p-group, metabelian, nilpotent (class 3), monomial
Aliases: 2+ 1+4⋊4C4, 2- 1+4⋊3C4, C4○D4.42D4, (C2×D4).69D4, (C2×Q8).67D4, (C22×C8)⋊4C22, C23.31(C2×D4), (C22×C4).59D4, C4.113C22≀C2, C42⋊C22⋊8C2, D4.12(C22⋊C4), C42⋊C2⋊1C22, C22.11C22≀C2, C23.24D4⋊1C2, C23.7(C22⋊C4), C2.C25.2C2, Q8.12(C22⋊C4), C2.19(C24⋊3C4), (C2×M4(2))⋊40C22, C23.C23⋊1C2, (C22×C4).659C23, C4○D4.3(C2×C4), C4.7(C2×C22⋊C4), (C2×D4).65(C2×C4), (C2×C4).977(C2×D4), (C2×C4).5(C22×C4), (C2×Q8).58(C2×C4), (C2×C4○D4).7C22, (C22×C8)⋊C2⋊21C2, (C2×C4).11(C22⋊C4), C22.13(C2×C22⋊C4), SmallGroup(128,526)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for 2+ 1+4⋊4C4
G = < a,b,c,d,e | a4=b2=e4=1, c2=d2=a2, bab=a-1, ebe-1=ac=ca, ad=da, eae-1=a2d, bc=cb, bd=db, dcd-1=a2c, ece-1=a2bd, ede-1=a >
Subgroups: 556 in 274 conjugacy classes, 66 normal (18 characteristic)
C1, C2, C2, C4, C4, C4, C22, C22, C22, C8, C2×C4, C2×C4, C2×C4, D4, D4, Q8, Q8, C23, C23, C23, C42, C22⋊C4, C4⋊C4, C2×C8, M4(2), C22×C4, C22×C4, C22×C4, C2×D4, C2×D4, C2×D4, C2×Q8, C2×Q8, C2×Q8, C4○D4, C4○D4, C22⋊C8, C23⋊C4, D4⋊C4, Q8⋊C4, C4≀C2, C42⋊C2, C22×C8, C2×M4(2), C2×C4○D4, C2×C4○D4, C2×C4○D4, 2+ 1+4, 2+ 1+4, 2- 1+4, 2- 1+4, (C22×C8)⋊C2, C23.C23, C23.24D4, C42⋊C22, C2.C25, 2+ 1+4⋊4C4
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⋊4C4
►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 2)(3 4)(5 6)(7 8)(9 12)(10 11)(13 28)(14 27)(15 26)(16 25)(17 18)(19 20)(21 32)(22 31)(23 30)(24 29)
(1 7 3 5)(2 8 4 6)(9 18 11 20)(10 19 12 17)(13 29 15 31)(14 30 16 32)(21 27 23 25)(22 28 24 26)
(1 12 3 10)(2 9 4 11)(5 17 7 19)(6 18 8 20)(13 30 15 32)(14 31 16 29)(21 28 23 26)(22 25 24 27)
(1 29 10 30)(2 14 9 13)(3 31 12 32)(4 16 11 15)(5 27 17 26)(6 22 20 23)(7 25 19 28)(8 24 18 21)
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), (1,2)(3,4)(5,6)(7,8)(9,12)(10,11)(13,28)(14,27)(15,26)(16,25)(17,18)(19,20)(21,32)(22,31)(23,30)(24,29), (1,7,3,5)(2,8,4,6)(9,18,11,20)(10,19,12,17)(13,29,15,31)(14,30,16,32)(21,27,23,25)(22,28,24,26), (1,12,3,10)(2,9,4,11)(5,17,7,19)(6,18,8,20)(13,30,15,32)(14,31,16,29)(21,28,23,26)(22,25,24,27), (1,29,10,30)(2,14,9,13)(3,31,12,32)(4,16,11,15)(5,27,17,26)(6,22,20,23)(7,25,19,28)(8,24,18,21)>;
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), (1,2)(3,4)(5,6)(7,8)(9,12)(10,11)(13,28)(14,27)(15,26)(16,25)(17,18)(19,20)(21,32)(22,31)(23,30)(24,29), (1,7,3,5)(2,8,4,6)(9,18,11,20)(10,19,12,17)(13,29,15,31)(14,30,16,32)(21,27,23,25)(22,28,24,26), (1,12,3,10)(2,9,4,11)(5,17,7,19)(6,18,8,20)(13,30,15,32)(14,31,16,29)(21,28,23,26)(22,25,24,27), (1,29,10,30)(2,14,9,13)(3,31,12,32)(4,16,11,15)(5,27,17,26)(6,22,20,23)(7,25,19,28)(8,24,18,21) );
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)], [(1,2),(3,4),(5,6),(7,8),(9,12),(10,11),(13,28),(14,27),(15,26),(16,25),(17,18),(19,20),(21,32),(22,31),(23,30),(24,29)], [(1,7,3,5),(2,8,4,6),(9,18,11,20),(10,19,12,17),(13,29,15,31),(14,30,16,32),(21,27,23,25),(22,28,24,26)], [(1,12,3,10),(2,9,4,11),(5,17,7,19),(6,18,8,20),(13,30,15,32),(14,31,16,29),(21,28,23,26),(22,25,24,27)], [(1,29,10,30),(2,14,9,13),(3,31,12,32),(4,16,11,15),(5,27,17,26),(6,22,20,23),(7,25,19,28),(8,24,18,21)]])
32 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | ··· | 2J | 4A | 4B | 4C | 4D | 4E | 4F | ··· | 4K | 4L | 4M | 4N | 4O | 8A | 8B | 8C | 8D | 8E | 8F |
| order | 1 | 2 | 2 | 2 | 2 | 2 | ··· | 2 | 4 | 4 | 4 | 4 | 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 | 8 | 8 | 8 | 8 | 4 | 4 | 4 | 4 | 8 | 8 |
32 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | |||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C4 | C4 | D4 | D4 | D4 | D4 | 2+ 1+4⋊4C4 |
| kernel | 2+ 1+4⋊4C4 | (C22×C8)⋊C2 | C23.C23 | C23.24D4 | C42⋊C22 | C2.C25 | 2+ 1+4 | 2- 1+4 | C22×C4 | C2×D4 | C2×Q8 | C4○D4 | C1 |
| # reps | 1 | 1 | 1 | 2 | 2 | 1 | 4 | 4 | 2 | 4 | 2 | 4 | 4 |
Matrix representation of 2+ 1+4⋊4C4 ►in GL4(𝔽17) generated by
| 16 | 0 | 0 | 8 |
| 4 | 0 | 13 | 1 |
| 0 | 13 | 0 | 4 |
| 4 | 0 | 0 | 1 |
| 1 | 0 | 0 | 9 |
| 0 | 0 | 13 | 16 |
| 0 | 4 | 0 | 13 |
| 0 | 0 | 0 | 16 |
| 1 | 0 | 15 | 0 |
| 13 | 0 | 4 | 16 |
| 1 | 0 | 16 | 0 |
| 0 | 1 | 4 | 0 |
| 13 | 0 | 8 | 0 |
| 0 | 0 | 1 | 13 |
| 0 | 0 | 4 | 0 |
| 0 | 13 | 1 | 0 |
| 8 | 2 | 15 | 2 |
| 0 | 14 | 5 | 12 |
| 5 | 2 | 15 | 0 |
| 5 | 14 | 5 | 14 |
G:=sub<GL(4,GF(17))| [16,4,0,4,0,0,13,0,0,13,0,0,8,1,4,1],[1,0,0,0,0,0,4,0,0,13,0,0,9,16,13,16],[1,13,1,0,0,0,0,1,15,4,16,4,0,16,0,0],[13,0,0,0,0,0,0,13,8,1,4,1,0,13,0,0],[8,0,5,5,2,14,2,14,15,5,15,5,2,12,0,14] >;
2+ 1+4⋊4C4 in GAP, Magma, Sage, TeX
2_+^{1+4}\rtimes_4C_4 % in TeX
G:=Group("ES+(2,2):4C4"); // GroupNames label
G:=SmallGroup(128,526);
// by ID
G=gap.SmallGroup(128,526);
# by ID
G:=PCGroup([7,-2,2,2,-2,2,2,-2,224,141,422,2019,1018,521,248,1411,124]);
// 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,e*b*e^-1=a*c=c*a,a*d=d*a,e*a*e^-1=a^2*d,b*c=c*b,b*d=d*b,d*c*d^-1=a^2*c,e*c*e^-1=a^2*b*d,e*d*e^-1=a>;
// generators/relations