C₂×#72268547 #72289192

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

Aliases

direct product

C₂×#72268547

Subgroups

	subgroup	G/H
center, socle	C₂⁵	C₂⁶
commutator, Frattini, cosocle	C₂⁴	C₂⁷
prime radical	C₂⁵×#120	C₂
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁵ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

2793640 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 26
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 7 generators on 26 points:

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

Conjugacy classes & character table

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

Relationships

maximal subgroup of	C₂×#72268588
maximal normal subgroup of	C₂×#72268588

GAP

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

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

C2×#72268547 #72289192