p-group, metabelian, nilpotent (class 2), monomial
Aliases: C42.171D4, C24.30C23, C23.448C24, C22.2332+ 1+4, (C2×Q8)⋊26D4, C4⋊4(C4.4D4), C4.62(C4⋊D4), C2.24(Q8⋊6D4), C23.10D4⋊42C2, (C22×C4).538C23, (C2×C42).553C22, C22.299(C22×D4), C24.3C22⋊56C2, (C22×D4).167C22, (C22×Q8).435C22, C2.23(C22.29C24), C2.C42.550C22, C2.17(C22.53C24), (C4×C4⋊C4)⋊91C2, (C2×C4×Q8)⋊24C2, (C2×C4).356(C2×D4), C2.40(C2×C4⋊D4), (C2×C4⋊1D4).17C2, (C2×C4.4D4)⋊16C2, C2.22(C2×C4.4D4), (C2×C4).895(C4○D4), (C2×C4⋊C4).870C22, C22.325(C2×C4○D4), (C2×C22⋊C4).54C22, SmallGroup(128,1280)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C42.171D4
G = < a,b,c,d | a4=b4=c4=d2=1, ab=ba, cac-1=a-1b2, dad=a-1, bc=cb, dbd=b-1, dcd=b2c-1 >
Subgroups: 724 in 338 conjugacy classes, 116 normal (18 characteristic)
C1, C2, C2, C2, C4, C4, C22, C22, C22, C2×C4, C2×C4, D4, Q8, C23, C23, C42, C42, C22⋊C4, C4⋊C4, C22×C4, C22×C4, C2×D4, C2×Q8, C2×Q8, C24, C2.C42, C2×C42, C2×C42, C2×C22⋊C4, C2×C4⋊C4, C2×C4⋊C4, C4×Q8, C4.4D4, C4⋊1D4, C22×D4, C22×Q8, C4×C4⋊C4, C24.3C22, C23.10D4, C2×C4×Q8, C2×C4.4D4, C2×C4⋊1D4, C42.171D4
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, C22.29C24, Q8⋊6D4, C22.53C24, C42.171D4
►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 11 55 16)(2 12 56 13)(3 9 53 14)(4 10 54 15)(5 19 63 24)(6 20 64 21)(7 17 61 22)(8 18 62 23)(25 48 30 42)(26 45 31 43)(27 46 32 44)(28 47 29 41)(33 59 38 52)(34 60 39 49)(35 57 40 50)(36 58 37 51)
(1 60 47 5)(2 52 48 62)(3 58 45 7)(4 50 46 64)(6 54 57 44)(8 56 59 42)(9 37 31 17)(10 35 32 21)(11 39 29 19)(12 33 30 23)(13 38 25 18)(14 36 26 22)(15 40 27 20)(16 34 28 24)(41 63 55 49)(43 61 53 51)
(1 16)(2 15)(3 14)(4 13)(5 39)(6 38)(7 37)(8 40)(9 53)(10 56)(11 55)(12 54)(17 58)(18 57)(19 60)(20 59)(21 52)(22 51)(23 50)(24 49)(25 46)(26 45)(27 48)(28 47)(29 41)(30 44)(31 43)(32 42)(33 64)(34 63)(35 62)(36 61)
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), (1,11,55,16)(2,12,56,13)(3,9,53,14)(4,10,54,15)(5,19,63,24)(6,20,64,21)(7,17,61,22)(8,18,62,23)(25,48,30,42)(26,45,31,43)(27,46,32,44)(28,47,29,41)(33,59,38,52)(34,60,39,49)(35,57,40,50)(36,58,37,51), (1,60,47,5)(2,52,48,62)(3,58,45,7)(4,50,46,64)(6,54,57,44)(8,56,59,42)(9,37,31,17)(10,35,32,21)(11,39,29,19)(12,33,30,23)(13,38,25,18)(14,36,26,22)(15,40,27,20)(16,34,28,24)(41,63,55,49)(43,61,53,51), (1,16)(2,15)(3,14)(4,13)(5,39)(6,38)(7,37)(8,40)(9,53)(10,56)(11,55)(12,54)(17,58)(18,57)(19,60)(20,59)(21,52)(22,51)(23,50)(24,49)(25,46)(26,45)(27,48)(28,47)(29,41)(30,44)(31,43)(32,42)(33,64)(34,63)(35,62)(36,61)>;
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), (1,11,55,16)(2,12,56,13)(3,9,53,14)(4,10,54,15)(5,19,63,24)(6,20,64,21)(7,17,61,22)(8,18,62,23)(25,48,30,42)(26,45,31,43)(27,46,32,44)(28,47,29,41)(33,59,38,52)(34,60,39,49)(35,57,40,50)(36,58,37,51), (1,60,47,5)(2,52,48,62)(3,58,45,7)(4,50,46,64)(6,54,57,44)(8,56,59,42)(9,37,31,17)(10,35,32,21)(11,39,29,19)(12,33,30,23)(13,38,25,18)(14,36,26,22)(15,40,27,20)(16,34,28,24)(41,63,55,49)(43,61,53,51), (1,16)(2,15)(3,14)(4,13)(5,39)(6,38)(7,37)(8,40)(9,53)(10,56)(11,55)(12,54)(17,58)(18,57)(19,60)(20,59)(21,52)(22,51)(23,50)(24,49)(25,46)(26,45)(27,48)(28,47)(29,41)(30,44)(31,43)(32,42)(33,64)(34,63)(35,62)(36,61) );
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)], [(1,11,55,16),(2,12,56,13),(3,9,53,14),(4,10,54,15),(5,19,63,24),(6,20,64,21),(7,17,61,22),(8,18,62,23),(25,48,30,42),(26,45,31,43),(27,46,32,44),(28,47,29,41),(33,59,38,52),(34,60,39,49),(35,57,40,50),(36,58,37,51)], [(1,60,47,5),(2,52,48,62),(3,58,45,7),(4,50,46,64),(6,54,57,44),(8,56,59,42),(9,37,31,17),(10,35,32,21),(11,39,29,19),(12,33,30,23),(13,38,25,18),(14,36,26,22),(15,40,27,20),(16,34,28,24),(41,63,55,49),(43,61,53,51)], [(1,16),(2,15),(3,14),(4,13),(5,39),(6,38),(7,37),(8,40),(9,53),(10,56),(11,55),(12,54),(17,58),(18,57),(19,60),(20,59),(21,52),(22,51),(23,50),(24,49),(25,46),(26,45),(27,48),(28,47),(29,41),(30,44),(31,43),(32,42),(33,64),(34,63),(35,62),(36,61)]])
38 conjugacy classes
| class | 1 | 2A | ··· | 2G | 2H | 2I | 2J | 2K | 4A | ··· | 4H | 4I | ··· | 4Z |
| order | 1 | 2 | ··· | 2 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 4 | ··· | 4 |
| size | 1 | 1 | ··· | 1 | 8 | 8 | 8 | 8 | 2 | ··· | 2 | 4 | ··· | 4 |
38 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | |
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | D4 | D4 | C4○D4 | 2+ 1+4 |
| kernel | C42.171D4 | C4×C4⋊C4 | C24.3C22 | C23.10D4 | C2×C4×Q8 | C2×C4.4D4 | C2×C4⋊1D4 | C42 | C2×Q8 | C2×C4 | C22 |
| # reps | 1 | 1 | 6 | 4 | 1 | 2 | 1 | 4 | 4 | 12 | 2 |
Matrix representation of C42.171D4 ►in GL6(𝔽5)
| 1 | 3 | 0 | 0 | 0 | 0 |
| 1 | 4 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 4 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 2 |
| 0 | 0 | 0 | 0 | 2 | 0 |
| 4 | 2 | 0 | 0 | 0 | 0 |
| 4 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 4 | 0 | 0 | 0 |
| 0 | 0 | 0 | 4 | 0 | 0 |
| 0 | 0 | 0 | 0 | 4 | 0 |
| 0 | 0 | 0 | 0 | 0 | 4 |
| 3 | 4 | 0 | 0 | 0 | 0 |
| 3 | 2 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 4 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 4 | 0 |
| 1 | 3 | 0 | 0 | 0 | 0 |
| 0 | 4 | 0 | 0 | 0 | 0 |
| 0 | 0 | 4 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 4 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
G:=sub<GL(6,GF(5))| [1,1,0,0,0,0,3,4,0,0,0,0,0,0,0,4,0,0,0,0,1,0,0,0,0,0,0,0,0,2,0,0,0,0,2,0],[4,4,0,0,0,0,2,1,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,4],[3,3,0,0,0,0,4,2,0,0,0,0,0,0,1,0,0,0,0,0,0,4,0,0,0,0,0,0,0,4,0,0,0,0,1,0],[1,0,0,0,0,0,3,4,0,0,0,0,0,0,4,0,0,0,0,0,0,1,0,0,0,0,0,0,4,0,0,0,0,0,0,1] >;
C42.171D4 in GAP, Magma, Sage, TeX
C_4^2._{171}D_4 % in TeX
G:=Group("C4^2.171D4"); // GroupNames label
G:=SmallGroup(128,1280);
// by ID
G=gap.SmallGroup(128,1280);
# by ID
G:=PCGroup([7,-2,2,2,2,-2,2,2,112,253,568,758,723,268,675,80]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^4=c^4=d^2=1,a*b=b*a,c*a*c^-1=a^-1*b^2,d*a*d=a^-1,b*c=c*b,d*b*d=b^-1,d*c*d=b^2*c^-1>;
// generators/relations