0 | 0 | 0 |
| 14 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 13 | 4 | 5 | 5 |
| 0 | 0 | 13 | 13 | 12 | 5 |
| 0 | 0 | 5 | 5 | 4 | 13 |
| 0 | 0 | 12 | 5 | 4 | 4 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 16 | 16 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 16 | 0 | 0 |
| 0 | 0 | 0 | 0 | 16 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
G:=sub<GL(6,GF(17))| [4,0,0,0,0,0,0,4,0,0,0,0,0,0,0,13,0,0,0,0,4,0,0,0,0,0,0,0,0,13,0,0,0,0,4,0],[13,0,0,0,0,0,0,13,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,1,0,0,0,0,0,0,1,0,0],[6,14,0,0,0,0,6,0,0,0,0,0,0,0,13,13,5,12,0,0,4,13,5,5,0,0,5,12,4,4,0,0,5,5,13,4],[1,16,0,0,0,0,0,16,0,0,0,0,0,0,1,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,1] >;
32 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 4A | ··· | 4J | 4K | 4L | ··· | 4R | 8A | ··· | 8H |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 4 | 4 | ··· | 4 | 8 | ··· | 8 |
| size | 1 | 1 | 1 | 1 | 4 | 8 | 2 | ··· | 2 | 4 | 8 | ··· | 8 | 4 | ··· | 4 |
32 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | - | ||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | D4 | D4 | C4○D8 | 2- (1+4) | D8⋊C22 |
| kernel | C42.285D4 | C42.12C4 | D4⋊Q8 | Q8⋊Q8 | D4⋊2Q8 | C4.Q16 | C23.19D4 | C23.20D4 | C42.78C22 | C8.5Q8 | C23.36C23 | C23.37C23 | C42 | C22×C4 | C4 | C4 | C2 |
| # reps | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 1 | 1 | 2 | 2 | 8 | 2 | 2 |
In GAP, Magma, Sage, TeX
C_4^2._{285}D_4 % in TeX
G:=Group("C4^2.285D4"); // GroupNames label
G:=SmallGroup(128,1965);
// by ID
G=gap.SmallGroup(128,1965);
# by ID
G:=PCGroup([7,-2,2,2,2,-2,2,-2,253,568,758,219,100,675,248,4037,1027,124]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^4=d^2=1,c^4=b^2,a*b=b*a,a*c=c*a,d*a*d=a^-1*b^2,c*b*c^-1=d*b*d=a^2*b^-1,d*c*d=a^2*c^3>;
// generators/relations