C₂×#72300631 #72331966

order	2048
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁷ · Schur multiplier (H₂) = C₂²⁰

Aliases

direct product

C₂×#72300631

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 22994 conjugacy classes of subgroups; only the 35360 normal subgroups are shown.

Full subgroup data →

Presentation

minimal degree	μ(G) = 26
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 7 generators on 26 points:

(8,10)(13,17)(16,18);(4,6)(7,8)(9,10)(11,12,14,15)(13,16,17,18)(19,20,23,25)(21,26,22,24);(4,6)(11,12,14,15)(13,16,17,18)(19,21,23,22)(20,24,25,26);(1,2);(3,4,5,6)(12,15)(16,18)(20,25)(21,22);(11,12)(13,16)(14,15)(17,18)(19,20)(21,26)(22,24)(23,25);(11,13)(12,16)(14,17)(15,18)(19,22,23,21)(20,26,25,24)

Conjugacy classes & character table

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

Relationships

maximal subgroup of	C₂×#72323339
maximal normal subgroup of	C₂×#72323339

GAP

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

gap> G := Group((8,10)(13,17)(16,18), (4,6)(7,8)(9,10)(11,12,14,15)(13,16,17,18)(19,20,23,25)(21,26,22,24), (4,6)(11,12,14,15)(13,16,17,18)(19,21,23,22)(20,24,25,26), (1,2), (3,4,5,6)(12,15)(16,18)(20,25)(21,22), (11,12)(13,16)(14,15)(17,18)(19,20)(21,26)(22,24)(23,25), (11,13)(12,16)(14,17)(15,18)(19,22,23,21)(20,26,25,24));

C2×#72300631 #72331966