C₂×#72334341 #72334575

order	2048
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁷ · Schur multiplier (H₂) = C₂¹⁹×C₄²

Aliases

direct product

C₂×#72334341

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

952624 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:

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

Conjugacy classes & character table

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

Relationships

maximal subgroup of	C₂×#72334342
maximal normal subgroup of	C₂×#72334342

GAP

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

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

C2×#72334341 #72334575