C₂×#72237461 #72333249

order	2048
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁷ · Schur multiplier (H₂) = C₂²⁰

Aliases

direct product

C₂×#72237461

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

935732 conjugacy classes of subgroups.

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:

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

Conjugacy classes & character table

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

Relationships

maximal subgroup of	C₂×#72323423, C₂×#72339320, C₂×#72237469
maximal normal subgroup of	C₂×#72323423, C₂×#72339320, C₂×#72237469

GAP

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

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

C2×#72237461 #72333249