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×C4.4D4)⋊16C2, (C2×C4⋊1D4).17C2, 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
Subgroups: 724 in 338 conjugacy classes, 116 normal (18 characteristic)
C1, C2 [×3], C2 [×4], C2 [×4], C4 [×8], C4 [×12], C22 [×3], C22 [×4], C22 [×28], C2×C4 [×18], C2×C4 [×24], D4 [×24], Q8 [×8], C23, C23 [×28], C42 [×4], C42 [×8], C22⋊C4 [×24], C4⋊C4 [×10], C22×C4 [×3], C22×C4 [×8], C2×D4 [×32], C2×Q8 [×4], C2×Q8 [×4], C24 [×4], C2.C42 [×2], C2×C42, C2×C42 [×4], C2×C22⋊C4 [×16], C2×C4⋊C4, C2×C4⋊C4 [×4], C4×Q8 [×4], C4.4D4 [×8], C4⋊1D4 [×4], C22×D4 [×6], C22×Q8, C4×C4⋊C4, C24.3C22 [×6], C23.10D4 [×4], C2×C4×Q8, C2×C4.4D4 [×2], C2×C4⋊1D4, C42.171D4
Quotients:
C1, C2 [×15], C22 [×35], D4 [×8], C23 [×15], C2×D4 [×12], C4○D4 [×6], C24, C4⋊D4 [×4], C4.4D4 [×4], C22×D4 [×2], C2×C4○D4 [×3], 2+ (1+4) [×2], C2×C4⋊D4, C2×C4.4D4, C22.29C24, Q8⋊6D4 [×2], C22.53C24 [×2], C42.171D4
Generators and relations
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 >
(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 16)(2 15)(3 14)(4 13)(5 63)(6 62)(7 61)(8 64)(9 58)(10 57)(11 60)(12 59)(17 49)(18 52)(19 51)(20 50)(21 27)(22 26)(23 25)(24 28)(29 47)(30 46)(31 45)(32 48)(33 40)(34 39)(35 38)(36 37)(41 55)(42 54)(43 53)(44 56)
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), (1,16)(2,15)(3,14)(4,13)(5,63)(6,62)(7,61)(8,64)(9,58)(10,57)(11,60)(12,59)(17,49)(18,52)(19,51)(20,50)(21,27)(22,26)(23,25)(24,28)(29,47)(30,46)(31,45)(32,48)(33,40)(34,39)(35,38)(36,37)(41,55)(42,54)(43,53)(44,56)>;
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), (1,16)(2,15)(3,14)(4,13)(5,63)(6,62)(7,61)(8,64)(9,58)(10,57)(11,60)(12,59)(17,49)(18,52)(19,51)(20,50)(21,27)(22,26)(23,25)(24,28)(29,47)(30,46)(31,45)(32,48)(33,40)(34,39)(35,38)(36,37)(41,55)(42,54)(43,53)(44,56) );
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)], [(1,16),(2,15),(3,14),(4,13),(5,63),(6,62),(7,61),(8,64),(9,58),(10,57),(11,60),(12,59),(17,49),(18,52),(19,51),(20,50),(21,27),(22,26),(23,25),(24,28),(29,47),(30,46),(31,45),(32,48),(33,40),(34,39),(35,38),(36,37),(41,55),(42,54),(43,53),(44,56)])
Matrix representation ►G ⊆ 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] >;
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 |
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