C₂×#72209998 #72235320

order	2048
nilpotency class	3
automorphism	?
homology	Abelianization (H₁) = C₂×C₄³ · Schur multiplier (H₂) = C₂⁷×C₄²

Aliases

direct product

C₂×#72209998

Subgroups

	subgroup	G/H
center	C₂²×C₄²	#127
commutator	C₄²	C₂×C₄³
Frattini, cosocle	C₂³×C₄²	C₂⁴
socle	C₂⁴	#1437
prime radical	C₂⁶	#120
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₄² ◁ C₂⁴×C₄² ◁ G
LECS	G ▷ C₂³×C₄² ▷ C₂² ▷ C₁

67728 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 4 generators on 22 points:

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

Conjugacy classes & character table

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

Relationships

maximal subgroup of	#72235339, C₂×#72210008, C₂×#72210007, …
maximal normal subgroup of	#72235339, #72235722, #72238833, …

GAP

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

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

C2×#72209998 #72235320