C₂×#72297578 #72334700

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

Aliases

direct product

C₂×#72297578

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 6422 conjugacy classes of subgroups; only the 42440 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:

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

Conjugacy classes & character table

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

Relationships

maximal subgroup of	C₂×#72334357, C₂×#72335137, C₂×#72339320
maximal normal subgroup of	C₂×#72334357, C₂×#72335137, C₂×#72339320

GAP

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

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

C2×#72297578 #72334700