p-group, metabelian, nilpotent (class 2), monomial
Aliases: C42.674C23, D4○(C4⋊C8), Q8○(C4⋊C8), C4⋊3(C8○D4), D4.8(C4⋊C4), C4○D4.58D4, (C4×D4).22C4, Q8.8(C4⋊C4), C4○D4.10Q8, (C4×Q8).21C4, C4.61(C22×Q8), C4⋊C8.353C22, C42.202(C2×C4), (C2×C4).635C24, (C2×C8).471C23, C4.187(C22×D4), C4⋊M4(2)⋊31C2, C2.9(Q8○M4(2)), C22.164(C23×C4), C23.139(C22×C4), (C22×C8).428C22, (C2×C42).753C22, (C22×C4).1503C23, C42.6C22⋊27C2, C42⋊C2.284C22, (C2×M4(2)).338C22, C4○D4○(C4⋊C8), (C2×Q8)○(C4⋊C8), (C2×C4⋊C8)⋊43C2, C4.20(C2×C4⋊C4), C2.9(C2×C8○D4), C22.2(C2×C4⋊C4), C4⋊C4.214(C2×C4), (C4×C4○D4).10C2, (C2×C8○D4).21C2, C2.21(C22×C4⋊C4), (C2×C4).313(C2×Q8), (C2×D4).247(C2×C4), (C2×C4).1080(C2×D4), C22⋊C4.65(C2×C4), (C2×Q8).224(C2×C4), (C22×C4).329(C2×C4), (C2×C4).249(C22×C4), (C2×C4○D4).339C22, C4⋊C8○(C2×C4○D4), SmallGroup(128,1638)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Subgroups: 316 in 242 conjugacy classes, 174 normal (18 characteristic)
C1, C2 [×3], C2 [×6], C4 [×2], C4 [×8], C4 [×7], C22, C22 [×6], C22 [×6], C8 [×8], C2×C4 [×3], C2×C4 [×21], C2×C4 [×9], D4 [×12], Q8 [×4], C23 [×3], C42, C42 [×9], C22⋊C4 [×6], C4⋊C4 [×6], C2×C8 [×8], C2×C8 [×12], M4(2) [×12], C22×C4 [×9], C2×D4 [×3], C2×Q8, C4○D4 [×8], C4⋊C8, C4⋊C8 [×15], C2×C42 [×3], C42⋊C2 [×3], C4×D4 [×6], C4×Q8 [×2], C22×C8 [×6], C2×M4(2) [×6], C8○D4 [×8], C2×C4○D4, C2×C4⋊C8 [×3], C4⋊M4(2) [×3], C42.6C22 [×6], C4×C4○D4, C2×C8○D4 [×2], C42.674C23
Quotients:
C1, C2 [×15], C4 [×8], C22 [×35], C2×C4 [×28], D4 [×4], Q8 [×4], C23 [×15], C4⋊C4 [×16], C22×C4 [×14], C2×D4 [×6], C2×Q8 [×6], C24, C2×C4⋊C4 [×12], C8○D4 [×2], C23×C4, C22×D4, C22×Q8, C22×C4⋊C4, C2×C8○D4, Q8○M4(2), C42.674C23
Generators and relations
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, cd=dc, ce=ec, ede=b2d >
►On 64 points
(1 63 55 47)(2 48 56 64)(3 57 49 41)(4 42 50 58)(5 59 51 43)(6 44 52 60)(7 61 53 45)(8 46 54 62)(9 26 38 18)(10 19 39 27)(11 28 40 20)(12 21 33 29)(13 30 34 22)(14 23 35 31)(15 32 36 24)(16 17 37 25)
(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)
(1 21)(2 22)(3 23)(4 24)(5 17)(6 18)(7 19)(8 20)(9 44)(10 45)(11 46)(12 47)(13 48)(14 41)(15 42)(16 43)(25 51)(26 52)(27 53)(28 54)(29 55)(30 56)(31 49)(32 50)(33 63)(34 64)(35 57)(36 58)(37 59)(38 60)(39 61)(40 62)
(1 23)(2 24)(3 17)(4 18)(5 19)(6 20)(7 21)(8 22)(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 61)(34 62)(35 63)(36 64)(37 57)(38 58)(39 59)(40 60)
G:=sub<Sym(64)| (1,63,55,47)(2,48,56,64)(3,57,49,41)(4,42,50,58)(5,59,51,43)(6,44,52,60)(7,61,53,45)(8,46,54,62)(9,26,38,18)(10,19,39,27)(11,28,40,20)(12,21,33,29)(13,30,34,22)(14,23,35,31)(15,32,36,24)(16,17,37,25), (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), (1,21)(2,22)(3,23)(4,24)(5,17)(6,18)(7,19)(8,20)(9,44)(10,45)(11,46)(12,47)(13,48)(14,41)(15,42)(16,43)(25,51)(26,52)(27,53)(28,54)(29,55)(30,56)(31,49)(32,50)(33,63)(34,64)(35,57)(36,58)(37,59)(38,60)(39,61)(40,62), (1,23)(2,24)(3,17)(4,18)(5,19)(6,20)(7,21)(8,22)(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)(33,61)(34,62)(35,63)(36,64)(37,57)(38,58)(39,59)(40,60)>;
G:=Group( (1,63,55,47)(2,48,56,64)(3,57,49,41)(4,42,50,58)(5,59,51,43)(6,44,52,60)(7,61,53,45)(8,46,54,62)(9,26,38,18)(10,19,39,27)(11,28,40,20)(12,21,33,29)(13,30,34,22)(14,23,35,31)(15,32,36,24)(16,17,37,25), (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), (1,21)(2,22)(3,23)(4,24)(5,17)(6,18)(7,19)(8,20)(9,44)(10,45)(11,46)(12,47)(13,48)(14,41)(15,42)(16,43)(25,51)(26,52)(27,53)(28,54)(29,55)(30,56)(31,49)(32,50)(33,63)(34,64)(35,57)(36,58)(37,59)(38,60)(39,61)(40,62), (1,23)(2,24)(3,17)(4,18)(5,19)(6,20)(7,21)(8,22)(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)(33,61)(34,62)(35,63)(36,64)(37,57)(38,58)(39,59)(40,60) );
G=PermutationGroup([(1,63,55,47),(2,48,56,64),(3,57,49,41),(4,42,50,58),(5,59,51,43),(6,44,52,60),(7,61,53,45),(8,46,54,62),(9,26,38,18),(10,19,39,27),(11,28,40,20),(12,21,33,29),(13,30,34,22),(14,23,35,31),(15,32,36,24),(16,17,37,25)], [(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)], [(1,21),(2,22),(3,23),(4,24),(5,17),(6,18),(7,19),(8,20),(9,44),(10,45),(11,46),(12,47),(13,48),(14,41),(15,42),(16,43),(25,51),(26,52),(27,53),(28,54),(29,55),(30,56),(31,49),(32,50),(33,63),(34,64),(35,57),(36,58),(37,59),(38,60),(39,61),(40,62)], [(1,23),(2,24),(3,17),(4,18),(5,19),(6,20),(7,21),(8,22),(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),(33,61),(34,62),(35,63),(36,64),(37,57),(38,58),(39,59),(40,60)])
Matrix representation ►G ⊆ GL4(𝔽17) generated by
| 16 | 0 | 0 | 0 |
| 0 | 16 | 0 | 0 |
| 0 | 0 | 13 | 0 |
| 0 | 0 | 9 | 4 |
| 4 | 0 | 0 | 0 |
| 0 | 4 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 2 | 0 | 0 | 0 |
| 0 | 2 | 0 | 0 |
| 0 | 0 | 1 | 16 |
| 0 | 0 | 0 | 16 |
| 13 | 9 | 0 | 0 |
| 4 | 4 | 0 | 0 |
| 0 | 0 | 16 | 0 |
| 0 | 0 | 0 | 16 |
| 16 | 15 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 16 | 0 |
| 0 | 0 | 0 | 16 |
G:=sub<GL(4,GF(17))| [16,0,0,0,0,16,0,0,0,0,13,9,0,0,0,4],[4,0,0,0,0,4,0,0,0,0,1,0,0,0,0,1],[2,0,0,0,0,2,0,0,0,0,1,0,0,0,16,16],[13,4,0,0,9,4,0,0,0,0,16,0,0,0,0,16],[16,0,0,0,15,1,0,0,0,0,16,0,0,0,0,16] >;
50 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | ··· | 2I | 4A | 4B | 4C | 4D | 4E | ··· | 4N | 4O | ··· | 4T | 8A | ··· | 8H | 8I | ··· | 8T |
| order | 1 | 2 | 2 | 2 | 2 | ··· | 2 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 4 | ··· | 4 | 8 | ··· | 8 | 8 | ··· | 8 |
| size | 1 | 1 | 1 | 1 | 2 | ··· | 2 | 1 | 1 | 1 | 1 | 2 | ··· | 2 | 4 | ··· | 4 | 2 | ··· | 2 | 4 | ··· | 4 |
50 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 4 |
| type | + | + | + | + | + | + | + | - | ||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C4 | C4 | D4 | Q8 | C8○D4 | Q8○M4(2) |
| kernel | C42.674C23 | C2×C4⋊C8 | C4⋊M4(2) | C42.6C22 | C4×C4○D4 | C2×C8○D4 | C4×D4 | C4×Q8 | C4○D4 | C4○D4 | C4 | C2 |
| # reps | 1 | 3 | 3 | 6 | 1 | 2 | 12 | 4 | 4 | 4 | 8 | 2 |
In GAP, Magma, Sage, TeX
C_4^2._{674}C_2^3 % in TeX
G:=Group("C4^2.674C2^3"); // GroupNames label
G:=SmallGroup(128,1638);
// by ID
G=gap.SmallGroup(128,1638);
# by ID
G:=PCGroup([7,-2,2,2,2,-2,2,-2,224,253,120,521,124]);
// 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,c*d=d*c,c*e=e*c,e*d*e=b^2*d>;
// generators/relations