C₂×#72323417 #72332971

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

Aliases

direct product

C₂×#72323417

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 11972 conjugacy classes of subgroups; only the 35176 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)(13,18)(16,17)(19,21)(20,23)(22,26)(24,25);(11,12,15,14)(13,16,18,17)(19,21)(20,23)(22,26)(24,25);(1,2);(4,6)(11,13,15,18)(12,16,14,17)(19,22,24,23)(20,25,26,21);(11,13)(12,17)(14,16)(15,18)(19,21)(20,22)(23,26)(24,25);(3,4)(5,6)(11,14,15,12)(13,17,18,16)(19,23,24,22)(20,25,26,21)

Conjugacy classes & character table

272 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)(13,18)(16,17)(19,21)(20,23)(22,26)(24,25), (11,12,15,14)(13,16,18,17)(19,21)(20,23)(22,26)(24,25), (1,2), (4,6)(11,13,15,18)(12,16,14,17)(19,22,24,23)(20,25,26,21), (11,13)(12,17)(14,16)(15,18)(19,21)(20,22)(23,26)(24,25), (3,4)(5,6)(11,14,15,12)(13,17,18,16)(19,23,24,22)(20,25,26,21));

C2×#72323417 #72332971