p-group, metabelian, nilpotent (class 3), monomial
Aliases: C24.67D4, C8⋊1(C22⋊C4), (C2×C8).100D4, C2.1(C8⋊D4), C2.1(C8⋊2D4), (C2×M4(2))⋊5C4, (C22×C4).43Q8, C23.26(C4⋊C4), C2.1(C8.D4), C23.746(C2×D4), (C22×C4).271D4, C4.71(C22⋊Q8), C22.4Q16⋊46C2, C4.54(C42⋊C2), C23.7Q8.8C2, C22.62(C8⋊C22), (C22×C8).212C22, (C23×C4).237C22, (C22×M4(2)).1C2, C22.111(C4⋊D4), C2.7(M4(2)⋊C4), (C22×C4).1331C23, C22.51(C8.C22), C2.18(C23.7Q8), (C2×C4.Q8)⋊1C2, (C2×C2.D8)⋊32C2, (C2×C4).42(C4⋊C4), (C2×C8).142(C2×C4), C4.88(C2×C22⋊C4), C22.92(C2×C4⋊C4), (C2×C4).189(C2×Q8), (C2×C4).1320(C2×D4), (C2×C4⋊C4).38C22, (C2×C4).553(C4○D4), (C22×C4).264(C2×C4), (C2×C4).529(C22×C4), SmallGroup(128,541)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C24.67D4
G = < a,b,c,d,e,f | a2=b2=c2=d2=1, e4=d, f2=dc=cd, ab=ba, ac=ca, eae-1=ad=da, faf-1=abd, bc=cb, bd=db, be=eb, bf=fb, ce=ec, cf=fc, de=ed, df=fd, fef-1=e3 >
Subgroups: 316 in 158 conjugacy classes, 68 normal (28 characteristic)
C1, C2, C2, C2, C4, C4, C22, C22, C22, C8, C8, C2×C4, C2×C4, C2×C4, C23, C23, C23, C22⋊C4, C4⋊C4, C2×C8, C2×C8, M4(2), C22×C4, C22×C4, C22×C4, C24, C2.C42, C4.Q8, C2.D8, C2×C22⋊C4, C2×C4⋊C4, C22×C8, C2×M4(2), C2×M4(2), C23×C4, C22.4Q16, C23.7Q8, C2×C4.Q8, C2×C2.D8, C22×M4(2), C24.67D4
Quotients: C1, C2, C4, C22, C2×C4, D4, Q8, C23, C22⋊C4, C4⋊C4, C22×C4, C2×D4, C2×Q8, C4○D4, C2×C22⋊C4, C2×C4⋊C4, C42⋊C2, C4⋊D4, C22⋊Q8, C8⋊C22, C8.C22, C23.7Q8, M4(2)⋊C4, C8⋊D4, C8⋊2D4, C8.D4, C24.67D4
►On 64 points
Generators in S64
(2 6)(4 8)(9 59)(10 64)(11 61)(12 58)(13 63)(14 60)(15 57)(16 62)(17 27)(18 32)(19 29)(20 26)(21 31)(22 28)(23 25)(24 30)(34 38)(36 40)(42 46)(44 48)(49 53)(51 55)
(1 56)(2 49)(3 50)(4 51)(5 52)(6 53)(7 54)(8 55)(9 59)(10 60)(11 61)(12 62)(13 63)(14 64)(15 57)(16 58)(17 27)(18 28)(19 29)(20 30)(21 31)(22 32)(23 25)(24 26)(33 45)(34 46)(35 47)(36 48)(37 41)(38 42)(39 43)(40 44)
(1 43)(2 44)(3 45)(4 46)(5 47)(6 48)(7 41)(8 42)(9 21)(10 22)(11 23)(12 24)(13 17)(14 18)(15 19)(16 20)(25 61)(26 62)(27 63)(28 64)(29 57)(30 58)(31 59)(32 60)(33 50)(34 51)(35 52)(36 53)(37 54)(38 55)(39 56)(40 49)
(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 47 16)(2 19 48 11)(3 22 41 14)(4 17 42 9)(5 20 43 12)(6 23 44 15)(7 18 45 10)(8 21 46 13)(25 40 57 53)(26 35 58 56)(27 38 59 51)(28 33 60 54)(29 36 61 49)(30 39 62 52)(31 34 63 55)(32 37 64 50)
G:=sub<Sym(64)| (2,6)(4,8)(9,59)(10,64)(11,61)(12,58)(13,63)(14,60)(15,57)(16,62)(17,27)(18,32)(19,29)(20,26)(21,31)(22,28)(23,25)(24,30)(34,38)(36,40)(42,46)(44,48)(49,53)(51,55), (1,56)(2,49)(3,50)(4,51)(5,52)(6,53)(7,54)(8,55)(9,59)(10,60)(11,61)(12,62)(13,63)(14,64)(15,57)(16,58)(17,27)(18,28)(19,29)(20,30)(21,31)(22,32)(23,25)(24,26)(33,45)(34,46)(35,47)(36,48)(37,41)(38,42)(39,43)(40,44), (1,43)(2,44)(3,45)(4,46)(5,47)(6,48)(7,41)(8,42)(9,21)(10,22)(11,23)(12,24)(13,17)(14,18)(15,19)(16,20)(25,61)(26,62)(27,63)(28,64)(29,57)(30,58)(31,59)(32,60)(33,50)(34,51)(35,52)(36,53)(37,54)(38,55)(39,56)(40,49), (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), (1,24,47,16)(2,19,48,11)(3,22,41,14)(4,17,42,9)(5,20,43,12)(6,23,44,15)(7,18,45,10)(8,21,46,13)(25,40,57,53)(26,35,58,56)(27,38,59,51)(28,33,60,54)(29,36,61,49)(30,39,62,52)(31,34,63,55)(32,37,64,50)>;
G:=Group( (2,6)(4,8)(9,59)(10,64)(11,61)(12,58)(13,63)(14,60)(15,57)(16,62)(17,27)(18,32)(19,29)(20,26)(21,31)(22,28)(23,25)(24,30)(34,38)(36,40)(42,46)(44,48)(49,53)(51,55), (1,56)(2,49)(3,50)(4,51)(5,52)(6,53)(7,54)(8,55)(9,59)(10,60)(11,61)(12,62)(13,63)(14,64)(15,57)(16,58)(17,27)(18,28)(19,29)(20,30)(21,31)(22,32)(23,25)(24,26)(33,45)(34,46)(35,47)(36,48)(37,41)(38,42)(39,43)(40,44), (1,43)(2,44)(3,45)(4,46)(5,47)(6,48)(7,41)(8,42)(9,21)(10,22)(11,23)(12,24)(13,17)(14,18)(15,19)(16,20)(25,61)(26,62)(27,63)(28,64)(29,57)(30,58)(31,59)(32,60)(33,50)(34,51)(35,52)(36,53)(37,54)(38,55)(39,56)(40,49), (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), (1,24,47,16)(2,19,48,11)(3,22,41,14)(4,17,42,9)(5,20,43,12)(6,23,44,15)(7,18,45,10)(8,21,46,13)(25,40,57,53)(26,35,58,56)(27,38,59,51)(28,33,60,54)(29,36,61,49)(30,39,62,52)(31,34,63,55)(32,37,64,50) );
G=PermutationGroup([[(2,6),(4,8),(9,59),(10,64),(11,61),(12,58),(13,63),(14,60),(15,57),(16,62),(17,27),(18,32),(19,29),(20,26),(21,31),(22,28),(23,25),(24,30),(34,38),(36,40),(42,46),(44,48),(49,53),(51,55)], [(1,56),(2,49),(3,50),(4,51),(5,52),(6,53),(7,54),(8,55),(9,59),(10,60),(11,61),(12,62),(13,63),(14,64),(15,57),(16,58),(17,27),(18,28),(19,29),(20,30),(21,31),(22,32),(23,25),(24,26),(33,45),(34,46),(35,47),(36,48),(37,41),(38,42),(39,43),(40,44)], [(1,43),(2,44),(3,45),(4,46),(5,47),(6,48),(7,41),(8,42),(9,21),(10,22),(11,23),(12,24),(13,17),(14,18),(15,19),(16,20),(25,61),(26,62),(27,63),(28,64),(29,57),(30,58),(31,59),(32,60),(33,50),(34,51),(35,52),(36,53),(37,54),(38,55),(39,56),(40,49)], [(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)], [(1,24,47,16),(2,19,48,11),(3,22,41,14),(4,17,42,9),(5,20,43,12),(6,23,44,15),(7,18,45,10),(8,21,46,13),(25,40,57,53),(26,35,58,56),(27,38,59,51),(28,33,60,54),(29,36,61,49),(30,39,62,52),(31,34,63,55),(32,37,64,50)]])
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 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | - | + | + | - | ||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C4 | D4 | D4 | Q8 | D4 | C4○D4 | C8⋊C22 | C8.C22 |
| kernel | C24.67D4 | C22.4Q16 | C23.7Q8 | C2×C4.Q8 | C2×C2.D8 | C22×M4(2) | C2×M4(2) | C2×C8 | C22×C4 | C22×C4 | C24 | C2×C4 | C22 | C22 |
| # reps | 1 | 2 | 2 | 1 | 1 | 1 | 8 | 4 | 1 | 2 | 1 | 4 | 2 | 2 |
Matrix representation of C24.67D4 ►in GL8(𝔽17)
| 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 8 | 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 | 13 | 4 | 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 | 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 |
| 16 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 16 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 4 | 2 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 13 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 13 | 4 | 1 | 15 |
| 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 | 0 | 13 |
| 13 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 4 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 16 | 8 | 0 | 0 | 0 | 0 |
| 0 | 0 | 4 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 11 | 6 | 9 | 14 |
| 0 | 0 | 0 | 0 | 13 | 4 | 11 | 15 |
| 0 | 0 | 0 | 0 | 1 | 7 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 | 4 | 2 |
G:=sub<GL(8,GF(17))| [1,8,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,1,0,0,13,0,0,0,0,0,1,0,4,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,16],[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],[16,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,16],[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],[16,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,2,13,0,0,0,0,0,0,0,0,0,13,0,1,0,0,0,0,0,4,1,0,0,0,0,0,1,1,0,0,0,0,0,0,0,15,0,13],[13,0,0,0,0,0,0,0,1,4,0,0,0,0,0,0,0,0,16,4,0,0,0,0,0,0,8,1,0,0,0,0,0,0,0,0,11,13,1,1,0,0,0,0,6,4,7,0,0,0,0,0,9,11,0,4,0,0,0,0,14,15,0,2] >;
C24.67D4 in GAP, Magma, Sage, TeX
C_2^4._{67}D_4 % in TeX
G:=Group("C2^4.67D4"); // GroupNames label
G:=SmallGroup(128,541);
// by ID
G=gap.SmallGroup(128,541);
# by ID
G:=PCGroup([7,-2,2,2,-2,2,2,-2,224,141,64,422,723,2019,248]);
// Polycyclic
G:=Group<a,b,c,d,e,f|a^2=b^2=c^2=d^2=1,e^4=d,f^2=d*c=c*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*e=e*c,c*f=f*c,d*e=e*d,d*f=f*d,f*e*f^-1=e^3>;
// generators/relations