C₂×#72213848 #72233889

order	2048
nilpotency class	3
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂¹²

Aliases

direct product

C₂×#72213848

Subgroups

	subgroup	G/H
center, socle	C₂³	#14715
commutator, Frattini, cosocle	C₂²×C₄²	C₂⁵
LCS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

At least 8985 conjugacy classes of subgroups; only the 1672 normal subgroups are shown.

Full subgroup data →

Presentation

minimal degree	μ(G) = 34
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 5 generators on 34 points:

(1,2);(4,9)(5,14)(6,17)(10,25)(11,22)(12,23)(13,24)(15,30)(18,29)(20,27)(21,28)(26,33);(5,15)(10,26)(14,30)(16,32)(20,21)(25,33)(27,28)(31,34);(4,10,12,26)(6,18)(9,20,23,21)(11,27,13,28)(16,32)(17,29)(22,25,24,33)(31,34);(3,4,6,9,8,13,17,24)(5,10,16,21,14,28,31,25)(7,12,18,23,19,11,29,22)(15,26,32,20,30,27,34,33)

Conjugacy classes & character table

164 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72215640
maximal normal subgroup of	C₂×#72215640

GAP

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

gap> G := Group((1,2), (4,9)(5,14)(6,17)(10,25)(11,22)(12,23)(13,24)(15,30)(18,29)(20,27)(21,28)(26,33), (5,15)(10,26)(14,30)(16,32)(20,21)(25,33)(27,28)(31,34), (4,10,12,26)(6,18)(9,20,23,21)(11,27,13,28)(16,32)(17,29)(22,25,24,33)(31,34), (3,4,6,9,8,13,17,24)(5,10,16,21,14,28,31,25)(7,12,18,23,19,11,29,22)(15,26,32,20,30,27,34,33));

C2×#72213848 #72233889