C₂×#72233716 #72334677

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

Aliases

direct product

C₂×#72233716

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 6892 conjugacy classes of subgroups; only the 40488 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)(11,12,15,16)(13,17,18,14)(19,20)(21,25)(22,23)(24,26);(11,12)(13,17)(14,18)(15,16)(19,21)(20,24)(22,25)(23,26);(12,16)(14,17)(19,21)(20,25)(22,24)(23,26);(4,6);(1,2);(8,10)(11,13)(12,17)(14,16)(15,18)(19,21)(20,25)(22,24)(23,26);(3,4)(5,6)(7,8,9,10)(11,14,15,17)(12,18,16,13)(19,22,23,20)(21,25,26,24)

Conjugacy classes & character table

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

Relationships

maximal subgroup of	C₂×#72233717, C₂×#72335129
maximal normal subgroup of	C₂×#72233717, C₂×#72335129

GAP

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

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

C2×#72233716 #72334677