p-group, metabelian, nilpotent (class 2), monomial
Aliases: C42.678C23, (C4×D4).25C4, (C4×Q8).24C4, C4.48(C8○D4), C4⋊C8.231C22, C42.207(C2×C4), (C2×C4).644C24, (C2×C8).400C23, (C4×C8).327C22, C8○2M4(2)⋊29C2, C4⋊M4(2)⋊33C2, C42.6C4⋊46C2, C8⋊C4.154C22, C42.12C4⋊48C2, C4.44(C42⋊C2), C2.13(Q8○M4(2)), C22⋊C8.139C22, (C2×C42).757C22, C23.102(C22×C4), (C22×C4).914C23, (C22×C8).432C22, C22.172(C23×C4), C42.7C22⋊22C2, C42⋊C2.350C22, C22.18(C42⋊C2), (C2×M4(2)).346C22, (C2×C4⋊C8)⋊46C2, C2.13(C2×C8○D4), C4⋊C4.220(C2×C4), C4⋊C8○(C42⋊C2), (C4×C4○D4).13C2, C4.295(C2×C4○D4), (C2×D4).229(C2×C4), C22⋊C4.70(C2×C4), (C2×Q8).207(C2×C4), (C2×C4).829(C4○D4), (C2×C4).260(C22×C4), (C22×C4).338(C2×C4), C2.44(C2×C42⋊C2), (C22×C8)⋊C2.19C2, (C2×C4○D4).282C22, SmallGroup(128,1657)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C42.678C23
G = < a,b,c,d,e | a4=b4=d2=e2=1, c2=b, ab=ba, cac-1=a-1, ad=da, ae=ea, bc=cb, bd=db, be=eb, dcd=a2c, ce=ec, ede=b2d >
Subgroups: 268 in 196 conjugacy classes, 134 normal (38 characteristic)
C1, C2, C2, C4, C4, C4, C22, C22, C22, C8, C2×C4, C2×C4, C2×C4, D4, Q8, C23, C23, C42, C42, C22⋊C4, C4⋊C4, C4⋊C4, C2×C8, C2×C8, M4(2), C22×C4, C22×C4, C2×D4, C2×D4, C2×Q8, C4○D4, C4×C8, C8⋊C4, C22⋊C8, C4⋊C8, C4⋊C8, C2×C42, C2×C42, C42⋊C2, C42⋊C2, C4×D4, C4×D4, C4×Q8, C22×C8, C2×M4(2), C2×C4○D4, C8○2M4(2), (C22×C8)⋊C2, C2×C4⋊C8, C4⋊M4(2), C42.12C4, C42.6C4, C42.7C22, C4×C4○D4, C42.678C23
Quotients: C1, C2, C4, C22, C2×C4, C23, C22×C4, C4○D4, C24, C42⋊C2, C8○D4, C23×C4, C2×C4○D4, C2×C42⋊C2, C2×C8○D4, Q8○M4(2), C42.678C23
►On 64 points
(1 63 55 16)(2 9 56 64)(3 57 49 10)(4 11 50 58)(5 59 51 12)(6 13 52 60)(7 61 53 14)(8 15 54 62)(17 39 31 41)(18 42 32 40)(19 33 25 43)(20 44 26 34)(21 35 27 45)(22 46 28 36)(23 37 29 47)(24 48 30 38)
(1 3 5 7)(2 4 6 8)(9 11 13 15)(10 12 14 16)(17 19 21 23)(18 20 22 24)(25 27 29 31)(26 28 30 32)(33 35 37 39)(34 36 38 40)(41 43 45 47)(42 44 46 48)(49 51 53 55)(50 52 54 56)(57 59 61 63)(58 60 62 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)
(2 56)(4 50)(6 52)(8 54)(9 64)(11 58)(13 60)(15 62)(17 21)(18 28)(19 23)(20 30)(22 32)(24 26)(25 29)(27 31)(33 37)(34 48)(35 39)(36 42)(38 44)(40 46)(41 45)(43 47)
(1 17)(2 18)(3 19)(4 20)(5 21)(6 22)(7 23)(8 24)(9 42)(10 43)(11 44)(12 45)(13 46)(14 47)(15 48)(16 41)(25 49)(26 50)(27 51)(28 52)(29 53)(30 54)(31 55)(32 56)(33 57)(34 58)(35 59)(36 60)(37 61)(38 62)(39 63)(40 64)
G:=sub<Sym(64)| (1,63,55,16)(2,9,56,64)(3,57,49,10)(4,11,50,58)(5,59,51,12)(6,13,52,60)(7,61,53,14)(8,15,54,62)(17,39,31,41)(18,42,32,40)(19,33,25,43)(20,44,26,34)(21,35,27,45)(22,46,28,36)(23,37,29,47)(24,48,30,38), (1,3,5,7)(2,4,6,8)(9,11,13,15)(10,12,14,16)(17,19,21,23)(18,20,22,24)(25,27,29,31)(26,28,30,32)(33,35,37,39)(34,36,38,40)(41,43,45,47)(42,44,46,48)(49,51,53,55)(50,52,54,56)(57,59,61,63)(58,60,62,64), (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), (2,56)(4,50)(6,52)(8,54)(9,64)(11,58)(13,60)(15,62)(17,21)(18,28)(19,23)(20,30)(22,32)(24,26)(25,29)(27,31)(33,37)(34,48)(35,39)(36,42)(38,44)(40,46)(41,45)(43,47), (1,17)(2,18)(3,19)(4,20)(5,21)(6,22)(7,23)(8,24)(9,42)(10,43)(11,44)(12,45)(13,46)(14,47)(15,48)(16,41)(25,49)(26,50)(27,51)(28,52)(29,53)(30,54)(31,55)(32,56)(33,57)(34,58)(35,59)(36,60)(37,61)(38,62)(39,63)(40,64)>;
G:=Group( (1,63,55,16)(2,9,56,64)(3,57,49,10)(4,11,50,58)(5,59,51,12)(6,13,52,60)(7,61,53,14)(8,15,54,62)(17,39,31,41)(18,42,32,40)(19,33,25,43)(20,44,26,34)(21,35,27,45)(22,46,28,36)(23,37,29,47)(24,48,30,38), (1,3,5,7)(2,4,6,8)(9,11,13,15)(10,12,14,16)(17,19,21,23)(18,20,22,24)(25,27,29,31)(26,28,30,32)(33,35,37,39)(34,36,38,40)(41,43,45,47)(42,44,46,48)(49,51,53,55)(50,52,54,56)(57,59,61,63)(58,60,62,64), (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), (2,56)(4,50)(6,52)(8,54)(9,64)(11,58)(13,60)(15,62)(17,21)(18,28)(19,23)(20,30)(22,32)(24,26)(25,29)(27,31)(33,37)(34,48)(35,39)(36,42)(38,44)(40,46)(41,45)(43,47), (1,17)(2,18)(3,19)(4,20)(5,21)(6,22)(7,23)(8,24)(9,42)(10,43)(11,44)(12,45)(13,46)(14,47)(15,48)(16,41)(25,49)(26,50)(27,51)(28,52)(29,53)(30,54)(31,55)(32,56)(33,57)(34,58)(35,59)(36,60)(37,61)(38,62)(39,63)(40,64) );
G=PermutationGroup([[(1,63,55,16),(2,9,56,64),(3,57,49,10),(4,11,50,58),(5,59,51,12),(6,13,52,60),(7,61,53,14),(8,15,54,62),(17,39,31,41),(18,42,32,40),(19,33,25,43),(20,44,26,34),(21,35,27,45),(22,46,28,36),(23,37,29,47),(24,48,30,38)], [(1,3,5,7),(2,4,6,8),(9,11,13,15),(10,12,14,16),(17,19,21,23),(18,20,22,24),(25,27,29,31),(26,28,30,32),(33,35,37,39),(34,36,38,40),(41,43,45,47),(42,44,46,48),(49,51,53,55),(50,52,54,56),(57,59,61,63),(58,60,62,64)], [(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)], [(2,56),(4,50),(6,52),(8,54),(9,64),(11,58),(13,60),(15,62),(17,21),(18,28),(19,23),(20,30),(22,32),(24,26),(25,29),(27,31),(33,37),(34,48),(35,39),(36,42),(38,44),(40,46),(41,45),(43,47)], [(1,17),(2,18),(3,19),(4,20),(5,21),(6,22),(7,23),(8,24),(9,42),(10,43),(11,44),(12,45),(13,46),(14,47),(15,48),(16,41),(25,49),(26,50),(27,51),(28,52),(29,53),(30,54),(31,55),(32,56),(33,57),(34,58),(35,59),(36,60),(37,61),(38,62),(39,63),(40,64)]])
50 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 4A | 4B | 4C | 4D | 4E | ··· | 4R | 4S | 4T | 4U | 4V | 8A | ··· | 8H | 8I | ··· | 8T |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 4 | 4 | 4 | 4 | 8 | ··· | 8 | 8 | ··· | 8 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 4 | 4 | 1 | 1 | 1 | 1 | 2 | ··· | 2 | 4 | 4 | 4 | 4 | 2 | ··· | 2 | 4 | ··· | 4 |
50 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 4 |
| type | + | + | + | + | + | + | + | + | + | |||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C4 | C4 | C4○D4 | C8○D4 | Q8○M4(2) |
| kernel | C42.678C23 | C8○2M4(2) | (C22×C8)⋊C2 | C2×C4⋊C8 | C4⋊M4(2) | C42.12C4 | C42.6C4 | C42.7C22 | C4×C4○D4 | C4×D4 | C4×Q8 | C2×C4 | C4 | C2 |
| # reps | 1 | 2 | 2 | 1 | 1 | 2 | 2 | 4 | 1 | 12 | 4 | 8 | 8 | 2 |
Matrix representation of C42.678C23 ►in GL4(𝔽17) generated by
| 13 | 0 | 0 | 0 |
| 13 | 4 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 16 | 0 | 0 | 0 |
| 0 | 16 | 0 | 0 |
| 0 | 0 | 13 | 0 |
| 0 | 0 | 0 | 13 |
| 1 | 15 | 0 | 0 |
| 1 | 16 | 0 | 0 |
| 0 | 0 | 8 | 0 |
| 0 | 0 | 0 | 8 |
| 1 | 0 | 0 | 0 |
| 1 | 16 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 16 |
| 16 | 0 | 0 | 0 |
| 0 | 16 | 0 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 0 | 1 | 0 |
G:=sub<GL(4,GF(17))| [13,13,0,0,0,4,0,0,0,0,1,0,0,0,0,1],[16,0,0,0,0,16,0,0,0,0,13,0,0,0,0,13],[1,1,0,0,15,16,0,0,0,0,8,0,0,0,0,8],[1,1,0,0,0,16,0,0,0,0,1,0,0,0,0,16],[16,0,0,0,0,16,0,0,0,0,0,1,0,0,1,0] >;
C42.678C23 in GAP, Magma, Sage, TeX
C_4^2._{678}C_2^3 % in TeX
G:=Group("C4^2.678C2^3"); // GroupNames label
G:=SmallGroup(128,1657);
// by ID
G=gap.SmallGroup(128,1657);
# by ID
G:=PCGroup([7,-2,2,2,2,-2,2,-2,224,253,100,1018,521,124]);
// Polycyclic
G:=Group<a,b,c,d,e|a^4=b^4=d^2=e^2=1,c^2=b,a*b=b*a,c*a*c^-1=a^-1,a*d=d*a,a*e=e*a,b*c=c*b,b*d=d*b,b*e=e*b,d*c*d=a^2*c,c*e=e*c,e*d*e=b^2*d>;
// generators/relations