p-group, metabelian, nilpotent (class 3), monomial
Aliases: C42.285D4, C42.735C23, C4.572- 1+4, C4.Q16⋊9C2, D4⋊Q8⋊9C2, C8.5Q8⋊4C2, Q8⋊Q8⋊38C2, D4⋊2Q8⋊36C2, C4.115(C4○D8), C4⋊C8.290C22, C4⋊C4.172C23, (C4×C8).116C22, (C2×C4).431C24, (C2×C8).335C23, C23.298(C2×D4), (C22×C4).513D4, C4⋊Q8.314C22, C4.Q8.88C22, C2.D8.40C22, (C2×D4).177C23, (C4×D4).115C22, C23.20D4⋊3C2, (C2×Q8).165C23, (C4×Q8).112C22, 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, C23.37C23⋊21C2, C42.78C22⋊10C2, C42⋊C2.165C22, 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
Generators and relations for C42.285D4
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 >
Subgroups: 292 in 166 conjugacy classes, 86 normal (44 characteristic)
C1, C2, C2, C4, C4, C4, C22, C22, C8, C2×C4, C2×C4, D4, Q8, C23, C23, C42, C42, C22⋊C4, C4⋊C4, C4⋊C4, C4⋊C4, C2×C8, C22×C4, C22×C4, C2×D4, C2×D4, C2×Q8, C2×Q8, C4×C8, C22⋊C8, D4⋊C4, Q8⋊C4, C4⋊C8, C4.Q8, C2.D8, C2×C42, C42⋊C2, C42⋊C2, C4×D4, C4×D4, C4×Q8, C4×Q8, C4⋊D4, C22⋊Q8, C22⋊Q8, C22.D4, C4.4D4, C42.C2, C42.C2, C42⋊2C2, C4⋊Q8, C42.12C4, D4⋊Q8, Q8⋊Q8, D4⋊2Q8, C4.Q16, C23.19D4, C23.20D4, C42.78C22, C8.5Q8, C23.36C23, C23.37C23, C42.285D4
Quotients: C1, C2, C22, D4, C23, C2×D4, C24, C4○D8, C22×D4, 2- 1+4, C23.38C23, C2×C4○D8, D8⋊C22, C42.285D4
►On 64 points
(1 57 25 43)(2 58 26 44)(3 59 27 45)(4 60 28 46)(5 61 29 47)(6 62 30 48)(7 63 31 41)(8 64 32 42)(9 24 33 50)(10 17 34 51)(11 18 35 52)(12 19 36 53)(13 20 37 54)(14 21 38 55)(15 22 39 56)(16 23 40 49)
(1 39 5 35)(2 12 6 16)(3 33 7 37)(4 14 8 10)(9 31 13 27)(11 25 15 29)(17 60 21 64)(18 43 22 47)(19 62 23 58)(20 45 24 41)(26 36 30 40)(28 38 32 34)(42 51 46 55)(44 53 48 49)(50 63 54 59)(52 57 56 61)
(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 33)(10 12)(11 39)(13 37)(14 16)(15 35)(17 49)(19 55)(20 24)(21 53)(23 51)(27 31)(34 36)(38 40)(41 63)(42 44)(43 61)(45 59)(46 48)(47 57)(50 54)(58 64)(60 62)
G:=sub<Sym(64)| (1,57,25,43)(2,58,26,44)(3,59,27,45)(4,60,28,46)(5,61,29,47)(6,62,30,48)(7,63,31,41)(8,64,32,42)(9,24,33,50)(10,17,34,51)(11,18,35,52)(12,19,36,53)(13,20,37,54)(14,21,38,55)(15,22,39,56)(16,23,40,49), (1,39,5,35)(2,12,6,16)(3,33,7,37)(4,14,8,10)(9,31,13,27)(11,25,15,29)(17,60,21,64)(18,43,22,47)(19,62,23,58)(20,45,24,41)(26,36,30,40)(28,38,32,34)(42,51,46,55)(44,53,48,49)(50,63,54,59)(52,57,56,61), (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)(9,33)(10,12)(11,39)(13,37)(14,16)(15,35)(17,49)(19,55)(20,24)(21,53)(23,51)(27,31)(34,36)(38,40)(41,63)(42,44)(43,61)(45,59)(46,48)(47,57)(50,54)(58,64)(60,62)>;
G:=Group( (1,57,25,43)(2,58,26,44)(3,59,27,45)(4,60,28,46)(5,61,29,47)(6,62,30,48)(7,63,31,41)(8,64,32,42)(9,24,33,50)(10,17,34,51)(11,18,35,52)(12,19,36,53)(13,20,37,54)(14,21,38,55)(15,22,39,56)(16,23,40,49), (1,39,5,35)(2,12,6,16)(3,33,7,37)(4,14,8,10)(9,31,13,27)(11,25,15,29)(17,60,21,64)(18,43,22,47)(19,62,23,58)(20,45,24,41)(26,36,30,40)(28,38,32,34)(42,51,46,55)(44,53,48,49)(50,63,54,59)(52,57,56,61), (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)(9,33)(10,12)(11,39)(13,37)(14,16)(15,35)(17,49)(19,55)(20,24)(21,53)(23,51)(27,31)(34,36)(38,40)(41,63)(42,44)(43,61)(45,59)(46,48)(47,57)(50,54)(58,64)(60,62) );
G=PermutationGroup([[(1,57,25,43),(2,58,26,44),(3,59,27,45),(4,60,28,46),(5,61,29,47),(6,62,30,48),(7,63,31,41),(8,64,32,42),(9,24,33,50),(10,17,34,51),(11,18,35,52),(12,19,36,53),(13,20,37,54),(14,21,38,55),(15,22,39,56),(16,23,40,49)], [(1,39,5,35),(2,12,6,16),(3,33,7,37),(4,14,8,10),(9,31,13,27),(11,25,15,29),(17,60,21,64),(18,43,22,47),(19,62,23,58),(20,45,24,41),(26,36,30,40),(28,38,32,34),(42,51,46,55),(44,53,48,49),(50,63,54,59),(52,57,56,61)], [(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),(9,33),(10,12),(11,39),(13,37),(14,16),(15,35),(17,49),(19,55),(20,24),(21,53),(23,51),(27,31),(34,36),(38,40),(41,63),(42,44),(43,61),(45,59),(46,48),(47,57),(50,54),(58,64),(60,62)]])
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 |
Matrix representation of C42.285D4 ►in 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] >;
C42.285D4 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