C₂×#72323422 #72332938

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

Aliases

direct product

C₂×#72323422

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 11442 conjugacy classes of subgroups; only the 36040 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)(19,20)(21,22)(23,25)(24,26);(7,8)(9,10)(13,17)(16,18)(19,21)(20,24)(22,25)(23,26);(11,12,15,14)(13,18,17,16)(19,21)(20,24)(22,25)(23,26);(1,2);(4,6)(21,26)(22,24);(11,13)(12,16)(14,18)(15,17)(19,20)(21,24)(22,26)(23,25);(3,4)(5,6)(11,14,15,12)(13,16,17,18)(19,22,23,24)(20,21,25,26)

Conjugacy classes & character table

320 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((8,10)(19,20)(21,22)(23,25)(24,26), (7,8)(9,10)(13,17)(16,18)(19,21)(20,24)(22,25)(23,26), (11,12,15,14)(13,18,17,16)(19,21)(20,24)(22,25)(23,26), (1,2), (4,6)(21,26)(22,24), (11,13)(12,16)(14,18)(15,17)(19,20)(21,24)(22,26)(23,25), (3,4)(5,6)(11,14,15,12)(13,16,17,18)(19,22,23,24)(20,21,25,26));

C2×#72323422 #72332938