p-group, metabelian, nilpotent (class 3), monomial
Aliases: 2- 1+4⋊2C4, C4○D4.2D4, (C2×Q8).66D4, C4.57C22≀C2, (C2×D4).266D4, (C22×C4).58D4, C23.546(C2×D4), D4.11(C22⋊C4), C22.76C22≀C2, C2.1(D4.8D4), Q8.11(C22⋊C4), C2.18(C24⋊3C4), C2.1(D4.10D4), (C22×Q8).7C22, C23.36D4⋊30C2, (C22×C4).658C23, (C2×C42).243C22, (C2×2- 1+4).1C2, C23.67C23⋊1C2, (C2×M4(2)).149C22, (C2×C4≀C2)⋊11C2, C4○D4.2(C2×C4), C4.6(C2×C22⋊C4), (C2×C4).976(C2×D4), (C2×C4).4(C22×C4), (C2×Q8).57(C2×C4), (C2×C4⋊C4).31C22, (C2×C4○D4).6C22, (C2×C4.10D4)⋊15C2, (C2×C4).10(C22⋊C4), C22.12(C2×C22⋊C4), SmallGroup(128,525)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for 2- 1+4⋊2C4
G = < a,b,c,d,e | a4=b2=e4=1, c2=d2=a2, bab=a-1, ac=ca, ad=da, eae-1=abc, bc=cb, bd=db, be=eb, dcd-1=ece-1=a2c, ede-1=a2bd >
Subgroups: 524 in 276 conjugacy classes, 68 normal (18 characteristic)
C1, C2, C2, C4, C4, C22, C22, C8, C2×C4, C2×C4, C2×C4, D4, D4, Q8, Q8, C23, C23, C42, C4⋊C4, C2×C8, M4(2), C22×C4, C22×C4, C22×C4, C2×D4, C2×D4, C2×Q8, C2×Q8, C4○D4, C4○D4, C2.C42, C4.10D4, D4⋊C4, Q8⋊C4, C4≀C2, C2×C42, C2×C4⋊C4, C2×M4(2), C22×Q8, C22×Q8, C2×C4○D4, C2×C4○D4, 2- 1+4, 2- 1+4, C23.67C23, C2×C4.10D4, C23.36D4, C2×C4≀C2, C2×2- 1+4, 2- 1+4⋊2C4
Quotients: C1, C2, C4, C22, C2×C4, D4, C23, C22⋊C4, C22×C4, C2×D4, C2×C22⋊C4, C22≀C2, C24⋊3C4, D4.8D4, D4.10D4, 2- 1+4⋊2C4
►On 32 points
(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)
(1 25)(2 28)(3 27)(4 26)(5 17)(6 20)(7 19)(8 18)(9 32)(10 31)(11 30)(12 29)(13 24)(14 23)(15 22)(16 21)
(1 14 3 16)(2 15 4 13)(5 32 7 30)(6 29 8 31)(9 19 11 17)(10 20 12 18)(21 25 23 27)(22 26 24 28)
(1 31 3 29)(2 32 4 30)(5 13 7 15)(6 14 8 16)(9 26 11 28)(10 27 12 25)(17 24 19 22)(18 21 20 23)
(1 10 25 31)(2 5 26 19)(3 12 27 29)(4 7 28 17)(6 16 20 21)(8 14 18 23)(9 15 30 24)(11 13 32 22)
G:=sub<Sym(32)| (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), (1,25)(2,28)(3,27)(4,26)(5,17)(6,20)(7,19)(8,18)(9,32)(10,31)(11,30)(12,29)(13,24)(14,23)(15,22)(16,21), (1,14,3,16)(2,15,4,13)(5,32,7,30)(6,29,8,31)(9,19,11,17)(10,20,12,18)(21,25,23,27)(22,26,24,28), (1,31,3,29)(2,32,4,30)(5,13,7,15)(6,14,8,16)(9,26,11,28)(10,27,12,25)(17,24,19,22)(18,21,20,23), (1,10,25,31)(2,5,26,19)(3,12,27,29)(4,7,28,17)(6,16,20,21)(8,14,18,23)(9,15,30,24)(11,13,32,22)>;
G:=Group( (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), (1,25)(2,28)(3,27)(4,26)(5,17)(6,20)(7,19)(8,18)(9,32)(10,31)(11,30)(12,29)(13,24)(14,23)(15,22)(16,21), (1,14,3,16)(2,15,4,13)(5,32,7,30)(6,29,8,31)(9,19,11,17)(10,20,12,18)(21,25,23,27)(22,26,24,28), (1,31,3,29)(2,32,4,30)(5,13,7,15)(6,14,8,16)(9,26,11,28)(10,27,12,25)(17,24,19,22)(18,21,20,23), (1,10,25,31)(2,5,26,19)(3,12,27,29)(4,7,28,17)(6,16,20,21)(8,14,18,23)(9,15,30,24)(11,13,32,22) );
G=PermutationGroup([[(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)], [(1,25),(2,28),(3,27),(4,26),(5,17),(6,20),(7,19),(8,18),(9,32),(10,31),(11,30),(12,29),(13,24),(14,23),(15,22),(16,21)], [(1,14,3,16),(2,15,4,13),(5,32,7,30),(6,29,8,31),(9,19,11,17),(10,20,12,18),(21,25,23,27),(22,26,24,28)], [(1,31,3,29),(2,32,4,30),(5,13,7,15),(6,14,8,16),(9,26,11,28),(10,27,12,25),(17,24,19,22),(18,21,20,23)], [(1,10,25,31),(2,5,26,19),(3,12,27,29),(4,7,28,17),(6,16,20,21),(8,14,18,23),(9,15,30,24),(11,13,32,22)]])
32 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 2H | 2I | 4A | 4B | 4C | 4D | 4E | ··· | 4P | 4Q | 4R | 8A | 8B | 8C | 8D |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 4 | 4 | 8 | 8 | 8 | 8 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 4 | 4 | 4 | 4 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 8 | 8 | 8 | 8 | 8 | 8 |
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 | D4 | D4.8D4 | D4.10D4 |
| kernel | 2- 1+4⋊2C4 | C23.67C23 | C2×C4.10D4 | C23.36D4 | C2×C4≀C2 | C2×2- 1+4 | 2- 1+4 | C22×C4 | C2×D4 | C2×Q8 | C4○D4 | C2 | C2 |
| # reps | 1 | 1 | 1 | 2 | 2 | 1 | 8 | 2 | 2 | 4 | 4 | 2 | 2 |
Matrix representation of 2- 1+4⋊2C4 ►in GL6(𝔽17)
| 1 | 1 | 0 | 0 | 0 | 0 |
| 0 | 16 | 0 | 0 | 0 | 0 |
| 0 | 0 | 16 | 0 | 2 | 0 |
| 0 | 0 | 0 | 0 | 1 | 16 |
| 0 | 0 | 16 | 0 | 1 | 0 |
| 0 | 0 | 16 | 1 | 1 | 0 |
| 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 | 16 | 0 | 1 | 0 |
| 0 | 0 | 16 | 0 | 0 | 1 |
| 16 | 0 | 0 | 0 | 0 | 0 |
| 0 | 16 | 0 | 0 | 0 | 0 |
| 0 | 0 | 4 | 0 | 0 | 0 |
| 0 | 0 | 4 | 13 | 0 | 0 |
| 0 | 0 | 0 | 0 | 4 | 0 |
| 0 | 0 | 4 | 0 | 0 | 13 |
| 16 | 0 | 0 | 0 | 0 | 0 |
| 0 | 16 | 0 | 0 | 0 | 0 |
| 0 | 0 | 16 | 2 | 0 | 0 |
| 0 | 0 | 16 | 1 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 16 |
| 0 | 0 | 16 | 1 | 1 | 0 |
| 13 | 0 | 0 | 0 | 0 | 0 |
| 8 | 4 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 15 | 0 | 0 |
| 0 | 0 | 1 | 16 | 0 | 0 |
| 0 | 0 | 7 | 16 | 0 | 4 |
| 0 | 0 | 7 | 16 | 4 | 0 |
G:=sub<GL(6,GF(17))| [1,0,0,0,0,0,1,16,0,0,0,0,0,0,16,0,16,16,0,0,0,0,0,1,0,0,2,1,1,1,0,0,0,16,0,0],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,16,0,16,16,0,0,0,16,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[16,0,0,0,0,0,0,16,0,0,0,0,0,0,4,4,0,4,0,0,0,13,0,0,0,0,0,0,4,0,0,0,0,0,0,13],[16,0,0,0,0,0,0,16,0,0,0,0,0,0,16,16,0,16,0,0,2,1,1,1,0,0,0,0,0,1,0,0,0,0,16,0],[13,8,0,0,0,0,0,4,0,0,0,0,0,0,1,1,7,7,0,0,15,16,16,16,0,0,0,0,0,4,0,0,0,0,4,0] >;
2- 1+4⋊2C4 in GAP, Magma, Sage, TeX
2_-^{1+4}\rtimes_2C_4 % in TeX
G:=Group("ES-(2,2):2C4"); // GroupNames label
G:=SmallGroup(128,525);
// by ID
G=gap.SmallGroup(128,525);
# by ID
G:=PCGroup([7,-2,2,2,-2,2,2,-2,224,141,422,352,2019,1018,521,248,1411]);
// Polycyclic
G:=Group<a,b,c,d,e|a^4=b^2=e^4=1,c^2=d^2=a^2,b*a*b=a^-1,a*c=c*a,a*d=d*a,e*a*e^-1=a*b*c,b*c=c*b,b*d=d*b,b*e=e*b,d*c*d^-1=e*c*e^-1=a^2*c,e*d*e^-1=a^2*b*d>;
// generators/relations