C₂×#72337292 #72340016

order	2048
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶×C₄ · Schur multiplier (H₂) = C₂²³

Aliases

direct product

C₂×#72337292

Subgroups

	subgroup	G/H
center, socle	C₂⁵	C₂⁶
commutator	C₂³	C₂⁶×C₄
Frattini, cosocle	C₂⁴	C₂⁷
prime radical	C₂⁵×H(5,𝔽₂)	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂⁵ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

4648920 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 30
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 7 generators on 30 points:

(4,6)(8,10)(15,16)(17,22)(18,21)(19,23)(20,25)(24,28)(26,29)(27,30);(15,17,19,25)(16,20,23,22)(18,24,26,30)(21,27,29,28);(4,6)(8,10)(12,14)(15,18,19,26)(16,21,23,29)(17,24,25,30)(20,27,22,28);(12,14)(15,18)(16,21)(17,24)(19,26)(20,27)(22,28)(23,29)(25,30);(15,16)(17,20)(18,21)(19,23)(22,25)(24,27)(26,29)(28,30);(1,2);(3,4)(5,6)(7,8,9,10)(11,12)(13,14)(15,17)(16,22)(18,24)(19,25)(20,23)(21,28)(26,30)(27,29)

Conjugacy classes & character table

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

Relationships

maximal quotient of

#72340031, C₂×#72337391

GAP

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

gap> G := Group((4,6)(8,10)(15,16)(17,22)(18,21)(19,23)(20,25)(24,28)(26,29)(27,30), (15,17,19,25)(16,20,23,22)(18,24,26,30)(21,27,29,28), (4,6)(8,10)(12,14)(15,18,19,26)(16,21,23,29)(17,24,25,30)(20,27,22,28), (12,14)(15,18)(16,21)(17,24)(19,26)(20,27)(22,28)(23,29)(25,30), (15,16)(17,20)(18,21)(19,23)(22,25)(24,27)(26,29)(28,30), (1,2), (3,4)(5,6)(7,8,9,10)(11,12)(13,14)(15,17)(16,22)(18,24)(19,25)(20,23)(21,28)(26,30)(27,29));

C2×#72337292 #72340016