p-group, metabelian, nilpotent (class 3), monomial
Aliases: C24.70D4, C22.27C4≀C2, C42⋊C2⋊19C4, C42⋊6C4⋊14C2, (C22×C4).679D4, C23.551(C2×D4), C4.21(C42⋊C2), C23.77(C22⋊C4), C24.4C4.20C2, (C23×C4).246C22, (C2×C42).252C22, (C22×C4).1337C23, C42⋊C2.268C22, C2.40(C42⋊C22), C4.130(C22.D4), C2.14(C23.34D4), (C2×M4(2)).162C22, C22.44(C22.D4), (C2×C4⋊C4)⋊30C4, C2.40(C2×C4≀C2), C4⋊C4.197(C2×C4), (C2×C4).1518(C2×D4), (C4×C22⋊C4).12C2, (C2×C4).743(C4○D4), (C22×C4).269(C2×C4), (C2×C4).370(C22×C4), (C2×C4).128(C22⋊C4), (C2×C42⋊C2).18C2, C22.260(C2×C22⋊C4), SmallGroup(128,558)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C24.70D4
G = < a,b,c,d,e,f | a2=b2=c2=d2=1, e4=d, f2=bcd, eae-1=faf-1=ab=ba, ac=ca, ad=da, bc=cb, bd=db, be=eb, bf=fb, ece-1=cd=dc, cf=fc, de=ed, df=fd, fef-1=ce3 >
Subgroups: 300 in 153 conjugacy classes, 54 normal (26 characteristic)
C1, C2, C2, C4, C4, C22, C22, C22, C8, C2×C4, C2×C4, C2×C4, C23, C23, C42, C22⋊C4, C4⋊C4, C4⋊C4, C2×C8, M4(2), C22×C4, C22×C4, C24, C2.C42, C22⋊C8, C2×C42, C2×C42, C2×C22⋊C4, C2×C4⋊C4, C42⋊C2, C42⋊C2, C2×M4(2), C23×C4, C42⋊6C4, C4×C22⋊C4, C24.4C4, C2×C42⋊C2, C24.70D4
Quotients: C1, C2, C4, C22, C2×C4, D4, C23, C22⋊C4, C22×C4, C2×D4, C4○D4, C4≀C2, C2×C22⋊C4, C42⋊C2, C22.D4, C23.34D4, C2×C4≀C2, C42⋊C22, C24.70D4
►On 32 points
(1 9)(2 23)(3 11)(4 17)(5 13)(6 19)(7 15)(8 21)(10 31)(12 25)(14 27)(16 29)(18 26)(20 28)(22 30)(24 32)
(1 30)(2 31)(3 32)(4 25)(5 26)(6 27)(7 28)(8 29)(9 22)(10 23)(11 24)(12 17)(13 18)(14 19)(15 20)(16 21)
(1 5)(3 7)(9 13)(11 15)(18 22)(20 24)(26 30)(28 32)
(1 5)(2 6)(3 7)(4 8)(9 13)(10 14)(11 15)(12 16)(17 21)(18 22)(19 23)(20 24)(25 29)(26 30)(27 31)(28 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 9 30 22)(2 12 27 21)(3 11 32 24)(4 14 29 23)(5 13 26 18)(6 16 31 17)(7 15 28 20)(8 10 25 19)
G:=sub<Sym(32)| (1,9)(2,23)(3,11)(4,17)(5,13)(6,19)(7,15)(8,21)(10,31)(12,25)(14,27)(16,29)(18,26)(20,28)(22,30)(24,32), (1,30)(2,31)(3,32)(4,25)(5,26)(6,27)(7,28)(8,29)(9,22)(10,23)(11,24)(12,17)(13,18)(14,19)(15,20)(16,21), (1,5)(3,7)(9,13)(11,15)(18,22)(20,24)(26,30)(28,32), (1,5)(2,6)(3,7)(4,8)(9,13)(10,14)(11,15)(12,16)(17,21)(18,22)(19,23)(20,24)(25,29)(26,30)(27,31)(28,32), (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), (1,9,30,22)(2,12,27,21)(3,11,32,24)(4,14,29,23)(5,13,26,18)(6,16,31,17)(7,15,28,20)(8,10,25,19)>;
G:=Group( (1,9)(2,23)(3,11)(4,17)(5,13)(6,19)(7,15)(8,21)(10,31)(12,25)(14,27)(16,29)(18,26)(20,28)(22,30)(24,32), (1,30)(2,31)(3,32)(4,25)(5,26)(6,27)(7,28)(8,29)(9,22)(10,23)(11,24)(12,17)(13,18)(14,19)(15,20)(16,21), (1,5)(3,7)(9,13)(11,15)(18,22)(20,24)(26,30)(28,32), (1,5)(2,6)(3,7)(4,8)(9,13)(10,14)(11,15)(12,16)(17,21)(18,22)(19,23)(20,24)(25,29)(26,30)(27,31)(28,32), (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), (1,9,30,22)(2,12,27,21)(3,11,32,24)(4,14,29,23)(5,13,26,18)(6,16,31,17)(7,15,28,20)(8,10,25,19) );
G=PermutationGroup([[(1,9),(2,23),(3,11),(4,17),(5,13),(6,19),(7,15),(8,21),(10,31),(12,25),(14,27),(16,29),(18,26),(20,28),(22,30),(24,32)], [(1,30),(2,31),(3,32),(4,25),(5,26),(6,27),(7,28),(8,29),(9,22),(10,23),(11,24),(12,17),(13,18),(14,19),(15,20),(16,21)], [(1,5),(3,7),(9,13),(11,15),(18,22),(20,24),(26,30),(28,32)], [(1,5),(2,6),(3,7),(4,8),(9,13),(10,14),(11,15),(12,16),(17,21),(18,22),(19,23),(20,24),(25,29),(26,30),(27,31),(28,32)], [(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)], [(1,9,30,22),(2,12,27,21),(3,11,32,24),(4,14,29,23),(5,13,26,18),(6,16,31,17),(7,15,28,20),(8,10,25,19)]])
38 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 2H | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | ··· | 4Y | 8A | 8B | 8C | 8D |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 8 | 8 | 8 | 8 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 4 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 8 | 8 | 8 | 8 |
38 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 4 |
| type | + | + | + | + | + | + | + | |||||
| image | C1 | C2 | C2 | C2 | C2 | C4 | C4 | D4 | D4 | C4○D4 | C4≀C2 | C42⋊C22 |
| kernel | C24.70D4 | C42⋊6C4 | C4×C22⋊C4 | C24.4C4 | C2×C42⋊C2 | C2×C4⋊C4 | C42⋊C2 | C22×C4 | C24 | C2×C4 | C22 | C2 |
| # reps | 1 | 4 | 1 | 1 | 1 | 4 | 4 | 3 | 1 | 8 | 8 | 2 |
Matrix representation of C24.70D4 ►in GL4(𝔽17) generated by
| 16 | 0 | 0 | 0 |
| 0 | 16 | 0 | 0 |
| 0 | 0 | 0 | 13 |
| 0 | 0 | 4 | 0 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 16 | 0 |
| 0 | 0 | 0 | 16 |
| 16 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 16 | 0 | 0 | 0 |
| 0 | 16 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 13 | 0 | 0 |
| 16 | 0 | 0 | 0 |
| 0 | 0 | 13 | 0 |
| 0 | 0 | 0 | 4 |
| 16 | 0 | 0 | 0 |
| 0 | 13 | 0 | 0 |
| 0 | 0 | 0 | 4 |
| 0 | 0 | 4 | 0 |
G:=sub<GL(4,GF(17))| [16,0,0,0,0,16,0,0,0,0,0,4,0,0,13,0],[1,0,0,0,0,1,0,0,0,0,16,0,0,0,0,16],[16,0,0,0,0,1,0,0,0,0,1,0,0,0,0,1],[16,0,0,0,0,16,0,0,0,0,1,0,0,0,0,1],[0,16,0,0,13,0,0,0,0,0,13,0,0,0,0,4],[16,0,0,0,0,13,0,0,0,0,0,4,0,0,4,0] >;
C24.70D4 in GAP, Magma, Sage, TeX
C_2^4._{70}D_4 % in TeX
G:=Group("C2^4.70D4"); // GroupNames label
G:=SmallGroup(128,558);
// by ID
G=gap.SmallGroup(128,558);
# by ID
G:=PCGroup([7,-2,2,2,-2,2,2,-2,224,141,422,58,2804,718,172,124]);
// Polycyclic
G:=Group<a,b,c,d,e,f|a^2=b^2=c^2=d^2=1,e^4=d,f^2=b*c*d,e*a*e^-1=f*a*f^-1=a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,b*e=e*b,b*f=f*b,e*c*e^-1=c*d=d*c,c*f=f*c,d*e=e*d,d*f=f*d,f*e*f^-1=c*e^3>;
// generators/relations