C₂×#72278705 #72297083

order	2048
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁵×C₄ · Schur multiplier (H₂) = C₂¹⁴×C₄²

Aliases

direct product

C₂×#72278705

Subgroups

	subgroup	G/H
center, socle	C₂⁶	C₂⁵
commutator	C₂⁴	C₂⁵×C₄
Frattini, cosocle	C₂⁵	C₂⁶
prime radical	C₂²×#15258	C₂
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁶ ◁ G
LECS	G ▷ C₂⁵ ▷ C₁

At least 8322 conjugacy classes of subgroups; only the 10592 normal subgroups are shown.

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:

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

Conjugacy classes & character table

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

Relationships

maximal subgroup of	C₂×#72278822
maximal normal subgroup of	C₂×#72278822

GAP

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

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

C2×#72278705 #72297083