p-group, metabelian, nilpotent (class 3), monomial
Aliases: (C2×C4).27D8, (C2×C8).59D4, (C2×D4).15Q8, C42⋊9C4⋊7C2, C22.92(C2×D8), C2.11(C8⋊4D4), C23.940(C2×D4), (C22×C4).324D4, C4.40(C22⋊Q8), C2.10(D4⋊Q8), C2.16(C8.2D4), C22.79(C4⋊1D4), (C22×C8).123C22, (C2×C42).391C22, (C22×D4).96C22, C2.6(C23.4Q8), C22.7C42⋊19C2, (C22×C4).1474C23, C4.34(C22.D4), C2.10(C22.D8), C22.117(C22⋊Q8), C22.147(C8.C22), C24.3C22.22C2, C22.127(C22.D4), (C2×C2.D8)⋊14C2, (C2×C4).749(C2×D4), (C2×C4).291(C2×Q8), (C2×D4⋊C4).23C2, (C2×C4).780(C4○D4), (C2×C4⋊C4).163C22, SmallGroup(128,825)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for (C2×C4).27D8
G = < a,b,c,d | a2=b4=c8=1, d2=ab2, ab=ba, ac=ca, ad=da, cbc-1=ab-1, dbd-1=b-1, dcd-1=c-1 >
Subgroups: 352 in 145 conjugacy classes, 54 normal (18 characteristic)
C1, C2, C2, C2, C4, C4, C22, C22, C22, C8, C2×C4, C2×C4, C2×C4, D4, C23, C23, C42, C22⋊C4, C4⋊C4, C2×C8, C2×C8, C22×C4, C22×C4, C22×C4, C2×D4, C2×D4, C24, D4⋊C4, C2.D8, C2×C42, C2×C22⋊C4, C2×C4⋊C4, C2×C4⋊C4, C2×C4⋊C4, C22×C8, C22×D4, C22.7C42, C42⋊9C4, C24.3C22, C2×D4⋊C4, C2×C2.D8, (C2×C4).27D8
Quotients: C1, C2, C22, D4, Q8, C23, D8, C2×D4, C2×Q8, C4○D4, C22⋊Q8, C22.D4, C4⋊1D4, C2×D8, C8.C22, C23.4Q8, D4⋊Q8, C22.D8, C8⋊4D4, C8.2D4, (C2×C4).27D8
►On 64 points
(1 24)(2 17)(3 18)(4 19)(5 20)(6 21)(7 22)(8 23)(9 53)(10 54)(11 55)(12 56)(13 49)(14 50)(15 51)(16 52)(25 41)(26 42)(27 43)(28 44)(29 45)(30 46)(31 47)(32 48)(33 61)(34 62)(35 63)(36 64)(37 57)(38 58)(39 59)(40 60)
(1 49 39 29)(2 46 40 14)(3 51 33 31)(4 48 34 16)(5 53 35 25)(6 42 36 10)(7 55 37 27)(8 44 38 12)(9 63 41 20)(11 57 43 22)(13 59 45 24)(15 61 47 18)(17 30 60 50)(19 32 62 52)(21 26 64 54)(23 28 58 56)
(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 28 59 12)(2 27 60 11)(3 26 61 10)(4 25 62 9)(5 32 63 16)(6 31 64 15)(7 30 57 14)(8 29 58 13)(17 43 40 55)(18 42 33 54)(19 41 34 53)(20 48 35 52)(21 47 36 51)(22 46 37 50)(23 45 38 49)(24 44 39 56)
G:=sub<Sym(64)| (1,24)(2,17)(3,18)(4,19)(5,20)(6,21)(7,22)(8,23)(9,53)(10,54)(11,55)(12,56)(13,49)(14,50)(15,51)(16,52)(25,41)(26,42)(27,43)(28,44)(29,45)(30,46)(31,47)(32,48)(33,61)(34,62)(35,63)(36,64)(37,57)(38,58)(39,59)(40,60), (1,49,39,29)(2,46,40,14)(3,51,33,31)(4,48,34,16)(5,53,35,25)(6,42,36,10)(7,55,37,27)(8,44,38,12)(9,63,41,20)(11,57,43,22)(13,59,45,24)(15,61,47,18)(17,30,60,50)(19,32,62,52)(21,26,64,54)(23,28,58,56), (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), (1,28,59,12)(2,27,60,11)(3,26,61,10)(4,25,62,9)(5,32,63,16)(6,31,64,15)(7,30,57,14)(8,29,58,13)(17,43,40,55)(18,42,33,54)(19,41,34,53)(20,48,35,52)(21,47,36,51)(22,46,37,50)(23,45,38,49)(24,44,39,56)>;
G:=Group( (1,24)(2,17)(3,18)(4,19)(5,20)(6,21)(7,22)(8,23)(9,53)(10,54)(11,55)(12,56)(13,49)(14,50)(15,51)(16,52)(25,41)(26,42)(27,43)(28,44)(29,45)(30,46)(31,47)(32,48)(33,61)(34,62)(35,63)(36,64)(37,57)(38,58)(39,59)(40,60), (1,49,39,29)(2,46,40,14)(3,51,33,31)(4,48,34,16)(5,53,35,25)(6,42,36,10)(7,55,37,27)(8,44,38,12)(9,63,41,20)(11,57,43,22)(13,59,45,24)(15,61,47,18)(17,30,60,50)(19,32,62,52)(21,26,64,54)(23,28,58,56), (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), (1,28,59,12)(2,27,60,11)(3,26,61,10)(4,25,62,9)(5,32,63,16)(6,31,64,15)(7,30,57,14)(8,29,58,13)(17,43,40,55)(18,42,33,54)(19,41,34,53)(20,48,35,52)(21,47,36,51)(22,46,37,50)(23,45,38,49)(24,44,39,56) );
G=PermutationGroup([[(1,24),(2,17),(3,18),(4,19),(5,20),(6,21),(7,22),(8,23),(9,53),(10,54),(11,55),(12,56),(13,49),(14,50),(15,51),(16,52),(25,41),(26,42),(27,43),(28,44),(29,45),(30,46),(31,47),(32,48),(33,61),(34,62),(35,63),(36,64),(37,57),(38,58),(39,59),(40,60)], [(1,49,39,29),(2,46,40,14),(3,51,33,31),(4,48,34,16),(5,53,35,25),(6,42,36,10),(7,55,37,27),(8,44,38,12),(9,63,41,20),(11,57,43,22),(13,59,45,24),(15,61,47,18),(17,30,60,50),(19,32,62,52),(21,26,64,54),(23,28,58,56)], [(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)], [(1,28,59,12),(2,27,60,11),(3,26,61,10),(4,25,62,9),(5,32,63,16),(6,31,64,15),(7,30,57,14),(8,29,58,13),(17,43,40,55),(18,42,33,54),(19,41,34,53),(20,48,35,52),(21,47,36,51),(22,46,37,50),(23,45,38,49),(24,44,39,56)]])
32 conjugacy classes
| class | 1 | 2A | ··· | 2G | 2H | 2I | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | ··· | 4N | 8A | ··· | 8H |
| order | 1 | 2 | ··· | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 8 | ··· | 8 |
| size | 1 | 1 | ··· | 1 | 8 | 8 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 8 | ··· | 8 | 4 | ··· | 4 |
32 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 4 |
| type | + | + | + | + | + | + | + | + | - | + | - | |
| image | C1 | C2 | C2 | C2 | C2 | C2 | D4 | D4 | Q8 | D8 | C4○D4 | C8.C22 |
| kernel | (C2×C4).27D8 | C22.7C42 | C42⋊9C4 | C24.3C22 | C2×D4⋊C4 | C2×C2.D8 | C2×C8 | C22×C4 | C2×D4 | C2×C4 | C2×C4 | C22 |
| # reps | 1 | 1 | 1 | 1 | 2 | 2 | 4 | 2 | 2 | 8 | 6 | 2 |
Matrix representation of (C2×C4).27D8 ►in GL6(𝔽17)
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 16 | 0 | 0 | 0 |
| 0 | 0 | 0 | 16 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 16 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 15 | 0 | 0 |
| 0 | 0 | 1 | 16 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 16 | 10 | 0 | 0 | 0 | 0 |
| 10 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 6 | 11 | 0 | 0 |
| 0 | 0 | 3 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 3 | 3 |
| 0 | 0 | 0 | 0 | 14 | 3 |
| 7 | 16 | 0 | 0 | 0 | 0 |
| 16 | 10 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 11 | 0 | 0 |
| 0 | 0 | 14 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 3 | 3 |
| 0 | 0 | 0 | 0 | 3 | 14 |
G:=sub<GL(6,GF(17))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[0,16,0,0,0,0,1,0,0,0,0,0,0,0,1,1,0,0,0,0,15,16,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[16,10,0,0,0,0,10,1,0,0,0,0,0,0,6,3,0,0,0,0,11,0,0,0,0,0,0,0,3,14,0,0,0,0,3,3],[7,16,0,0,0,0,16,10,0,0,0,0,0,0,0,14,0,0,0,0,11,0,0,0,0,0,0,0,3,3,0,0,0,0,3,14] >;
(C2×C4).27D8 in GAP, Magma, Sage, TeX
(C_2\times C_4)._{27}D_8 % in TeX
G:=Group("(C2xC4).27D8"); // GroupNames label
G:=SmallGroup(128,825);
// by ID
G=gap.SmallGroup(128,825);
# by ID
G:=PCGroup([7,-2,2,2,-2,2,2,-2,56,141,624,422,387,1018,521,248,1411,718,172]);
// Polycyclic
G:=Group<a,b,c,d|a^2=b^4=c^8=1,d^2=a*b^2,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c^-1=a*b^-1,d*b*d^-1=b^-1,d*c*d^-1=c^-1>;
// generators/relations