direct product, p-group, metabelian, nilpotent (class 2), monomial
Aliases: C2×C24.4C4, C25.8C4, C23⋊6M4(2), (C2×C8)⋊9C23, (C23×C4).38C4, (C24×C4).12C2, C22⋊C8⋊71C22, C4○(C24.4C4), (C2×C4).627C24, C24.125(C2×C4), (C22×C8)⋊44C22, (C22×C4).772D4, C4.176(C22×D4), C22⋊3(C2×M4(2)), C2.6(C22×M4(2)), (C22×M4(2))⋊13C2, (C2×M4(2))⋊62C22, (C23×C4).687C22, C22.157(C23×C4), C23.219(C22×C4), C23.206(C22⋊C4), (C22×C4).1491C23, (C2×C22⋊C8)⋊39C2, C4.97(C2×C22⋊C4), (C2×C4).1562(C2×D4), (C2×C4)○(C24.4C4), (C22×C4).492(C2×C4), (C2×C4).496(C22×C4), C22.76(C2×C22⋊C4), C2.20(C22×C22⋊C4), (C2×C4).366(C22⋊C4), SmallGroup(128,1609)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C2×C24.4C4
G = < a,b,c,d,e,f | a2=b2=c2=d2=e2=1, f4=e, ab=ba, ac=ca, ad=da, ae=ea, af=fa, bc=cb, fbf-1=bd=db, be=eb, cd=dc, fcf-1=ce=ec, de=ed, df=fd, ef=fe >
Subgroups: 812 in 504 conjugacy classes, 196 normal (14 characteristic)
C1, C2, C2, C2, C4, C4, C22, C22, C22, C8, C2×C4, C2×C4, C2×C4, C23, C23, C23, C2×C8, C2×C8, M4(2), C22×C4, C22×C4, C22×C4, C24, C24, C24, C22⋊C8, C22×C8, C2×M4(2), C2×M4(2), C23×C4, C23×C4, C23×C4, C25, C2×C22⋊C8, C24.4C4, C22×M4(2), C24×C4, C2×C24.4C4
Quotients: C1, C2, C4, C22, C2×C4, D4, C23, C22⋊C4, M4(2), C22×C4, C2×D4, C24, C2×C22⋊C4, C2×M4(2), C23×C4, C22×D4, C24.4C4, C22×C22⋊C4, C22×M4(2), C2×C24.4C4
►On 32 points
(1 19)(2 20)(3 21)(4 22)(5 23)(6 24)(7 17)(8 18)(9 28)(10 29)(11 30)(12 31)(13 32)(14 25)(15 26)(16 27)
(2 28)(4 30)(6 32)(8 26)(9 20)(11 22)(13 24)(15 18)
(1 31)(2 28)(3 25)(4 30)(5 27)(6 32)(7 29)(8 26)(9 20)(10 17)(11 22)(12 19)(13 24)(14 21)(15 18)(16 23)
(1 27)(2 28)(3 29)(4 30)(5 31)(6 32)(7 25)(8 26)(9 20)(10 21)(11 22)(12 23)(13 24)(14 17)(15 18)(16 19)
(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)
G:=sub<Sym(32)| (1,19)(2,20)(3,21)(4,22)(5,23)(6,24)(7,17)(8,18)(9,28)(10,29)(11,30)(12,31)(13,32)(14,25)(15,26)(16,27), (2,28)(4,30)(6,32)(8,26)(9,20)(11,22)(13,24)(15,18), (1,31)(2,28)(3,25)(4,30)(5,27)(6,32)(7,29)(8,26)(9,20)(10,17)(11,22)(12,19)(13,24)(14,21)(15,18)(16,23), (1,27)(2,28)(3,29)(4,30)(5,31)(6,32)(7,25)(8,26)(9,20)(10,21)(11,22)(12,23)(13,24)(14,17)(15,18)(16,19), (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)>;
G:=Group( (1,19)(2,20)(3,21)(4,22)(5,23)(6,24)(7,17)(8,18)(9,28)(10,29)(11,30)(12,31)(13,32)(14,25)(15,26)(16,27), (2,28)(4,30)(6,32)(8,26)(9,20)(11,22)(13,24)(15,18), (1,31)(2,28)(3,25)(4,30)(5,27)(6,32)(7,29)(8,26)(9,20)(10,17)(11,22)(12,19)(13,24)(14,21)(15,18)(16,23), (1,27)(2,28)(3,29)(4,30)(5,31)(6,32)(7,25)(8,26)(9,20)(10,21)(11,22)(12,23)(13,24)(14,17)(15,18)(16,19), (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) );
G=PermutationGroup([[(1,19),(2,20),(3,21),(4,22),(5,23),(6,24),(7,17),(8,18),(9,28),(10,29),(11,30),(12,31),(13,32),(14,25),(15,26),(16,27)], [(2,28),(4,30),(6,32),(8,26),(9,20),(11,22),(13,24),(15,18)], [(1,31),(2,28),(3,25),(4,30),(5,27),(6,32),(7,29),(8,26),(9,20),(10,17),(11,22),(12,19),(13,24),(14,21),(15,18),(16,23)], [(1,27),(2,28),(3,29),(4,30),(5,31),(6,32),(7,25),(8,26),(9,20),(10,21),(11,22),(12,23),(13,24),(14,17),(15,18),(16,19)], [(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)]])
56 conjugacy classes
| class | 1 | 2A | ··· | 2G | 2H | ··· | 2S | 4A | ··· | 4H | 4I | ··· | 4T | 8A | ··· | 8P |
| order | 1 | 2 | ··· | 2 | 2 | ··· | 2 | 4 | ··· | 4 | 4 | ··· | 4 | 8 | ··· | 8 |
| size | 1 | 1 | ··· | 1 | 2 | ··· | 2 | 1 | ··· | 1 | 2 | ··· | 2 | 4 | ··· | 4 |
56 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 |
| type | + | + | + | + | + | + | |||
| image | C1 | C2 | C2 | C2 | C2 | C4 | C4 | D4 | M4(2) |
| kernel | C2×C24.4C4 | C2×C22⋊C8 | C24.4C4 | C22×M4(2) | C24×C4 | C23×C4 | C25 | C22×C4 | C23 |
| # reps | 1 | 4 | 8 | 2 | 1 | 14 | 2 | 8 | 16 |
Matrix representation of C2×C24.4C4 ►in GL5(𝔽17)
| 16 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 16 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 14 | 16 |
| 16 | 0 | 0 | 0 | 0 |
| 0 | 16 | 0 | 0 | 0 |
| 0 | 0 | 16 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 14 | 16 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 16 | 0 | 0 | 0 |
| 0 | 0 | 16 | 0 | 0 |
| 0 | 0 | 0 | 16 | 0 |
| 0 | 0 | 0 | 0 | 16 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 16 | 0 |
| 0 | 0 | 0 | 0 | 16 |
| 13 | 0 | 0 | 0 | 0 |
| 0 | 0 | 16 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 14 | 15 |
| 0 | 0 | 0 | 15 | 3 |
G:=sub<GL(5,GF(17))| [16,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1],[1,0,0,0,0,0,1,0,0,0,0,0,16,0,0,0,0,0,1,14,0,0,0,0,16],[16,0,0,0,0,0,16,0,0,0,0,0,16,0,0,0,0,0,1,14,0,0,0,0,16],[1,0,0,0,0,0,16,0,0,0,0,0,16,0,0,0,0,0,16,0,0,0,0,0,16],[1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,16,0,0,0,0,0,16],[13,0,0,0,0,0,0,1,0,0,0,16,0,0,0,0,0,0,14,15,0,0,0,15,3] >;
C2×C24.4C4 in GAP, Magma, Sage, TeX
C_2\times C_2^4._4C_4
% in TeX
G:=Group("C2xC2^4.4C4"); // GroupNames label
G:=SmallGroup(128,1609);
// by ID
G=gap.SmallGroup(128,1609);
# by ID
G:=PCGroup([7,-2,2,2,2,-2,2,-2,224,253,1430,124]);
// Polycyclic
G:=Group<a,b,c,d,e,f|a^2=b^2=c^2=d^2=e^2=1,f^4=e,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,a*f=f*a,b*c=c*b,f*b*f^-1=b*d=d*b,b*e=e*b,c*d=d*c,f*c*f^-1=c*e=e*c,d*e=e*d,d*f=f*d,e*f=f*e>;
// generators/relations