C₂×#72297852 #72334554

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

Aliases

direct product

C₂×#72297852

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 7506 conjugacy classes of subgroups; only the 42872 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:

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

Conjugacy classes & character table

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

Relationships

maximal subgroup of	C₂×D₄×#34980, C₂×#72334357, C₂×#72335137
maximal normal subgroup of	C₂×D₄×#34980, C₂×#72334357, C₂×#72335137

GAP

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

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

C2×#72297852 #72334554