C₂⁴×D₄² #72231697

order	1024
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁸ · Schur multiplier (H₂) = C₂²⁸

Aliases

direct product

C₂⁴×D₄²

Subgroups

	subgroup	G/H
center, socle	C₂⁶	C₂⁴
commutator, Frattini, cosocle	C₂²	C₂⁸
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂⁶ ◁ G
LECS	G ▷ C₂² ▷ C₁

1567813 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 16
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 8 generators on 16 points:

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

Conjugacy classes & character table

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

Relationships

maximal subgroup of	#72231721, C₂×D₄×#3003, #72233580, …
maximal normal subgroup of	#72231721, C₂×D₄×#3003, #72233580, …
maximal quotient of	#72234844, C₂²×#7684106, C₂×D₄×#3484, …
automorphism group of	#1621, #1622, #1625, …

GAP

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

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

C24×D42 #72231697