C₂×#72232056 #72324486

order	2048
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁷ · Schur multiplier (H₂) = C₂²¹

Aliases

direct product

C₂×#72232056

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 7332 conjugacy classes of subgroups; only the 40632 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:

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

Conjugacy classes & character table

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

Relationships

maximal subgroup of	C₂×#72232064
maximal normal subgroup of	C₂×#72232064

GAP

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

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

C2×#72232056 #72324486