C₂×#72256612 #72331773

order	2048
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁸ · Schur multiplier (H₂) = C₂²⁶

Aliases

direct product

C₂×#72256612

Subgroups

	subgroup	G/H
center, socle	C₂⁴	C₂⁷
commutator, Frattini, cosocle	C₂³	C₂⁸
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂³ ▷ C₁

At least 11188 conjugacy classes of subgroups; only the 444488 normal subgroups are shown.

Full subgroup data →

Presentation

minimal degree	μ(G) = 30
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 8 generators on 30 points:

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

Conjugacy classes & character table

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

Relationships

maximal quotient of

C₂×#72323339

GAP

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

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

C2×#72256612 #72331773