p-group, metabelian, nilpotent (class 3), monomial
Aliases: C24.76D4, (C2×C8)⋊23D4, C4⋊D4⋊13C4, C4.115(C4×D4), C4.92C22≀C2, C2.3(C8⋊D4), C2.2(C8⋊2D4), C23.775(C2×D4), (C22×C4).291D4, C23.7Q8⋊7C2, C22.4Q16⋊48C2, C23.82(C22⋊C4), C22.80(C8⋊C22), (C22×M4(2))⋊11C2, (C22×C8).390C22, (C23×C4).260C22, (C22×D4).25C22, C22.121(C4⋊D4), (C22×C4).1371C23, C4.87(C22.D4), C22.69(C8.C22), C2.34(C23.23D4), C2.27(C23.36D4), C2.21(C23.37D4), C4⋊C4.74(C2×C4), (C2×D4).82(C2×C4), (C2×C4⋊D4).9C2, (C2×D4⋊C4)⋊44C2, (C2×C4).1334(C2×D4), (C2×C4⋊C4).63C22, (C2×C4).568(C4○D4), (C22×C4).281(C2×C4), (C2×C4).389(C22×C4), (C2×C4).133(C22⋊C4), C22.270(C2×C22⋊C4), SmallGroup(128,627)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C24.76D4
G = < a,b,c,d,e,f | a2=b2=c2=d2=f2=1, e4=d, ab=ba, ac=ca, eae-1=ad=da, faf=abd, bc=cb, bd=db, be=eb, bf=fb, cd=dc, ce=ec, cf=fc, de=ed, df=fd, fef=cde3 >
Subgroups: 484 in 206 conjugacy classes, 64 normal (22 characteristic)
C1, C2, C2, C2, C4, C4, C4, C22, C22, C22, C8, C2×C4, C2×C4, C2×C4, D4, C23, C23, C23, C22⋊C4, C4⋊C4, C4⋊C4, C2×C8, C2×C8, M4(2), C22×C4, C22×C4, C22×C4, C2×D4, C2×D4, C24, C24, C2.C42, D4⋊C4, C2×C22⋊C4, C2×C4⋊C4, C2×C4⋊C4, C4⋊D4, C4⋊D4, C22×C8, C2×M4(2), C23×C4, C22×D4, C22×D4, C22.4Q16, C23.7Q8, C2×D4⋊C4, C2×C4⋊D4, C22×M4(2), C24.76D4
Quotients: C1, C2, C4, C22, C2×C4, D4, C23, C22⋊C4, C22×C4, C2×D4, C4○D4, C2×C22⋊C4, C4×D4, C22≀C2, C4⋊D4, C22.D4, C8⋊C22, C8.C22, C23.23D4, C23.36D4, C23.37D4, C8⋊D4, C8⋊2D4, C24.76D4
►On 64 points
Generators in S64
(1 55)(2 52)(3 49)(4 54)(5 51)(6 56)(7 53)(8 50)(9 45)(10 42)(11 47)(12 44)(13 41)(14 46)(15 43)(16 48)(17 30)(18 27)(19 32)(20 29)(21 26)(22 31)(23 28)(24 25)(33 63)(34 60)(35 57)(36 62)(37 59)(38 64)(39 61)(40 58)
(1 14)(2 15)(3 16)(4 9)(5 10)(6 11)(7 12)(8 13)(17 60)(18 61)(19 62)(20 63)(21 64)(22 57)(23 58)(24 59)(25 37)(26 38)(27 39)(28 40)(29 33)(30 34)(31 35)(32 36)(41 50)(42 51)(43 52)(44 53)(45 54)(46 55)(47 56)(48 49)
(1 18)(2 19)(3 20)(4 21)(5 22)(6 23)(7 24)(8 17)(9 64)(10 57)(11 58)(12 59)(13 60)(14 61)(15 62)(16 63)(25 53)(26 54)(27 55)(28 56)(29 49)(30 50)(31 51)(32 52)(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 24)(2 6)(3 22)(5 20)(7 18)(10 63)(11 15)(12 61)(14 59)(16 57)(19 23)(25 42)(26 34)(27 48)(28 40)(29 46)(30 38)(31 44)(32 36)(33 55)(35 53)(37 51)(39 49)(41 54)(43 52)(45 50)(47 56)(58 62)
G:=sub<Sym(64)| (1,55)(2,52)(3,49)(4,54)(5,51)(6,56)(7,53)(8,50)(9,45)(10,42)(11,47)(12,44)(13,41)(14,46)(15,43)(16,48)(17,30)(18,27)(19,32)(20,29)(21,26)(22,31)(23,28)(24,25)(33,63)(34,60)(35,57)(36,62)(37,59)(38,64)(39,61)(40,58), (1,14)(2,15)(3,16)(4,9)(5,10)(6,11)(7,12)(8,13)(17,60)(18,61)(19,62)(20,63)(21,64)(22,57)(23,58)(24,59)(25,37)(26,38)(27,39)(28,40)(29,33)(30,34)(31,35)(32,36)(41,50)(42,51)(43,52)(44,53)(45,54)(46,55)(47,56)(48,49), (1,18)(2,19)(3,20)(4,21)(5,22)(6,23)(7,24)(8,17)(9,64)(10,57)(11,58)(12,59)(13,60)(14,61)(15,62)(16,63)(25,53)(26,54)(27,55)(28,56)(29,49)(30,50)(31,51)(32,52)(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), (1,24)(2,6)(3,22)(5,20)(7,18)(10,63)(11,15)(12,61)(14,59)(16,57)(19,23)(25,42)(26,34)(27,48)(28,40)(29,46)(30,38)(31,44)(32,36)(33,55)(35,53)(37,51)(39,49)(41,54)(43,52)(45,50)(47,56)(58,62)>;
G:=Group( (1,55)(2,52)(3,49)(4,54)(5,51)(6,56)(7,53)(8,50)(9,45)(10,42)(11,47)(12,44)(13,41)(14,46)(15,43)(16,48)(17,30)(18,27)(19,32)(20,29)(21,26)(22,31)(23,28)(24,25)(33,63)(34,60)(35,57)(36,62)(37,59)(38,64)(39,61)(40,58), (1,14)(2,15)(3,16)(4,9)(5,10)(6,11)(7,12)(8,13)(17,60)(18,61)(19,62)(20,63)(21,64)(22,57)(23,58)(24,59)(25,37)(26,38)(27,39)(28,40)(29,33)(30,34)(31,35)(32,36)(41,50)(42,51)(43,52)(44,53)(45,54)(46,55)(47,56)(48,49), (1,18)(2,19)(3,20)(4,21)(5,22)(6,23)(7,24)(8,17)(9,64)(10,57)(11,58)(12,59)(13,60)(14,61)(15,62)(16,63)(25,53)(26,54)(27,55)(28,56)(29,49)(30,50)(31,51)(32,52)(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), (1,24)(2,6)(3,22)(5,20)(7,18)(10,63)(11,15)(12,61)(14,59)(16,57)(19,23)(25,42)(26,34)(27,48)(28,40)(29,46)(30,38)(31,44)(32,36)(33,55)(35,53)(37,51)(39,49)(41,54)(43,52)(45,50)(47,56)(58,62) );
G=PermutationGroup([[(1,55),(2,52),(3,49),(4,54),(5,51),(6,56),(7,53),(8,50),(9,45),(10,42),(11,47),(12,44),(13,41),(14,46),(15,43),(16,48),(17,30),(18,27),(19,32),(20,29),(21,26),(22,31),(23,28),(24,25),(33,63),(34,60),(35,57),(36,62),(37,59),(38,64),(39,61),(40,58)], [(1,14),(2,15),(3,16),(4,9),(5,10),(6,11),(7,12),(8,13),(17,60),(18,61),(19,62),(20,63),(21,64),(22,57),(23,58),(24,59),(25,37),(26,38),(27,39),(28,40),(29,33),(30,34),(31,35),(32,36),(41,50),(42,51),(43,52),(44,53),(45,54),(46,55),(47,56),(48,49)], [(1,18),(2,19),(3,20),(4,21),(5,22),(6,23),(7,24),(8,17),(9,64),(10,57),(11,58),(12,59),(13,60),(14,61),(15,62),(16,63),(25,53),(26,54),(27,55),(28,56),(29,49),(30,50),(31,51),(32,52),(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)], [(1,24),(2,6),(3,22),(5,20),(7,18),(10,63),(11,15),(12,61),(14,59),(16,57),(19,23),(25,42),(26,34),(27,48),(28,40),(29,46),(30,38),(31,44),(32,36),(33,55),(35,53),(37,51),(39,49),(41,54),(43,52),(45,50),(47,56),(58,62)]])
32 conjugacy classes
| class | 1 | 2A | ··· | 2G | 2H | 2I | 2J | 2K | 4A | 4B | 4C | 4D | 4E | 4F | 4G | ··· | 4L | 8A | ··· | 8H |
| order | 1 | 2 | ··· | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 8 | ··· | 8 |
| size | 1 | 1 | ··· | 1 | 4 | 4 | 8 | 8 | 2 | 2 | 2 | 2 | 4 | 4 | 8 | ··· | 8 | 4 | ··· | 4 |
32 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | - | ||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C4 | D4 | D4 | D4 | C4○D4 | C8⋊C22 | C8.C22 |
| kernel | C24.76D4 | C22.4Q16 | C23.7Q8 | C2×D4⋊C4 | C2×C4⋊D4 | C22×M4(2) | C4⋊D4 | C2×C8 | C22×C4 | C24 | C2×C4 | C22 | C22 |
| # reps | 1 | 2 | 1 | 2 | 1 | 1 | 8 | 4 | 3 | 1 | 4 | 3 | 1 |
Matrix representation of C24.76D4 ►in GL8(𝔽17)
| 16 | 9 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 |
| 16 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 16 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 |
| 16 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 16 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 16 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 16 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 16 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 16 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 16 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 16 |
| 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 16 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 16 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 16 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 16 |
| 13 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 13 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 11 | 2 | 0 | 0 | 0 | 0 |
| 0 | 0 | 8 | 6 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 4 | 13 | 4 |
| 0 | 0 | 0 | 0 | 15 | 4 | 15 | 0 |
| 0 | 0 | 0 | 0 | 4 | 13 | 0 | 13 |
| 0 | 0 | 0 | 0 | 2 | 0 | 2 | 13 |
| 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 4 | 16 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 16 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 11 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 16 | 2 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 1 | 15 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 16 |
G:=sub<GL(8,GF(17))| [16,0,0,0,0,0,0,0,9,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0],[16,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[16,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,16],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,16],[13,0,0,0,0,0,0,0,0,13,0,0,0,0,0,0,0,0,11,8,0,0,0,0,0,0,2,6,0,0,0,0,0,0,0,0,0,15,4,2,0,0,0,0,4,4,13,0,0,0,0,0,13,15,0,2,0,0,0,0,4,0,13,13],[1,4,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,16,11,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,2,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,15,16] >;
C24.76D4 in GAP, Magma, Sage, TeX
C_2^4._{76}D_4 % in TeX
G:=Group("C2^4.76D4"); // GroupNames label
G:=SmallGroup(128,627);
// by ID
G=gap.SmallGroup(128,627);
# by ID
G:=PCGroup([7,-2,2,2,-2,2,2,-2,224,141,736,422,723,2019,248]);
// Polycyclic
G:=Group<a,b,c,d,e,f|a^2=b^2=c^2=d^2=f^2=1,e^4=d,a*b=b*a,a*c=c*a,e*a*e^-1=a*d=d*a,f*a*f=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=c*d*e^3>;
// generators/relations