p-group, metabelian, nilpotent (class 3), monomial
Aliases: 2+ 1+4.2C4, (C2×D4).68D4, (C2×Q8).65D4, C4.55C22≀C2, (C22×C8)⋊3C22, (C22×C4).57D4, C23.4(C22×C4), C23.5(C22⋊C4), C2.C25.1C2, C2.16(C24⋊3C4), (C2×M4(2))⋊38C22, (C22×C4).656C23, M4(2).8C22⋊7C2, (C2×C4).31(C2×D4), (C2×D4).63(C2×C4), (C2×C4).9(C22⋊C4), (C2×C4○D4).4C22, (C22×C8)⋊C2⋊20C2, C22.10(C2×C22⋊C4), SmallGroup(128,523)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for 2+ 1+4.2C4
G = < a,b,c,d,e | a4=b2=d2=1, c2=e4=a2, bab=a-1, ac=ca, ad=da, eae-1=a-1cd, ece-1=bc=cb, bd=db, be=eb, dcd=a2c, ede-1=a2bd >
Subgroups: 540 in 274 conjugacy classes, 66 normal (8 characteristic)
C1, C2, C2, C4, C4, C4, C22, C22, C8, C2×C4, C2×C4, D4, Q8, C23, C23, C23, C2×C8, M4(2), C22×C4, C22×C4, C22×C4, C2×D4, C2×D4, C2×Q8, C2×Q8, C4○D4, C22⋊C8, C4.D4, C4.10D4, C22×C8, C2×M4(2), C2×C4○D4, C2×C4○D4, 2+ 1+4, 2+ 1+4, 2- 1+4, (C22×C8)⋊C2, M4(2).8C22, C2.C25, 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, 2+ 1+4.2C4
►On 32 points
(1 25 5 29)(2 10 6 14)(3 31 7 27)(4 16 8 12)(9 23 13 19)(11 21 15 17)(18 28 22 32)(20 26 24 30)
(1 31)(2 32)(3 25)(4 26)(5 27)(6 28)(7 29)(8 30)(9 21)(10 22)(11 23)(12 24)(13 17)(14 18)(15 19)(16 20)
(1 19 5 23)(2 16 6 12)(3 21 7 17)(4 10 8 14)(9 29 13 25)(11 31 15 27)(18 26 22 30)(20 28 24 32)
(2 28)(4 30)(6 32)(8 26)(9 13)(10 22)(11 15)(12 24)(14 18)(16 20)(17 21)(19 23)
(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)
G:=sub<Sym(32)| (1,25,5,29)(2,10,6,14)(3,31,7,27)(4,16,8,12)(9,23,13,19)(11,21,15,17)(18,28,22,32)(20,26,24,30), (1,31)(2,32)(3,25)(4,26)(5,27)(6,28)(7,29)(8,30)(9,21)(10,22)(11,23)(12,24)(13,17)(14,18)(15,19)(16,20), (1,19,5,23)(2,16,6,12)(3,21,7,17)(4,10,8,14)(9,29,13,25)(11,31,15,27)(18,26,22,30)(20,28,24,32), (2,28)(4,30)(6,32)(8,26)(9,13)(10,22)(11,15)(12,24)(14,18)(16,20)(17,21)(19,23), (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)>;
G:=Group( (1,25,5,29)(2,10,6,14)(3,31,7,27)(4,16,8,12)(9,23,13,19)(11,21,15,17)(18,28,22,32)(20,26,24,30), (1,31)(2,32)(3,25)(4,26)(5,27)(6,28)(7,29)(8,30)(9,21)(10,22)(11,23)(12,24)(13,17)(14,18)(15,19)(16,20), (1,19,5,23)(2,16,6,12)(3,21,7,17)(4,10,8,14)(9,29,13,25)(11,31,15,27)(18,26,22,30)(20,28,24,32), (2,28)(4,30)(6,32)(8,26)(9,13)(10,22)(11,15)(12,24)(14,18)(16,20)(17,21)(19,23), (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) );
G=PermutationGroup([[(1,25,5,29),(2,10,6,14),(3,31,7,27),(4,16,8,12),(9,23,13,19),(11,21,15,17),(18,28,22,32),(20,26,24,30)], [(1,31),(2,32),(3,25),(4,26),(5,27),(6,28),(7,29),(8,30),(9,21),(10,22),(11,23),(12,24),(13,17),(14,18),(15,19),(16,20)], [(1,19,5,23),(2,16,6,12),(3,21,7,17),(4,10,8,14),(9,29,13,25),(11,31,15,27),(18,26,22,30),(20,28,24,32)], [(2,28),(4,30),(6,32),(8,26),(9,13),(10,22),(11,15),(12,24),(14,18),(16,20),(17,21),(19,23)], [(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)]])
32 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | ··· | 2J | 4A | 4B | 4C | 4D | 4E | 4F | ··· | 4K | 8A | 8B | 8C | 8D | 8E | ··· | 8J |
| order | 1 | 2 | 2 | 2 | 2 | 2 | ··· | 2 | 4 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 8 | 8 | 8 | 8 | 8 | ··· | 8 |
| size | 1 | 1 | 2 | 2 | 2 | 4 | ··· | 4 | 1 | 1 | 2 | 2 | 2 | 4 | ··· | 4 | 4 | 4 | 4 | 4 | 8 | ··· | 8 |
32 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 4 |
| type | + | + | + | + | + | + | + | ||
| image | C1 | C2 | C2 | C2 | C4 | D4 | D4 | D4 | 2+ 1+4.2C4 |
| kernel | 2+ 1+4.2C4 | (C22×C8)⋊C2 | M4(2).8C22 | C2.C25 | 2+ 1+4 | C22×C4 | C2×D4 | C2×Q8 | C1 |
| # reps | 1 | 3 | 3 | 1 | 8 | 6 | 3 | 3 | 4 |
Matrix representation of 2+ 1+4.2C4 ►in GL4(𝔽17) generated by
| 0 | 0 | 16 | 2 |
| 0 | 0 | 0 | 1 |
| 1 | 15 | 0 | 0 |
| 0 | 16 | 0 | 0 |
| 4 | 9 | 0 | 0 |
| 4 | 13 | 0 | 0 |
| 0 | 0 | 4 | 9 |
| 0 | 0 | 4 | 13 |
| 0 | 0 | 16 | 0 |
| 0 | 0 | 0 | 16 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 13 | 8 | 0 | 0 |
| 13 | 4 | 0 | 0 |
| 0 | 0 | 4 | 9 |
| 0 | 0 | 4 | 13 |
| 14 | 8 | 5 | 9 |
| 13 | 5 | 4 | 14 |
| 12 | 8 | 3 | 9 |
| 13 | 3 | 4 | 12 |
G:=sub<GL(4,GF(17))| [0,0,1,0,0,0,15,16,16,0,0,0,2,1,0,0],[4,4,0,0,9,13,0,0,0,0,4,4,0,0,9,13],[0,0,1,0,0,0,0,1,16,0,0,0,0,16,0,0],[13,13,0,0,8,4,0,0,0,0,4,4,0,0,9,13],[14,13,12,13,8,5,8,3,5,4,3,4,9,14,9,12] >;
2+ 1+4.2C4 in GAP, Magma, Sage, TeX
2_+^{1+4}._2C_4 % in TeX
G:=Group("ES+(2,2).2C4"); // GroupNames label
G:=SmallGroup(128,523);
// by ID
G=gap.SmallGroup(128,523);
# by ID
G:=PCGroup([7,-2,2,2,-2,2,2,-2,224,141,422,2019,521,1411,2028,124]);
// Polycyclic
G:=Group<a,b,c,d,e|a^4=b^2=d^2=1,c^2=e^4=a^2,b*a*b=a^-1,a*c=c*a,a*d=d*a,e*a*e^-1=a^-1*c*d,e*c*e^-1=b*c=c*b,b*d=d*b,b*e=e*b,d*c*d=a^2*c,e*d*e^-1=a^2*b*d>;
// generators/relations