C₂×#72323324 #72331877

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

Aliases

direct product

C₂×#72323324

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 13122 conjugacy classes of subgroups; only the 35320 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)(11,12)(13,17)(14,18)(15,16);(7,8)(9,10)(21,25)(24,26);(11,13)(12,14)(15,18)(16,17)(19,20,23,22)(21,26,25,24);(1,2);(4,6)(12,16)(14,17)(20,22)(21,25);(11,13)(12,17)(14,16)(15,18)(19,21)(20,24)(22,26)(23,25);(3,4)(5,6)(11,14,15,17)(12,18,16,13)(19,22,23,20)(21,24,25,26)

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

C2×#72323324 #72331877