p-group, metabelian, nilpotent (class 3), monomial
Aliases: C24.69D4, C42⋊C2⋊18C4, C22.4Q16⋊7C2, (C22×C4).678D4, C23.749(C2×D4), C22.45(C4○D8), (C22×C8).18C22, C4.20(C42⋊C2), C22.64(C8⋊C22), C23.76(C22⋊C4), (C23×C4).245C22, C23.7Q8.11C2, (C22×C4).1336C23, C2.1(C23.20D4), C2.1(C23.19D4), C22.53(C8.C22), C2.23(C23.36D4), C2.23(C23.24D4), C2.13(C23.34D4), C4.105(C22.D4), C22.80(C22.D4), C4⋊C4.196(C2×C4), (C2×C4).1326(C2×D4), (C2×C22⋊C8).19C2, (C2×C4⋊C4).43C22, (C2×C4).742(C4○D4), (C22×C4).268(C2×C4), (C2×C4).369(C22×C4), (C2×C4).335(C22⋊C4), (C2×C42⋊C2).17C2, C22.259(C2×C22⋊C4), SmallGroup(128,557)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C24.69D4
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=bcde3 >
Subgroups: 300 in 150 conjugacy classes, 60 normal (22 characteristic)
C1, C2, C2, C2, C4, C4, C4, C22, C22, C22, C8, C2×C4, C2×C4, C2×C4, C23, C23, C23, C42, C22⋊C4, C4⋊C4, C4⋊C4, C2×C8, C22×C4, C22×C4, C22×C4, C24, C2.C42, C22⋊C8, C2×C42, C2×C22⋊C4, C2×C4⋊C4, C2×C4⋊C4, C42⋊C2, C42⋊C2, C22×C8, C23×C4, C22.4Q16, C23.7Q8, C2×C22⋊C8, C2×C42⋊C2, C24.69D4
Quotients: C1, C2, C4, C22, C2×C4, D4, C23, C22⋊C4, C22×C4, C2×D4, C4○D4, C2×C22⋊C4, C42⋊C2, C22.D4, C4○D8, C8⋊C22, C8.C22, C23.34D4, C23.24D4, C23.36D4, C23.19D4, C23.20D4, C24.69D4
►On 64 points
Generators in S64
(2 57)(4 59)(6 61)(8 63)(10 27)(12 29)(14 31)(16 25)(17 21)(18 33)(19 23)(20 35)(22 37)(24 39)(34 38)(36 40)(41 45)(42 55)(43 47)(44 49)(46 51)(48 53)(50 54)(52 56)
(1 64)(2 57)(3 58)(4 59)(5 60)(6 61)(7 62)(8 63)(9 26)(10 27)(11 28)(12 29)(13 30)(14 31)(15 32)(16 25)(17 36)(18 37)(19 38)(20 39)(21 40)(22 33)(23 34)(24 35)(41 50)(42 51)(43 52)(44 53)(45 54)(46 55)(47 56)(48 49)
(1 28)(2 29)(3 30)(4 31)(5 32)(6 25)(7 26)(8 27)(9 62)(10 63)(11 64)(12 57)(13 58)(14 59)(15 60)(16 61)(17 52)(18 53)(19 54)(20 55)(21 56)(22 49)(23 50)(24 51)(33 48)(34 41)(35 42)(36 43)(37 44)(38 45)(39 46)(40 47)
(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 20 28 55)(2 45 29 38)(3 18 30 53)(4 43 31 36)(5 24 32 51)(6 41 25 34)(7 22 26 49)(8 47 27 40)(9 48 62 33)(10 21 63 56)(11 46 64 39)(12 19 57 54)(13 44 58 37)(14 17 59 52)(15 42 60 35)(16 23 61 50)
G:=sub<Sym(64)| (2,57)(4,59)(6,61)(8,63)(10,27)(12,29)(14,31)(16,25)(17,21)(18,33)(19,23)(20,35)(22,37)(24,39)(34,38)(36,40)(41,45)(42,55)(43,47)(44,49)(46,51)(48,53)(50,54)(52,56), (1,64)(2,57)(3,58)(4,59)(5,60)(6,61)(7,62)(8,63)(9,26)(10,27)(11,28)(12,29)(13,30)(14,31)(15,32)(16,25)(17,36)(18,37)(19,38)(20,39)(21,40)(22,33)(23,34)(24,35)(41,50)(42,51)(43,52)(44,53)(45,54)(46,55)(47,56)(48,49), (1,28)(2,29)(3,30)(4,31)(5,32)(6,25)(7,26)(8,27)(9,62)(10,63)(11,64)(12,57)(13,58)(14,59)(15,60)(16,61)(17,52)(18,53)(19,54)(20,55)(21,56)(22,49)(23,50)(24,51)(33,48)(34,41)(35,42)(36,43)(37,44)(38,45)(39,46)(40,47), (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), (1,20,28,55)(2,45,29,38)(3,18,30,53)(4,43,31,36)(5,24,32,51)(6,41,25,34)(7,22,26,49)(8,47,27,40)(9,48,62,33)(10,21,63,56)(11,46,64,39)(12,19,57,54)(13,44,58,37)(14,17,59,52)(15,42,60,35)(16,23,61,50)>;
G:=Group( (2,57)(4,59)(6,61)(8,63)(10,27)(12,29)(14,31)(16,25)(17,21)(18,33)(19,23)(20,35)(22,37)(24,39)(34,38)(36,40)(41,45)(42,55)(43,47)(44,49)(46,51)(48,53)(50,54)(52,56), (1,64)(2,57)(3,58)(4,59)(5,60)(6,61)(7,62)(8,63)(9,26)(10,27)(11,28)(12,29)(13,30)(14,31)(15,32)(16,25)(17,36)(18,37)(19,38)(20,39)(21,40)(22,33)(23,34)(24,35)(41,50)(42,51)(43,52)(44,53)(45,54)(46,55)(47,56)(48,49), (1,28)(2,29)(3,30)(4,31)(5,32)(6,25)(7,26)(8,27)(9,62)(10,63)(11,64)(12,57)(13,58)(14,59)(15,60)(16,61)(17,52)(18,53)(19,54)(20,55)(21,56)(22,49)(23,50)(24,51)(33,48)(34,41)(35,42)(36,43)(37,44)(38,45)(39,46)(40,47), (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), (1,20,28,55)(2,45,29,38)(3,18,30,53)(4,43,31,36)(5,24,32,51)(6,41,25,34)(7,22,26,49)(8,47,27,40)(9,48,62,33)(10,21,63,56)(11,46,64,39)(12,19,57,54)(13,44,58,37)(14,17,59,52)(15,42,60,35)(16,23,61,50) );
G=PermutationGroup([[(2,57),(4,59),(6,61),(8,63),(10,27),(12,29),(14,31),(16,25),(17,21),(18,33),(19,23),(20,35),(22,37),(24,39),(34,38),(36,40),(41,45),(42,55),(43,47),(44,49),(46,51),(48,53),(50,54),(52,56)], [(1,64),(2,57),(3,58),(4,59),(5,60),(6,61),(7,62),(8,63),(9,26),(10,27),(11,28),(12,29),(13,30),(14,31),(15,32),(16,25),(17,36),(18,37),(19,38),(20,39),(21,40),(22,33),(23,34),(24,35),(41,50),(42,51),(43,52),(44,53),(45,54),(46,55),(47,56),(48,49)], [(1,28),(2,29),(3,30),(4,31),(5,32),(6,25),(7,26),(8,27),(9,62),(10,63),(11,64),(12,57),(13,58),(14,59),(15,60),(16,61),(17,52),(18,53),(19,54),(20,55),(21,56),(22,49),(23,50),(24,51),(33,48),(34,41),(35,42),(36,43),(37,44),(38,45),(39,46),(40,47)], [(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)], [(1,20,28,55),(2,45,29,38),(3,18,30,53),(4,43,31,36),(5,24,32,51),(6,41,25,34),(7,22,26,49),(8,47,27,40),(9,48,62,33),(10,21,63,56),(11,46,64,39),(12,19,57,54),(13,44,58,37),(14,17,59,52),(15,42,60,35),(16,23,61,50)]])
38 conjugacy classes
| class | 1 | 2A | ··· | 2G | 2H | 2I | 4A | ··· | 4H | 4I | ··· | 4P | 4Q | 4R | 4S | 4T | 8A | ··· | 8H |
| order | 1 | 2 | ··· | 2 | 2 | 2 | 4 | ··· | 4 | 4 | ··· | 4 | 4 | 4 | 4 | 4 | 8 | ··· | 8 |
| size | 1 | 1 | ··· | 1 | 4 | 4 | 2 | ··· | 2 | 4 | ··· | 4 | 8 | 8 | 8 | 8 | 4 | ··· | 4 |
38 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | - | |||
| image | C1 | C2 | C2 | C2 | C2 | C4 | D4 | D4 | C4○D4 | C4○D8 | C8⋊C22 | C8.C22 |
| kernel | C24.69D4 | C22.4Q16 | C23.7Q8 | C2×C22⋊C8 | C2×C42⋊C2 | C42⋊C2 | C22×C4 | C24 | C2×C4 | C22 | C22 | C22 |
| # reps | 1 | 4 | 1 | 1 | 1 | 8 | 3 | 1 | 8 | 8 | 1 | 1 |
Matrix representation of C24.69D4 ►in GL6(𝔽17)
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 7 | 16 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 16 |
| 1 | 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 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 16 | 0 | 0 | 0 | 0 | 0 |
| 0 | 16 | 0 | 0 | 0 | 0 |
| 0 | 0 | 16 | 0 | 0 | 0 |
| 0 | 0 | 0 | 16 | 0 | 0 |
| 0 | 0 | 0 | 0 | 16 | 0 |
| 0 | 0 | 0 | 0 | 0 | 16 |
| 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 | 16 | 0 |
| 0 | 0 | 0 | 0 | 0 | 16 |
| 10 | 2 | 0 | 0 | 0 | 0 |
| 10 | 7 | 0 | 0 | 0 | 0 |
| 0 | 0 | 11 | 9 | 0 | 0 |
| 0 | 0 | 11 | 6 | 0 | 0 |
| 0 | 0 | 0 | 0 | 9 | 0 |
| 0 | 0 | 0 | 0 | 0 | 15 |
| 7 | 5 | 0 | 0 | 0 | 0 |
| 7 | 10 | 0 | 0 | 0 | 0 |
| 0 | 0 | 7 | 15 | 0 | 0 |
| 0 | 0 | 8 | 10 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 15 |
| 0 | 0 | 0 | 0 | 9 | 0 |
G:=sub<GL(6,GF(17))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,7,0,0,0,0,0,16,0,0,0,0,0,0,1,0,0,0,0,0,0,16],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[16,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,16],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,16,0,0,0,0,0,0,16],[10,10,0,0,0,0,2,7,0,0,0,0,0,0,11,11,0,0,0,0,9,6,0,0,0,0,0,0,9,0,0,0,0,0,0,15],[7,7,0,0,0,0,5,10,0,0,0,0,0,0,7,8,0,0,0,0,15,10,0,0,0,0,0,0,0,9,0,0,0,0,15,0] >;
C24.69D4 in GAP, Magma, Sage, TeX
C_2^4._{69}D_4 % in TeX
G:=Group("C2^4.69D4"); // GroupNames label
G:=SmallGroup(128,557);
// by ID
G=gap.SmallGroup(128,557);
# by ID
G:=PCGroup([7,-2,2,2,-2,2,2,-2,224,141,456,422,58,2804,718,172]);
// 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=b*c*d*e^3>;
// generators/relations