C₂×#72232405 #72334481

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

Aliases

direct product

C₂×#72232405

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 11488 conjugacy classes of subgroups; only the 40240 normal subgroups are shown.

Full subgroup data →

Presentation

minimal degree	μ(G) = 26
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 7 generators on 26 points:

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

Conjugacy classes & character table

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

Relationships

maximal subgroup of	C₂×#72237114, C₂×#72237469
maximal normal subgroup of	C₂×#72237114, C₂×#72237469

GAP

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

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

C2×#72232405 #72334481