C₂×#115348 #72225616

order	1024
nilpotency class	3
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂¹⁰

Aliases

direct product

C₂×#115348

Subgroups

	subgroup	G/H
center, socle	C₂⁴	#379
commutator	C₂⁵	C₂³×C₄
Frattini, cosocle	C₂⁶	C₂⁴
prime radical	C₂×#34980	C₂
LCS	G ▷ C₂⁵ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ C₂⁷ ◁ G
LECS	G ▷ C₂⁶ ▷ C₂² ▷ C₁

27163 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 22
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 22 points:

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

Conjugacy classes & character table

136 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72225629, #72225645, #72225664, …
maximal normal subgroup of	#72225629, #72225645, #72225664, …
maximal quotient of	#72312063

GAP

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

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

C2×#115348 #72225616