p-group, metabelian, nilpotent (class 3), monomial
Aliases: C24.75D4, (C2×C8).196D4, C4.114(C4×D4), C22⋊Q8⋊13C4, C4.91C22≀C2, C2.2(C8⋊D4), C2.2(C8.D4), (C22×C4).290D4, C23.774(C2×D4), C22.4Q16⋊47C2, C22.79(C8⋊C22), C23.81(C22⋊C4), (C23×C4).259C22, (C22×C8).389C22, C23.7Q8.16C2, (C22×Q8).19C22, C22.120(C4⋊D4), (C22×C4).1370C23, C4.86(C22.D4), C22.68(C8.C22), (C22×M4(2)).21C2, C2.26(C23.36D4), C2.33(C23.23D4), C2.21(C23.38D4), C4⋊C4.73(C2×C4), (C2×Q8).70(C2×C4), (C2×C22⋊Q8).9C2, (C2×Q8⋊C4)⋊45C2, (C2×C4).1333(C2×D4), (C2×C4⋊C4).62C22, (C2×C4).567(C4○D4), (C22×C4).280(C2×C4), (C2×C4).388(C22×C4), (C2×C4).132(C22⋊C4), C22.269(C2×C22⋊C4), SmallGroup(128,626)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C24.75D4
G = < a,b,c,d,e,f | a2=b2=c2=d2=1, e4=f2=d, ab=ba, ac=ca, eae-1=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=ce3 >
Subgroups: 356 in 180 conjugacy classes, 64 normal (22 characteristic)
C1, C2, C2, C2, C4, C4, C4, C22, C22, C22, C8, C2×C4, C2×C4, C2×C4, Q8, C23, C23, C23, C22⋊C4, C4⋊C4, C4⋊C4, C2×C8, C2×C8, M4(2), C22×C4, C22×C4, C22×C4, C2×Q8, C2×Q8, C24, C2.C42, Q8⋊C4, C2×C22⋊C4, C2×C4⋊C4, C2×C4⋊C4, C2×C4⋊C4, C22⋊Q8, C22⋊Q8, C22×C8, C2×M4(2), C23×C4, C22×Q8, C22.4Q16, C23.7Q8, C2×Q8⋊C4, C2×C22⋊Q8, C22×M4(2), C24.75D4
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, C23.38D4, C8⋊D4, C8.D4, C24.75D4
►On 64 points
Generators in S64
(2 6)(4 8)(9 18)(10 23)(11 20)(12 17)(13 22)(14 19)(15 24)(16 21)(25 62)(26 59)(27 64)(28 61)(29 58)(30 63)(31 60)(32 57)(34 38)(36 40)(41 45)(43 47)(49 53)(51 55)
(1 39)(2 40)(3 33)(4 34)(5 35)(6 36)(7 37)(8 38)(9 18)(10 19)(11 20)(12 21)(13 22)(14 23)(15 24)(16 17)(25 62)(26 63)(27 64)(28 57)(29 58)(30 59)(31 60)(32 61)(41 51)(42 52)(43 53)(44 54)(45 55)(46 56)(47 49)(48 50)
(1 48)(2 41)(3 42)(4 43)(5 44)(6 45)(7 46)(8 47)(9 31)(10 32)(11 25)(12 26)(13 27)(14 28)(15 29)(16 30)(17 59)(18 60)(19 61)(20 62)(21 63)(22 64)(23 57)(24 58)(33 52)(34 53)(35 54)(36 55)(37 56)(38 49)(39 50)(40 51)
(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 23 5 19)(2 60 6 64)(3 21 7 17)(4 58 8 62)(9 55 13 51)(10 39 14 35)(11 53 15 49)(12 37 16 33)(18 45 22 41)(20 43 24 47)(25 34 29 38)(26 56 30 52)(27 40 31 36)(28 54 32 50)(42 63 46 59)(44 61 48 57)
G:=sub<Sym(64)| (2,6)(4,8)(9,18)(10,23)(11,20)(12,17)(13,22)(14,19)(15,24)(16,21)(25,62)(26,59)(27,64)(28,61)(29,58)(30,63)(31,60)(32,57)(34,38)(36,40)(41,45)(43,47)(49,53)(51,55), (1,39)(2,40)(3,33)(4,34)(5,35)(6,36)(7,37)(8,38)(9,18)(10,19)(11,20)(12,21)(13,22)(14,23)(15,24)(16,17)(25,62)(26,63)(27,64)(28,57)(29,58)(30,59)(31,60)(32,61)(41,51)(42,52)(43,53)(44,54)(45,55)(46,56)(47,49)(48,50), (1,48)(2,41)(3,42)(4,43)(5,44)(6,45)(7,46)(8,47)(9,31)(10,32)(11,25)(12,26)(13,27)(14,28)(15,29)(16,30)(17,59)(18,60)(19,61)(20,62)(21,63)(22,64)(23,57)(24,58)(33,52)(34,53)(35,54)(36,55)(37,56)(38,49)(39,50)(40,51), (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), (1,23,5,19)(2,60,6,64)(3,21,7,17)(4,58,8,62)(9,55,13,51)(10,39,14,35)(11,53,15,49)(12,37,16,33)(18,45,22,41)(20,43,24,47)(25,34,29,38)(26,56,30,52)(27,40,31,36)(28,54,32,50)(42,63,46,59)(44,61,48,57)>;
G:=Group( (2,6)(4,8)(9,18)(10,23)(11,20)(12,17)(13,22)(14,19)(15,24)(16,21)(25,62)(26,59)(27,64)(28,61)(29,58)(30,63)(31,60)(32,57)(34,38)(36,40)(41,45)(43,47)(49,53)(51,55), (1,39)(2,40)(3,33)(4,34)(5,35)(6,36)(7,37)(8,38)(9,18)(10,19)(11,20)(12,21)(13,22)(14,23)(15,24)(16,17)(25,62)(26,63)(27,64)(28,57)(29,58)(30,59)(31,60)(32,61)(41,51)(42,52)(43,53)(44,54)(45,55)(46,56)(47,49)(48,50), (1,48)(2,41)(3,42)(4,43)(5,44)(6,45)(7,46)(8,47)(9,31)(10,32)(11,25)(12,26)(13,27)(14,28)(15,29)(16,30)(17,59)(18,60)(19,61)(20,62)(21,63)(22,64)(23,57)(24,58)(33,52)(34,53)(35,54)(36,55)(37,56)(38,49)(39,50)(40,51), (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), (1,23,5,19)(2,60,6,64)(3,21,7,17)(4,58,8,62)(9,55,13,51)(10,39,14,35)(11,53,15,49)(12,37,16,33)(18,45,22,41)(20,43,24,47)(25,34,29,38)(26,56,30,52)(27,40,31,36)(28,54,32,50)(42,63,46,59)(44,61,48,57) );
G=PermutationGroup([[(2,6),(4,8),(9,18),(10,23),(11,20),(12,17),(13,22),(14,19),(15,24),(16,21),(25,62),(26,59),(27,64),(28,61),(29,58),(30,63),(31,60),(32,57),(34,38),(36,40),(41,45),(43,47),(49,53),(51,55)], [(1,39),(2,40),(3,33),(4,34),(5,35),(6,36),(7,37),(8,38),(9,18),(10,19),(11,20),(12,21),(13,22),(14,23),(15,24),(16,17),(25,62),(26,63),(27,64),(28,57),(29,58),(30,59),(31,60),(32,61),(41,51),(42,52),(43,53),(44,54),(45,55),(46,56),(47,49),(48,50)], [(1,48),(2,41),(3,42),(4,43),(5,44),(6,45),(7,46),(8,47),(9,31),(10,32),(11,25),(12,26),(13,27),(14,28),(15,29),(16,30),(17,59),(18,60),(19,61),(20,62),(21,63),(22,64),(23,57),(24,58),(33,52),(34,53),(35,54),(36,55),(37,56),(38,49),(39,50),(40,51)], [(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)], [(1,23,5,19),(2,60,6,64),(3,21,7,17),(4,58,8,62),(9,55,13,51),(10,39,14,35),(11,53,15,49),(12,37,16,33),(18,45,22,41),(20,43,24,47),(25,34,29,38),(26,56,30,52),(27,40,31,36),(28,54,32,50),(42,63,46,59),(44,61,48,57)]])
32 conjugacy classes
| class | 1 | 2A | ··· | 2G | 2H | 2I | 4A | 4B | 4C | 4D | 4E | 4F | 4G | ··· | 4N | 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 | 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 | C24.75D4 | C22.4Q16 | C23.7Q8 | C2×Q8⋊C4 | C2×C22⋊Q8 | C22×M4(2) | C22⋊Q8 | C2×C8 | C22×C4 | C24 | C2×C4 | C22 | C22 |
| # reps | 1 | 2 | 1 | 2 | 1 | 1 | 8 | 4 | 3 | 1 | 4 | 1 | 3 |
Matrix representation of C24.75D4 ►in GL8(𝔽17)
| 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 6 | 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 | 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 |
| 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 | 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 |
| 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 |
| 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 6 | 16 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 4 | 9 | 0 | 0 | 0 | 0 |
| 0 | 0 | 4 | 13 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 16 | 0 | 0 | 0 |
| 11 | 2 | 0 | 0 | 0 | 0 | 0 | 0 |
| 8 | 6 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 6 | 11 | 0 | 0 | 0 | 0 |
| 0 | 0 | 3 | 11 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 6 | 4 |
| 0 | 0 | 0 | 0 | 0 | 0 | 4 | 11 |
| 0 | 0 | 0 | 0 | 11 | 13 | 0 | 0 |
| 0 | 0 | 0 | 0 | 13 | 6 | 0 | 0 |
G:=sub<GL(8,GF(17))| [1,6,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,16],[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],[16,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[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],[1,6,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,4,4,0,0,0,0,0,0,9,13,0,0,0,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0],[11,8,0,0,0,0,0,0,2,6,0,0,0,0,0,0,0,0,6,3,0,0,0,0,0,0,11,11,0,0,0,0,0,0,0,0,0,0,11,13,0,0,0,0,0,0,13,6,0,0,0,0,6,4,0,0,0,0,0,0,4,11,0,0] >;
C24.75D4 in GAP, Magma, Sage, TeX
C_2^4._{75}D_4 % in TeX
G:=Group("C2^4.75D4"); // GroupNames label
G:=SmallGroup(128,626);
// by ID
G=gap.SmallGroup(128,626);
# by ID
G:=PCGroup([7,-2,2,2,-2,2,2,-2,224,141,288,422,723,2019,248]);
// Polycyclic
G:=Group<a,b,c,d,e,f|a^2=b^2=c^2=d^2=1,e^4=f^2=d,a*b=b*a,a*c=c*a,e*a*e^-1=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*e^3>;
// generators/relations