C₂⁴×H(3,ℤ/4) #72302727

order	1024
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄² · Schur multiplier (H₂) = C₂¹⁵×C₄

Aliases

direct product

C₂⁴×H(3,ℤ/4)

Subgroups

	subgroup	G/H
center	C₂⁴×C₄	C₄²
commutator	C₄	C₂⁴×C₄²
Frattini, cosocle	C₂²×C₄	C₂⁶
socle	C₂⁵	#95
prime radical	C₂⁷	D₄
LCS	G ▷ C₄ ▷ C₁
UCS	C₁ ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

87029 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 24
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 24 points:

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

Conjugacy classes & character table

352 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	C₂×#72237270, C₂⁴×#2010, C₂×#72239235
maximal normal subgroup of	C₂×#72237270, C₂⁴×#2010, C₂×#72239235

GAP

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

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

C24×H(3,ℤ/4) #72302727