C₂×#72252090 #72334768

order	2048
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁵×C₄ · Schur multiplier (H₂) = C₂¹⁹

Aliases

direct product

C₂×#72252090

Subgroups

	subgroup	G/H
center, socle	C₂⁶	C₂⁵
commutator	C₂⁴	C₂⁵×C₄
Frattini, cosocle	C₂⁵	C₂⁶
prime radical	C₂⁴×D₄²	C₂
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁶ ◁ G
LECS	G ▷ C₂⁵ ▷ C₁

At least 6464 conjugacy classes of subgroups; only the 19680 normal subgroups are shown.

Full subgroup data →

Presentation

minimal degree	μ(G) = 22
transitive degree	μ_t(G) = 512

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

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

Conjugacy classes & character table

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

Relationships

maximal subgroup of	C₂×#72334421
maximal normal subgroup of	C₂×#72334421

GAP

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

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

C2×#72252090 #72334768