p-group, metabelian, nilpotent (class 3), monomial
Aliases: C24.65D4, C4.54C22≀C2, D4⋊4(C22⋊C4), (C2×D4).264D4, Q8⋊4(C22⋊C4), (C2×Q8).207D4, C2.1(D4⋊D4), (C22×C4).675D4, C23.739(C2×D4), (C22×C8).7C22, C23.7Q8⋊4C2, C22.74C22≀C2, C2.1(D4.7D4), C22.40(C4○D8), C2.13(C24⋊3C4), C23.72(C22⋊C4), C22.58(C8⋊C22), (C23×C4).233C22, (C22×C4).1322C23, (C22×D4).453C22, C22.47(C8.C22), (C22×Q8).381C22, C2.20(C23.36D4), C2.20(C23.24D4), (C2×C4○D4)⋊11C4, C4.3(C2×C22⋊C4), (C2×D4⋊C4)⋊2C2, (C2×C22⋊C8)⋊12C2, (C2×Q8⋊C4)⋊2C2, (C2×C4).973(C2×D4), (C2×D4).201(C2×C4), (C2×C4⋊C4).30C22, (C2×Q8).184(C2×C4), (C22×C4○D4).4C2, (C2×C4).360(C22×C4), (C22×C4).262(C2×C4), (C2×C4).330(C22⋊C4), C22.241(C2×C22⋊C4), SmallGroup(128,520)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C24.65D4
G = < a,b,c,d,e,f | a2=b2=c2=d2=1, e4=d, f2=c, eae-1=ab=ba, ac=ca, ad=da, faf-1=abd, bc=cb, bd=db, be=eb, bf=fb, cd=dc, ce=ec, cf=fc, de=ed, df=fd, fef-1=cde3 >
Subgroups: 620 in 306 conjugacy classes, 76 normal (22 characteristic)
C1, C2, C2, C2, C4, C4, C4, C22, C22, C22, C8, C2×C4, C2×C4, C2×C4, D4, D4, Q8, Q8, C23, C23, C23, C22⋊C4, C4⋊C4, C2×C8, C22×C4, C22×C4, C22×C4, C2×D4, C2×D4, C2×Q8, C2×Q8, C4○D4, C24, C24, C2.C42, C22⋊C8, D4⋊C4, Q8⋊C4, C2×C22⋊C4, C2×C4⋊C4, C22×C8, C23×C4, C23×C4, C22×D4, C22×D4, C22×Q8, C2×C4○D4, C2×C4○D4, C23.7Q8, C2×C22⋊C8, C2×D4⋊C4, C2×Q8⋊C4, C22×C4○D4, C24.65D4
Quotients: C1, C2, C4, C22, C2×C4, D4, C23, C22⋊C4, C22×C4, C2×D4, C2×C22⋊C4, C22≀C2, C4○D8, C8⋊C22, C8.C22, C24⋊3C4, C23.24D4, C23.36D4, D4⋊D4, D4.7D4, C24.65D4
►On 64 points
(2 63)(4 57)(6 59)(8 61)(9 21)(11 23)(13 17)(15 19)(25 48)(26 30)(27 42)(28 32)(29 44)(31 46)(33 56)(34 38)(35 50)(36 40)(37 52)(39 54)(41 45)(43 47)(49 53)(51 55)
(1 62)(2 63)(3 64)(4 57)(5 58)(6 59)(7 60)(8 61)(9 21)(10 22)(11 23)(12 24)(13 17)(14 18)(15 19)(16 20)(25 44)(26 45)(27 46)(28 47)(29 48)(30 41)(31 42)(32 43)(33 52)(34 53)(35 54)(36 55)(37 56)(38 49)(39 50)(40 51)
(1 20)(2 21)(3 22)(4 23)(5 24)(6 17)(7 18)(8 19)(9 63)(10 64)(11 57)(12 58)(13 59)(14 60)(15 61)(16 62)(25 37)(26 38)(27 39)(28 40)(29 33)(30 34)(31 35)(32 36)(41 53)(42 54)(43 55)(44 56)(45 49)(46 50)(47 51)(48 52)
(1 5)(2 6)(3 7)(4 8)(9 13)(10 14)(11 15)(12 16)(17 21)(18 22)(19 23)(20 24)(25 29)(26 30)(27 31)(28 32)(33 37)(34 38)(35 39)(36 40)(41 45)(42 46)(43 47)(44 48)(49 53)(50 54)(51 55)(52 56)(57 61)(58 62)(59 63)(60 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 25 20 37)(2 36 21 32)(3 31 22 35)(4 34 23 30)(5 29 24 33)(6 40 17 28)(7 27 18 39)(8 38 19 26)(9 43 63 55)(10 54 64 42)(11 41 57 53)(12 52 58 48)(13 47 59 51)(14 50 60 46)(15 45 61 49)(16 56 62 44)
G:=sub<Sym(64)| (2,63)(4,57)(6,59)(8,61)(9,21)(11,23)(13,17)(15,19)(25,48)(26,30)(27,42)(28,32)(29,44)(31,46)(33,56)(34,38)(35,50)(36,40)(37,52)(39,54)(41,45)(43,47)(49,53)(51,55), (1,62)(2,63)(3,64)(4,57)(5,58)(6,59)(7,60)(8,61)(9,21)(10,22)(11,23)(12,24)(13,17)(14,18)(15,19)(16,20)(25,44)(26,45)(27,46)(28,47)(29,48)(30,41)(31,42)(32,43)(33,52)(34,53)(35,54)(36,55)(37,56)(38,49)(39,50)(40,51), (1,20)(2,21)(3,22)(4,23)(5,24)(6,17)(7,18)(8,19)(9,63)(10,64)(11,57)(12,58)(13,59)(14,60)(15,61)(16,62)(25,37)(26,38)(27,39)(28,40)(29,33)(30,34)(31,35)(32,36)(41,53)(42,54)(43,55)(44,56)(45,49)(46,50)(47,51)(48,52), (1,5)(2,6)(3,7)(4,8)(9,13)(10,14)(11,15)(12,16)(17,21)(18,22)(19,23)(20,24)(25,29)(26,30)(27,31)(28,32)(33,37)(34,38)(35,39)(36,40)(41,45)(42,46)(43,47)(44,48)(49,53)(50,54)(51,55)(52,56)(57,61)(58,62)(59,63)(60,64), (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), (1,25,20,37)(2,36,21,32)(3,31,22,35)(4,34,23,30)(5,29,24,33)(6,40,17,28)(7,27,18,39)(8,38,19,26)(9,43,63,55)(10,54,64,42)(11,41,57,53)(12,52,58,48)(13,47,59,51)(14,50,60,46)(15,45,61,49)(16,56,62,44)>;
G:=Group( (2,63)(4,57)(6,59)(8,61)(9,21)(11,23)(13,17)(15,19)(25,48)(26,30)(27,42)(28,32)(29,44)(31,46)(33,56)(34,38)(35,50)(36,40)(37,52)(39,54)(41,45)(43,47)(49,53)(51,55), (1,62)(2,63)(3,64)(4,57)(5,58)(6,59)(7,60)(8,61)(9,21)(10,22)(11,23)(12,24)(13,17)(14,18)(15,19)(16,20)(25,44)(26,45)(27,46)(28,47)(29,48)(30,41)(31,42)(32,43)(33,52)(34,53)(35,54)(36,55)(37,56)(38,49)(39,50)(40,51), (1,20)(2,21)(3,22)(4,23)(5,24)(6,17)(7,18)(8,19)(9,63)(10,64)(11,57)(12,58)(13,59)(14,60)(15,61)(16,62)(25,37)(26,38)(27,39)(28,40)(29,33)(30,34)(31,35)(32,36)(41,53)(42,54)(43,55)(44,56)(45,49)(46,50)(47,51)(48,52), (1,5)(2,6)(3,7)(4,8)(9,13)(10,14)(11,15)(12,16)(17,21)(18,22)(19,23)(20,24)(25,29)(26,30)(27,31)(28,32)(33,37)(34,38)(35,39)(36,40)(41,45)(42,46)(43,47)(44,48)(49,53)(50,54)(51,55)(52,56)(57,61)(58,62)(59,63)(60,64), (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), (1,25,20,37)(2,36,21,32)(3,31,22,35)(4,34,23,30)(5,29,24,33)(6,40,17,28)(7,27,18,39)(8,38,19,26)(9,43,63,55)(10,54,64,42)(11,41,57,53)(12,52,58,48)(13,47,59,51)(14,50,60,46)(15,45,61,49)(16,56,62,44) );
G=PermutationGroup([[(2,63),(4,57),(6,59),(8,61),(9,21),(11,23),(13,17),(15,19),(25,48),(26,30),(27,42),(28,32),(29,44),(31,46),(33,56),(34,38),(35,50),(36,40),(37,52),(39,54),(41,45),(43,47),(49,53),(51,55)], [(1,62),(2,63),(3,64),(4,57),(5,58),(6,59),(7,60),(8,61),(9,21),(10,22),(11,23),(12,24),(13,17),(14,18),(15,19),(16,20),(25,44),(26,45),(27,46),(28,47),(29,48),(30,41),(31,42),(32,43),(33,52),(34,53),(35,54),(36,55),(37,56),(38,49),(39,50),(40,51)], [(1,20),(2,21),(3,22),(4,23),(5,24),(6,17),(7,18),(8,19),(9,63),(10,64),(11,57),(12,58),(13,59),(14,60),(15,61),(16,62),(25,37),(26,38),(27,39),(28,40),(29,33),(30,34),(31,35),(32,36),(41,53),(42,54),(43,55),(44,56),(45,49),(46,50),(47,51),(48,52)], [(1,5),(2,6),(3,7),(4,8),(9,13),(10,14),(11,15),(12,16),(17,21),(18,22),(19,23),(20,24),(25,29),(26,30),(27,31),(28,32),(33,37),(34,38),(35,39),(36,40),(41,45),(42,46),(43,47),(44,48),(49,53),(50,54),(51,55),(52,56),(57,61),(58,62),(59,63),(60,64)], [(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)], [(1,25,20,37),(2,36,21,32),(3,31,22,35),(4,34,23,30),(5,29,24,33),(6,40,17,28),(7,27,18,39),(8,38,19,26),(9,43,63,55),(10,54,64,42),(11,41,57,53),(12,52,58,48),(13,47,59,51),(14,50,60,46),(15,45,61,49),(16,56,62,44)]])
38 conjugacy classes
| class | 1 | 2A | ··· | 2G | 2H | ··· | 2M | 4A | ··· | 4H | 4I | 4J | 4K | 4L | 4M | 4N | 4O | 4P | 8A | ··· | 8H |
| order | 1 | 2 | ··· | 2 | 2 | ··· | 2 | 4 | ··· | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 8 | ··· | 8 |
| size | 1 | 1 | ··· | 1 | 4 | ··· | 4 | 2 | ··· | 2 | 4 | 4 | 4 | 4 | 8 | 8 | 8 | 8 | 4 | ··· | 4 |
38 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | - | ||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C4 | D4 | D4 | D4 | D4 | C4○D8 | C8⋊C22 | C8.C22 |
| kernel | C24.65D4 | C23.7Q8 | C2×C22⋊C8 | C2×D4⋊C4 | C2×Q8⋊C4 | C22×C4○D4 | C2×C4○D4 | C22×C4 | C2×D4 | C2×Q8 | C24 | C22 | C22 | C22 |
| # reps | 1 | 1 | 1 | 2 | 2 | 1 | 8 | 3 | 4 | 4 | 1 | 8 | 1 | 1 |
Matrix representation of C24.65D4 ►in GL5(𝔽17)
| 16 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 16 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 8 | 16 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 16 | 0 |
| 0 | 0 | 0 | 0 | 16 |
| 16 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 16 | 0 | 0 | 0 |
| 0 | 0 | 16 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 1 |
| 13 | 0 | 0 | 0 | 0 |
| 0 | 15 | 0 | 0 | 0 |
| 0 | 0 | 8 | 0 | 0 |
| 0 | 0 | 0 | 8 | 15 |
| 0 | 0 | 0 | 7 | 9 |
| 13 | 0 | 0 | 0 | 0 |
| 0 | 0 | 8 | 0 | 0 |
| 0 | 15 | 0 | 0 | 0 |
| 0 | 0 | 0 | 9 | 2 |
| 0 | 0 | 0 | 11 | 8 |
G:=sub<GL(5,GF(17))| [16,0,0,0,0,0,1,0,0,0,0,0,16,0,0,0,0,0,1,8,0,0,0,0,16],[1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,16,0,0,0,0,0,16],[16,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1],[1,0,0,0,0,0,16,0,0,0,0,0,16,0,0,0,0,0,1,0,0,0,0,0,1],[13,0,0,0,0,0,15,0,0,0,0,0,8,0,0,0,0,0,8,7,0,0,0,15,9],[13,0,0,0,0,0,0,15,0,0,0,8,0,0,0,0,0,0,9,11,0,0,0,2,8] >;
C24.65D4 in GAP, Magma, Sage, TeX
C_2^4._{65}D_4 % in TeX
G:=Group("C2^4.65D4"); // GroupNames label
G:=SmallGroup(128,520);
// by ID
G=gap.SmallGroup(128,520);
# by ID
G:=PCGroup([7,-2,2,2,-2,2,2,-2,224,141,422,352,2019,1018,248]);
// Polycyclic
G:=Group<a,b,c,d,e,f|a^2=b^2=c^2=d^2=1,e^4=d,f^2=c,e*a*e^-1=a*b=b*a,a*c=c*a,a*d=d*a,f*a*f^-1=a*b*d,b*c=c*b,b*d=d*b,b*e=e*b,b*f=f*b,c*d=d*c,c*e=e*c,c*f=f*c,d*e=e*d,d*f=f*d,f*e*f^-1=c*d*e^3>;
// generators/relations