p-group, metabelian, nilpotent (class 2), monomial
Aliases: C42.170D4, C24.325C23, C23.447C24, C22.1802- 1+4, (C2×D4).213D4, C4⋊3(C4.4D4), C23.51(C2×D4), C4.61(C4⋊D4), C2.51(D4⋊6D4), (C22×C4).96C23, C23.7Q8⋊67C2, C23.11D4⋊42C2, (C23×C4).396C22, (C2×C42).552C22, C22.298(C22×D4), (C22×D4).530C22, (C22×Q8).131C22, C23.67C23⋊60C2, C2.C42.185C22, C2.38(C22.50C24), C2.24(C23.38C23), (C4×C4⋊C4)⋊90C2, (C2×C4⋊Q8)⋊14C2, (C2×C4×D4).61C2, (C2×C4).355(C2×D4), C2.39(C2×C4⋊D4), C2.21(C2×C4.4D4), (C2×C4).822(C4○D4), (C2×C4⋊C4).302C22, (C2×C4.4D4).25C2, C22.324(C2×C4○D4), (C2×C22⋊C4).180C22, SmallGroup(128,1279)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C42.170D4
G = < a,b,c,d | a4=b4=c4=1, d2=b2, ab=ba, cac-1=a-1b2, dad-1=a-1, bc=cb, dbd-1=b-1, dcd-1=b2c-1 >
Subgroups: 564 in 302 conjugacy classes, 116 normal (18 characteristic)
C1, C2, C2, C2, C4, C4, C22, C22, C22, C2×C4, C2×C4, D4, Q8, C23, C23, C23, C42, C42, C22⋊C4, C4⋊C4, C22×C4, C22×C4, C22×C4, C2×D4, C2×D4, C2×Q8, C24, C2.C42, C2×C42, C2×C42, C2×C22⋊C4, C2×C4⋊C4, C2×C4⋊C4, C4×D4, C4.4D4, C4⋊Q8, C23×C4, C22×D4, C22×Q8, C4×C4⋊C4, C23.7Q8, C23.67C23, C23.11D4, C2×C4×D4, C2×C4.4D4, C2×C4⋊Q8, C42.170D4
Quotients: C1, C2, C22, D4, C23, C2×D4, C4○D4, C24, C4⋊D4, C4.4D4, C22×D4, C2×C4○D4, 2- 1+4, C2×C4⋊D4, C2×C4.4D4, C23.38C23, D4⋊6D4, C22.50C24, C42.170D4
►On 64 points
(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 41 55 16)(2 42 56 13)(3 43 53 14)(4 44 54 15)(5 17 39 28)(6 18 40 25)(7 19 37 26)(8 20 38 27)(9 29 47 58)(10 30 48 59)(11 31 45 60)(12 32 46 57)(21 35 50 64)(22 36 51 61)(23 33 52 62)(24 34 49 63)
(1 17 9 34)(2 27 10 62)(3 19 11 36)(4 25 12 64)(5 58 24 16)(6 32 21 44)(7 60 22 14)(8 30 23 42)(13 38 59 52)(15 40 57 50)(18 46 35 54)(20 48 33 56)(26 45 61 53)(28 47 63 55)(29 49 41 39)(31 51 43 37)
(1 5 55 39)(2 8 56 38)(3 7 53 37)(4 6 54 40)(9 24 47 49)(10 23 48 52)(11 22 45 51)(12 21 46 50)(13 20 42 27)(14 19 43 26)(15 18 44 25)(16 17 41 28)(29 63 58 34)(30 62 59 33)(31 61 60 36)(32 64 57 35)
G:=sub<Sym(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), (1,41,55,16)(2,42,56,13)(3,43,53,14)(4,44,54,15)(5,17,39,28)(6,18,40,25)(7,19,37,26)(8,20,38,27)(9,29,47,58)(10,30,48,59)(11,31,45,60)(12,32,46,57)(21,35,50,64)(22,36,51,61)(23,33,52,62)(24,34,49,63), (1,17,9,34)(2,27,10,62)(3,19,11,36)(4,25,12,64)(5,58,24,16)(6,32,21,44)(7,60,22,14)(8,30,23,42)(13,38,59,52)(15,40,57,50)(18,46,35,54)(20,48,33,56)(26,45,61,53)(28,47,63,55)(29,49,41,39)(31,51,43,37), (1,5,55,39)(2,8,56,38)(3,7,53,37)(4,6,54,40)(9,24,47,49)(10,23,48,52)(11,22,45,51)(12,21,46,50)(13,20,42,27)(14,19,43,26)(15,18,44,25)(16,17,41,28)(29,63,58,34)(30,62,59,33)(31,61,60,36)(32,64,57,35)>;
G:=Group( (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), (1,41,55,16)(2,42,56,13)(3,43,53,14)(4,44,54,15)(5,17,39,28)(6,18,40,25)(7,19,37,26)(8,20,38,27)(9,29,47,58)(10,30,48,59)(11,31,45,60)(12,32,46,57)(21,35,50,64)(22,36,51,61)(23,33,52,62)(24,34,49,63), (1,17,9,34)(2,27,10,62)(3,19,11,36)(4,25,12,64)(5,58,24,16)(6,32,21,44)(7,60,22,14)(8,30,23,42)(13,38,59,52)(15,40,57,50)(18,46,35,54)(20,48,33,56)(26,45,61,53)(28,47,63,55)(29,49,41,39)(31,51,43,37), (1,5,55,39)(2,8,56,38)(3,7,53,37)(4,6,54,40)(9,24,47,49)(10,23,48,52)(11,22,45,51)(12,21,46,50)(13,20,42,27)(14,19,43,26)(15,18,44,25)(16,17,41,28)(29,63,58,34)(30,62,59,33)(31,61,60,36)(32,64,57,35) );
G=PermutationGroup([[(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)], [(1,41,55,16),(2,42,56,13),(3,43,53,14),(4,44,54,15),(5,17,39,28),(6,18,40,25),(7,19,37,26),(8,20,38,27),(9,29,47,58),(10,30,48,59),(11,31,45,60),(12,32,46,57),(21,35,50,64),(22,36,51,61),(23,33,52,62),(24,34,49,63)], [(1,17,9,34),(2,27,10,62),(3,19,11,36),(4,25,12,64),(5,58,24,16),(6,32,21,44),(7,60,22,14),(8,30,23,42),(13,38,59,52),(15,40,57,50),(18,46,35,54),(20,48,33,56),(26,45,61,53),(28,47,63,55),(29,49,41,39),(31,51,43,37)], [(1,5,55,39),(2,8,56,38),(3,7,53,37),(4,6,54,40),(9,24,47,49),(10,23,48,52),(11,22,45,51),(12,21,46,50),(13,20,42,27),(14,19,43,26),(15,18,44,25),(16,17,41,28),(29,63,58,34),(30,62,59,33),(31,61,60,36),(32,64,57,35)]])
38 conjugacy classes
| class | 1 | 2A | ··· | 2G | 2H | 2I | 2J | 2K | 4A | ··· | 4H | 4I | ··· | 4V | 4W | 4X | 4Y | 4Z |
| order | 1 | 2 | ··· | 2 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 4 | ··· | 4 | 4 | 4 | 4 | 4 |
| size | 1 | 1 | ··· | 1 | 4 | 4 | 4 | 4 | 2 | ··· | 2 | 4 | ··· | 4 | 8 | 8 | 8 | 8 |
38 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | - | |
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | D4 | D4 | C4○D4 | 2- 1+4 |
| kernel | C42.170D4 | C4×C4⋊C4 | C23.7Q8 | C23.67C23 | C23.11D4 | C2×C4×D4 | C2×C4.4D4 | C2×C4⋊Q8 | C42 | C2×D4 | C2×C4 | C22 |
| # reps | 1 | 1 | 4 | 2 | 4 | 1 | 2 | 1 | 4 | 4 | 12 | 2 |
Matrix representation of C42.170D4 ►in GL6(𝔽5)
| 4 | 0 | 0 | 0 | 0 | 0 |
| 0 | 4 | 0 | 0 | 0 | 0 |
| 0 | 0 | 2 | 0 | 0 | 0 |
| 0 | 0 | 0 | 3 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 3 |
| 0 | 0 | 0 | 0 | 1 | 4 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 3 |
| 0 | 0 | 0 | 0 | 1 | 4 |
| 4 | 4 | 0 | 0 | 0 | 0 |
| 2 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 4 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 2 | 1 |
| 0 | 0 | 0 | 0 | 2 | 3 |
| 1 | 1 | 0 | 0 | 0 | 0 |
| 0 | 4 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 4 | 0 | 0 |
| 0 | 0 | 4 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 2 | 0 |
| 0 | 0 | 0 | 0 | 2 | 3 |
G:=sub<GL(6,GF(5))| [4,0,0,0,0,0,0,4,0,0,0,0,0,0,2,0,0,0,0,0,0,3,0,0,0,0,0,0,1,1,0,0,0,0,3,4],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,1,0,0,0,0,3,4],[4,2,0,0,0,0,4,1,0,0,0,0,0,0,0,4,0,0,0,0,1,0,0,0,0,0,0,0,2,2,0,0,0,0,1,3],[1,0,0,0,0,0,1,4,0,0,0,0,0,0,0,4,0,0,0,0,4,0,0,0,0,0,0,0,2,2,0,0,0,0,0,3] >;
C42.170D4 in GAP, Magma, Sage, TeX
C_4^2._{170}D_4 % in TeX
G:=Group("C4^2.170D4"); // GroupNames label
G:=SmallGroup(128,1279);
// by ID
G=gap.SmallGroup(128,1279);
# by ID
G:=PCGroup([7,-2,2,2,2,-2,2,2,253,568,758,723,268,675,80]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^4=c^4=1,d^2=b^2,a*b=b*a,c*a*c^-1=a^-1*b^2,d*a*d^-1=a^-1,b*c=c*b,d*b*d^-1=b^-1,d*c*d^-1=b^2*c^-1>;
// generators/relations