C₂×#72297911 #72324115

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

Aliases

direct product

C₂×#72297911

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 14774 conjugacy classes of subgroups; only the 35728 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:

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

Conjugacy classes & character table

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

Relationships

maximal subgroup of	C₂×#72323275
maximal normal subgroup of	C₂×#72323275

GAP

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

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

C2×#72297911 #72324115