C₂⁵×D₄² #72289263

order	2048
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁹ · Schur multiplier (H₂) = C₂³⁶

Aliases

direct product

C₂⁵×D₄²

Subgroups

	subgroup	G/H
center, socle	C₂⁷	C₂⁴
commutator, Frattini, cosocle	C₂²	C₂⁹
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂⁷ ◁ G
LECS	G ▷ C₂² ▷ C₁

31687016 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 18
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 9 generators on 18 points:

(11,12,13,14)(16,18);(11,12)(13,14)(15,16)(17,18);(11,12,13,14);(11,12)(13,14);(3,4);(5,6);(7,8);(9,10);(1,2)(3,4)(5,6)(7,8)(9,10)

Conjugacy classes & character table

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

Relationships

maximal subgroup of	C₂²×#72219693, C₂³×#7684106, C₂³×D₄×#392, …
maximal normal subgroup of	C₂²×#72219693, C₂³×#7684106, C₂³×D₄×#392, …
maximal quotient of	C₂³×#7684106, C₂⁴×D₄×#120, C₂³×D₄³, …
automorphism group of	#8436, #8437, #8438, …

GAP

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

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

C25×D42 #72289263