C₂×#72237164 #72309332

order	2048
nilpotency class	3
automorphism	?
homology	Abelianization (H₁) = C₂⁵×C₄ · Schur multiplier (H₂) = C₂¹⁵×C₄

Aliases

direct product

C₂×#72237164

Subgroups

	subgroup	G/H
center	C₂³×C₄	C₂²×C₂²⋊C₄
commutator	C₂⁴	C₂⁵×C₄
Frattini, cosocle	C₂³×C₄	C₂⁶
socle	C₂⁴	C₂³×C₂²⋊C₄
prime radical	C₂⁶×D₄	C₄
LCS	G ▷ C₂⁴ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₄ ◁ C₂⁵×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

803424 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 22
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 22 points:

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

Conjugacy classes & character table

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

Relationships

maximal subgroup of	#72309372
maximal normal subgroup of	#72309372

GAP

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

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

C2×#72237164 #72309332