C₂×#7677134 #72302334

order	1024
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁷ · Schur multiplier (H₂) = C₂¹⁹

Aliases

direct product

C₂×#7677134

Subgroups

	subgroup	G/H
center, socle	C₂⁴	C₂⁶
commutator, Frattini, cosocle	C₂³	C₂⁷
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂³ ▷ C₁

101477 conjugacy classes of subgroups.

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:

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

Conjugacy classes & character table

220 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72302346, C₂×#72260861
maximal normal subgroup of	#72302346, C₂×#72260861

GAP

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

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

C2×#7677134 #72302334