p-group, metabelian, nilpotent (class 3), monomial
Aliases: C25.3C4, C23⋊4M4(2), C23⋊C8⋊12C2, (C23×C4).8C4, C24.21(C2×C4), C22⋊C8⋊40C22, (C22×C4).198D4, C24.4C4⋊15C2, C22.25(C23⋊C4), C2.6(C24.4C4), (C22×C4).426C23, (C23×C4).197C22, C23.163(C22×C4), C22.15(C2×M4(2)), C23.165(C22⋊C4), C22.16(C4.D4), C2.6(C2×C23⋊C4), C2.5(C2×C4.D4), (C2×C4).1123(C2×D4), (C2×C22⋊C4).34C4, (C2×C4).68(C22⋊C4), (C22×C4).175(C2×C4), (C22×C22⋊C4).14C2, C22.144(C2×C22⋊C4), (C2×C22⋊C4).402C22, SmallGroup(128,194)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C25.3C4
G = < a,b,c,d,e,f | a2=b2=c2=d2=e2=1, f4=e, ab=ba, ac=ca, ad=da, ae=ea, faf-1=acd, bc=cb, bd=db, fbf-1=be=eb, fcf-1=cd=dc, ce=ec, de=ed, df=fd, ef=fe >
Subgroups: 564 in 220 conjugacy classes, 52 normal (16 characteristic)
C1, C2, C2, C4, C22, C22, C22, C8, C2×C4, C2×C4, C23, C23, C23, C22⋊C4, C2×C8, M4(2), C22×C4, C22×C4, C22×C4, C24, C24, C24, C22⋊C8, C22⋊C8, C2×C22⋊C4, C2×C22⋊C4, C2×M4(2), C23×C4, C25, C23⋊C8, C24.4C4, C22×C22⋊C4, C25.3C4
Quotients: C1, C2, C4, C22, C2×C4, D4, C23, C22⋊C4, M4(2), C22×C4, C2×D4, C23⋊C4, C4.D4, C2×C22⋊C4, C2×M4(2), C24.4C4, C2×C23⋊C4, C2×C4.D4, C25.3C4
(1 5)(2 14)(3 11)(6 10)(7 15)(9 13)
(1 9)(2 14)(3 11)(4 16)(5 13)(6 10)(7 15)(8 12)
(1 5)(2 10)(3 7)(4 12)(6 14)(8 16)(9 13)(11 15)
(1 13)(2 14)(3 15)(4 16)(5 9)(6 10)(7 11)(8 12)
(1 5)(2 6)(3 7)(4 8)(9 13)(10 14)(11 15)(12 16)
(1 2 3 4 5 6 7 8)(9 10 11 12 13 14 15 16)
G:=sub<Sym(16)| (1,5)(2,14)(3,11)(6,10)(7,15)(9,13), (1,9)(2,14)(3,11)(4,16)(5,13)(6,10)(7,15)(8,12), (1,5)(2,10)(3,7)(4,12)(6,14)(8,16)(9,13)(11,15), (1,13)(2,14)(3,15)(4,16)(5,9)(6,10)(7,11)(8,12), (1,5)(2,6)(3,7)(4,8)(9,13)(10,14)(11,15)(12,16), (1,2,3,4,5,6,7,8)(9,10,11,12,13,14,15,16)>;
G:=Group( (1,5)(2,14)(3,11)(6,10)(7,15)(9,13), (1,9)(2,14)(3,11)(4,16)(5,13)(6,10)(7,15)(8,12), (1,5)(2,10)(3,7)(4,12)(6,14)(8,16)(9,13)(11,15), (1,13)(2,14)(3,15)(4,16)(5,9)(6,10)(7,11)(8,12), (1,5)(2,6)(3,7)(4,8)(9,13)(10,14)(11,15)(12,16), (1,2,3,4,5,6,7,8)(9,10,11,12,13,14,15,16) );
G=PermutationGroup([[(1,5),(2,14),(3,11),(6,10),(7,15),(9,13)], [(1,9),(2,14),(3,11),(4,16),(5,13),(6,10),(7,15),(8,12)], [(1,5),(2,10),(3,7),(4,12),(6,14),(8,16),(9,13),(11,15)], [(1,13),(2,14),(3,15),(4,16),(5,9),(6,10),(7,11),(8,12)], [(1,5),(2,6),(3,7),(4,8),(9,13),(10,14),(11,15),(12,16)], [(1,2,3,4,5,6,7,8),(9,10,11,12,13,14,15,16)]])
G:=TransitiveGroup(16,252);
32 conjugacy classes
class | 1 | 2A | 2B | 2C | 2D | ··· | 2I | 2J | 2K | 2L | 2M | 4A | 4B | 4C | 4D | 4E | ··· | 4J | 8A | ··· | 8H |
order | 1 | 2 | 2 | 2 | 2 | ··· | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 8 | ··· | 8 |
size | 1 | 1 | 1 | 1 | 2 | ··· | 2 | 4 | 4 | 4 | 4 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 8 | ··· | 8 |
32 irreducible representations
dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 4 | 4 |
type | + | + | + | + | + | + | + | ||||
image | C1 | C2 | C2 | C2 | C4 | C4 | C4 | D4 | M4(2) | C23⋊C4 | C4.D4 |
kernel | C25.3C4 | C23⋊C8 | C24.4C4 | C22×C22⋊C4 | C2×C22⋊C4 | C23×C4 | C25 | C22×C4 | C23 | C22 | C22 |
# reps | 1 | 4 | 2 | 1 | 4 | 2 | 2 | 4 | 8 | 2 | 2 |
Matrix representation of C25.3C4 ►in GL6(𝔽17)
1 | 0 | 0 | 0 | 0 | 0 |
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 | 0 | 0 | 1 |
1 | 0 | 0 | 0 | 0 | 0 |
0 | 16 | 0 | 0 | 0 | 0 |
0 | 0 | 1 | 0 | 0 | 0 |
0 | 0 | 0 | 1 | 0 | 0 |
0 | 0 | 0 | 0 | 16 | 0 |
0 | 0 | 0 | 0 | 0 | 16 |
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 | 0 | 0 | 1 | 0 |
0 | 0 | 0 | 0 | 0 | 1 |
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 | 0 | 0 | 16 | 0 |
0 | 0 | 0 | 0 | 0 | 16 |
16 | 0 | 0 | 0 | 0 | 0 |
0 | 16 | 0 | 0 | 0 | 0 |
0 | 0 | 16 | 0 | 0 | 0 |
0 | 0 | 0 | 16 | 0 | 0 |
0 | 0 | 0 | 0 | 16 | 0 |
0 | 0 | 0 | 0 | 0 | 16 |
0 | 1 | 0 | 0 | 0 | 0 |
4 | 0 | 0 | 0 | 0 | 0 |
0 | 0 | 0 | 0 | 1 | 0 |
0 | 0 | 0 | 0 | 0 | 1 |
0 | 0 | 0 | 1 | 0 | 0 |
0 | 0 | 16 | 0 | 0 | 0 |
G:=sub<GL(6,GF(17))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,16,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,16,0,0,0,0,0,0,16],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,16],[16,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,16],[0,4,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,16,0,0,0,0,1,0,0,0,1,0,0,0,0,0,0,1,0,0] >;
C25.3C4 in GAP, Magma, Sage, TeX
C_2^5._3C_4
% in TeX
G:=Group("C2^5.3C4");
// GroupNames label
G:=SmallGroup(128,194);
// by ID
G=gap.SmallGroup(128,194);
# by ID
G:=PCGroup([7,-2,2,2,-2,2,-2,2,112,141,1430,1123,851,172]);
// 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,f*a*f^-1=a*c*d,b*c=c*b,b*d=d*b,f*b*f^-1=b*e=e*b,f*c*f^-1=c*d=d*c,c*e=e*c,d*e=e*d,d*f=f*d,e*f=f*e>;
// generators/relations