C₂×#72237236 #72302874

order	2048
nilpotency class	3
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂¹⁶

Aliases

direct product

C₂×#72237236

Subgroups

	subgroup	G/H
center	C₂³×C₄	C₂×#127
commutator	C₂³×C₄	C₂⁴×C₄
Frattini, cosocle	C₂⁴×C₄	C₂⁵
socle	C₂⁴	C₂×#392
prime radical	C₂⁴×#525	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₄ ◁ C₂⁶×C₄ ◁ G
LECS	G ▷ C₂⁴×C₄ ▷ C₂ ▷ C₁

At least 7416 conjugacy classes of subgroups; only the 9240 normal subgroups are shown.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 5 generators on 26 points:

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

Conjugacy classes & character table

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

Relationships

maximal quotient of

#72303335

GAP

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

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

C2×#72237236 #72302874