C₂×#72232907 #72296398

order	2048
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁷ · Schur multiplier (H₂) = C₂²⁰

Aliases

direct product

C₂×#72232907

Subgroups

	subgroup	G/H
center, socle	C₂⁵	C₂⁶
commutator, Frattini, cosocle	C₂⁴	C₂⁷
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁵ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

At least 11612 conjugacy classes of subgroups; only the 32736 normal subgroups are shown.

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:

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

Conjugacy classes & character table

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

Relationships

maximal quotient of

C₂×#72278229

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

C2×#72232907 #72296398