C₂×#72260984 #72341857

order	2048
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁷ · Schur multiplier (H₂) = C₂²¹

Aliases

direct product

C₂×#72260984

Subgroups

	subgroup	G/H
center, socle	C₂⁵	C₂⁶
commutator, Frattini, cosocle	C₂⁴	C₂⁷
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁵ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

At least 10144 conjugacy classes of subgroups; only the 43624 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)(12,14)(15,16,19,22)(17,21,20,18);(4,6);(8,10)(12,14);(7,8)(9,10)(11,12,13,14)(15,17)(16,18)(19,20)(21,22);(7,8)(9,10)(11,12,13,14)(15,18,19,21)(16,20,22,17);(7,8)(9,10)(11,12)(13,14)(15,18)(16,20)(17,22)(19,21);(3,4)(5,6)(15,17)(16,21)(18,22)(19,20)

Conjugacy classes & character table

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

Relationships

maximal subgroup of	C₂×#120×#525
maximal normal subgroup of	C₂×#120×#525

GAP

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

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

C2×#72260984 #72341857