C₂×#72323419 #72333252

order	2048
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁷ · Schur multiplier (H₂) = C₂¹⁹

Aliases

direct product

C₂×#72323419

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 8568 conjugacy classes of subgroups; only the 37320 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:

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

Conjugacy classes & character table

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

Relationships

maximal subgroup of	C₂×#72323423
maximal normal subgroup of	C₂×#72323423

GAP

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

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

C2×#72323419 #72333252