C₂×#72232039 #72324472

order	2048
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶×C₄ · Schur multiplier (H₂) = C₂²¹

Aliases

direct product

C₂×#72232039

Subgroups

	subgroup	G/H
center, socle	C₂⁵	C₂⁶
commutator	C₂³	C₂⁶×C₄
Frattini, cosocle	C₂⁴	C₂⁷
prime radical	C₂⁴×D₄²	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂⁵ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

976688 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:

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

Conjugacy classes & character table

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

Relationships

maximal subgroup of	C₂×#72232064, C₂×#72237114, #72340056
maximal normal subgroup of	C₂×#72232064, C₂×#72237114, #72340056
maximal quotient of	#72340056, C₂×#72337376

GAP

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

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

C2×#72232039 #72324472