C₂×#72323416 #72333178

order	2048
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁷ · Schur multiplier (H₂) = C₂¹⁸×C₄

Aliases

direct product

C₂×#72323416

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 9248 conjugacy classes of subgroups; only the 39464 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)(7,8)(9,10)(11,12,14,15)(13,17,18,16)(20,22)(24,25);(4,6)(8,9)(11,12,14,15)(13,16,18,17)(19,20)(21,25)(22,23)(24,26);(19,21)(20,24)(22,25)(23,26);(1,2);(3,4,5,6)(7,9,10,8)(12,15)(13,18)(19,21)(20,25)(22,24)(23,26);(11,12)(13,16)(14,15)(17,18)(19,21)(20,25)(22,24)(23,26);(11,13)(12,16)(14,18)(15,17)(19,22,23,20)(21,25,26,24)

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

C2×#72323416 #72333178