C₄×#13037 #72309604

order	1024
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂⁶×C₄

Aliases

direct product

C₄×#13037

Subgroups

	subgroup	G/H
center	C₂²×C₄	#442
commutator	C₂×C₈	C₂²×C₄²
Frattini, cosocle	C₂³×C₈	C₂⁴
socle	C₂³	C₂×#442
prime radical	C₂²×#509	C₂²
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×C₄² ◁ C₂×C₄×#97 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

3379 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 24
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 24 points:

(10,14,16,21)(11,17)(12,18)(15,20,23,22)(19,24);(5,6,7,8);(2,4)(10,15,16,23)(12,19)(14,20,21,22)(18,24);(1,2,3,4)(9,10,12,14,11,15,18,22,13,16,19,21,17,23,24,20)

Conjugacy classes & character table

160 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	D₄×#13037, C₄×#495725, #72310080
maximal normal subgroup of	D₄×#13037, C₄×#495725, #72310080

GAP

Past the SmallGroups library coverage — no SmallGroups id. Construct it from a minimal faithful permutation representation:

gap> G := Group((10,14,16,21)(11,17)(12,18)(15,20,23,22)(19,24), (5,6,7,8), (2,4)(10,15,16,23)(12,19)(14,20,21,22)(18,24), (1,2,3,4)(9,10,12,14,11,15,18,22,13,16,19,21,17,23,24,20));

C4×#13037 #72309604