C₂×#72237160 #72238810

order	2048
nilpotency class	3
automorphism	?
homology	Abelianization (H₁) = C₂⁷ · Schur multiplier (H₂) = C₂¹⁹

Aliases

direct product

C₂×#72237160

Subgroups

	subgroup	G/H
center, socle	C₂⁴	C₂⁴×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁷
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ C₂×C₄×#525 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

At least 12905 conjugacy classes of subgroups; only the 35744 normal subgroups are shown.

Full subgroup data →

Presentation

minimal degree	μ(G) = 22
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 7 generators on 22 points:

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

Conjugacy classes & character table

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

Relationships

maximal subgroup of	#72306287, C₂²×#72237160
maximal normal subgroup of	#72306287, C₂²×#72237160
maximal quotient of	C₂×#72237167, #72306287, C₂²×#72237160
automorphism group of	#2139, #2159, #2206, …

GAP

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

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

C2×#72237160 #72238810