C₂×#7646028 #72323780

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

Aliases

direct product

C₂×#7646028

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

99349 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

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

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

Conjugacy classes & character table

196 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72238224
maximal normal subgroup of	C₂×#72238224
maximal quotient of	C₂×#72323246, C₂×#72323386

GAP

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

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

C2×#7646028 #72323780