C₂×#72323644 #72339375

order	2048
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁵×C₄ · Schur multiplier (H₂) = C₂¹⁸×C₄

Aliases

direct product

C₂×#72323644

Subgroups

	subgroup	G/H
center, socle	C₂⁶	C₂⁵
commutator	C₂⁴	C₂⁵×C₄
Frattini, cosocle	C₂⁵	C₂⁶
prime radical	C₂⁵×#120	C₂
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁶ ◁ G
LECS	G ▷ C₂⁵ ▷ C₁

2680984 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 6 generators on 30 points:

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

Conjugacy classes & character table

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

Relationships

maximal subgroup of	C₂×#72337376
maximal normal subgroup of	C₂×#72337376

GAP

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

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

C2×#72323644 #72339375