C₂×#72317974 #72331868

order	2048
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁷ · Schur multiplier (H₂) = C₂¹⁹×C₄

Aliases

direct product

C₂×#72317974

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 14824 conjugacy classes of subgroups; only the 35560 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)(11,12,15,14)(13,16,17,18)(19,20,23,25)(21,24,26,22);(7,8)(9,10)(19,20)(21,22)(23,25)(24,26);(19,21)(20,24)(22,25)(23,26);(1,2);(4,6)(12,14)(16,18)(20,25)(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

272 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)(11,12,15,14)(13,16,17,18)(19,20,23,25)(21,24,26,22), (7,8)(9,10)(19,20)(21,22)(23,25)(24,26), (19,21)(20,24)(22,25)(23,26), (1,2), (4,6)(12,14)(16,18)(20,25)(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×#72317974 #72331868