C₂×#72323614 #72339379

order	2048
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁵×C₄ · Schur multiplier (H₂) = C₂¹⁹

Aliases

direct product

C₂×#72323614

Subgroups

	subgroup	G/H
center, socle	C₂⁶	C₂⁵
commutator	C₂⁴	C₂⁵×C₄
Frattini, cosocle	C₂⁵	C₂⁶
prime radical	C₂⁵×#121	C₂
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁶ ◁ G
LECS	G ▷ C₂⁵ ▷ C₁

879496 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 30
transitive degree	μ_t(G) = 512

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

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

Conjugacy classes & character table

368 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((3,4,5,6)(7,8,9,10)(12,14)(15,16)(17,21)(18,22)(19,20)(23,24,27,28)(25,26,29,30), (4,6)(7,8,9,10)(12,14)(17,22)(18,21)(25,29)(26,30), (4,6)(8,10)(11,12,13,14)(16,20)(18,21)(23,25,27,29)(24,26,28,30), (3,4)(5,6)(7,8)(9,10)(11,12)(13,14)(15,17)(16,18)(19,22)(20,21), (1,2), (15,18)(16,17)(19,21)(20,22)(23,26)(24,25)(27,30)(28,29));

C2×#72323614 #72339379