Copied to
clipboard

G = (C2×C42)⋊C4order 128 = 27

8th semidirect product of C2×C42 and C4 acting faithfully

p-group, metabelian, nilpotent (class 3), monomial

Series: Derived Chief Lower central Upper central Jennings

 Derived series C1 — C2×C4 — (C2×C42)⋊C4
 Chief series C1 — C2 — C22 — C2×C4 — C22×C4 — C42⋊C2 — C4×C4○D4 — (C2×C42)⋊C4
 Lower central C1 — C2 — C2×C4 — (C2×C42)⋊C4
 Upper central C1 — C4 — C42⋊C2 — (C2×C42)⋊C4
 Jennings C1 — C2 — C2 — C22×C4 — (C2×C42)⋊C4

Generators and relations for (C2×C42)⋊C4
G = < a,b,c,d | a2=b4=c4=d4=1, ab=ba, ac=ca, dad-1=ac2, bc=cb, dbd-1=bc2, dcd-1=abc >

Subgroups: 260 in 131 conjugacy classes, 50 normal (20 characteristic)
C1, C2, C2 [×5], C4 [×2], C4 [×2], C4 [×9], C22, C22 [×2], C22 [×5], C8 [×2], C2×C4 [×2], C2×C4 [×6], C2×C4 [×16], D4 [×6], Q8 [×2], C23, C23 [×2], C42 [×2], C42 [×2], C42 [×5], C22⋊C4 [×5], C4⋊C4 [×5], C2×C8 [×2], M4(2) [×4], C22×C4, C22×C4 [×2], C22×C4 [×3], C2×D4, C2×D4 [×2], C2×Q8, C4○D4 [×4], C23⋊C4 [×2], C4.D4, C4.10D4, C2×C42, C2×C42 [×2], C42⋊C2, C42⋊C2 [×4], C4×D4 [×3], C4×Q8, C2×M4(2) [×2], C2×C4○D4, C4.9C42 [×2], C426C4 [×2], C23.C23, M4(2).8C22, C4×C4○D4, (C2×C42)⋊C4
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C2×C4 [×6], D4 [×4], C23, C22⋊C4 [×4], C22×C4, C2×D4 [×2], C4○D4 [×4], C2×C22⋊C4, C42⋊C2 [×2], C22.D4 [×4], C23.34D4, (C2×C42)⋊C4

Permutation representations of (C2×C42)⋊C4
On 16 points - transitive group 16T291
Generators in S16
(1 10)(2 11)(3 12)(4 9)(5 13)(6 14)(7 15)(8 16)
(1 2 3 4)(5 6 7 8)(9 10 11 12)(13 14 15 16)
(1 2 3 4)(5 13)(6 14)(7 15)(8 16)(9 10 11 12)
(1 16)(2 15 4 13)(3 14)(5 11)(6 12 8 10)(7 9)

G:=sub<Sym(16)| (1,10)(2,11)(3,12)(4,9)(5,13)(6,14)(7,15)(8,16), (1,2,3,4)(5,6,7,8)(9,10,11,12)(13,14,15,16), (1,2,3,4)(5,13)(6,14)(7,15)(8,16)(9,10,11,12), (1,16)(2,15,4,13)(3,14)(5,11)(6,12,8,10)(7,9)>;

G:=Group( (1,10)(2,11)(3,12)(4,9)(5,13)(6,14)(7,15)(8,16), (1,2,3,4)(5,6,7,8)(9,10,11,12)(13,14,15,16), (1,2,3,4)(5,13)(6,14)(7,15)(8,16)(9,10,11,12), (1,16)(2,15,4,13)(3,14)(5,11)(6,12,8,10)(7,9) );

G=PermutationGroup([(1,10),(2,11),(3,12),(4,9),(5,13),(6,14),(7,15),(8,16)], [(1,2,3,4),(5,6,7,8),(9,10,11,12),(13,14,15,16)], [(1,2,3,4),(5,13),(6,14),(7,15),(8,16),(9,10,11,12)], [(1,16),(2,15,4,13),(3,14),(5,11),(6,12,8,10),(7,9)])

G:=TransitiveGroup(16,291);

On 16 points - transitive group 16T314
Generators in S16
(1 3)(2 4)(5 7)(6 8)
(1 2 3 4)(5 6 7 8)(9 10 11 12)(13 14 15 16)
(5 8 7 6)(9 11)(10 12)(13 16 15 14)
(1 5 9 14)(2 6 12 13)(3 7 11 16)(4 8 10 15)

G:=sub<Sym(16)| (1,3)(2,4)(5,7)(6,8), (1,2,3,4)(5,6,7,8)(9,10,11,12)(13,14,15,16), (5,8,7,6)(9,11)(10,12)(13,16,15,14), (1,5,9,14)(2,6,12,13)(3,7,11,16)(4,8,10,15)>;

G:=Group( (1,3)(2,4)(5,7)(6,8), (1,2,3,4)(5,6,7,8)(9,10,11,12)(13,14,15,16), (5,8,7,6)(9,11)(10,12)(13,16,15,14), (1,5,9,14)(2,6,12,13)(3,7,11,16)(4,8,10,15) );

G=PermutationGroup([(1,3),(2,4),(5,7),(6,8)], [(1,2,3,4),(5,6,7,8),(9,10,11,12),(13,14,15,16)], [(5,8,7,6),(9,11),(10,12),(13,16,15,14)], [(1,5,9,14),(2,6,12,13),(3,7,11,16),(4,8,10,15)])

G:=TransitiveGroup(16,314);

32 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 4A 4B 4C ··· 4I 4J ··· 4Q 4R 4S 4T 4U 8A 8B 8C 8D order 1 2 2 2 2 2 2 4 4 4 ··· 4 4 ··· 4 4 4 4 4 8 8 8 8 size 1 1 2 2 2 4 4 1 1 2 ··· 2 4 ··· 4 8 8 8 8 8 8 8 8

32 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 2 4 type + + + + + + + + image C1 C2 C2 C2 C2 C2 C4 C4 D4 D4 C4○D4 (C2×C42)⋊C4 kernel (C2×C42)⋊C4 C4.9C42 C42⋊6C4 C23.C23 M4(2).8C22 C4×C4○D4 C2×C42 C42⋊C2 C2×D4 C2×Q8 C2×C4 C1 # reps 1 2 2 1 1 1 4 4 3 1 8 4

Matrix representation of (C2×C42)⋊C4 in GL4(𝔽5) generated by

 0 0 1 3 0 0 4 0 0 4 0 0 2 2 0 0
,
 0 0 2 1 0 0 3 0 0 3 0 0 4 4 0 0
,
 0 0 4 0 0 0 0 4 2 2 0 0 1 3 0 0
,
 0 0 0 4 0 0 3 0 2 1 0 0 3 3 0 0
G:=sub<GL(4,GF(5))| [0,0,0,2,0,0,4,2,1,4,0,0,3,0,0,0],[0,0,0,4,0,0,3,4,2,3,0,0,1,0,0,0],[0,0,2,1,0,0,2,3,4,0,0,0,0,4,0,0],[0,0,2,3,0,0,1,3,0,3,0,0,4,0,0,0] >;

(C2×C42)⋊C4 in GAP, Magma, Sage, TeX

(C_2\times C_4^2)\rtimes C_4
% in TeX

G:=Group("(C2xC4^2):C4");
// GroupNames label

G:=SmallGroup(128,559);
// by ID

G=gap.SmallGroup(128,559);
# by ID

G:=PCGroup([7,-2,2,2,-2,2,2,-2,224,141,422,58,2019,172,2028,1027]);
// Polycyclic

G:=Group<a,b,c,d|a^2=b^4=c^4=d^4=1,a*b=b*a,a*c=c*a,d*a*d^-1=a*c^2,b*c=c*b,d*b*d^-1=b*c^2,d*c*d^-1=a*b*c>;
// generators/relations

׿
×
𝔽