C₂×#72232143 #72325717

order	2048
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁷ · Schur multiplier (H₂) = C₂¹⁹

Aliases

direct product

C₂×#72232143

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 23316 conjugacy classes of subgroups; only the 35688 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:

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

Conjugacy classes & character table

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

Relationships

maximal subgroup of	C₂×#72232191
maximal normal subgroup of	C₂×#72232191

GAP

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

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

C2×#72232143 #72325717