C₂×#72297844 #72324147

order	2048
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁷ · Schur multiplier (H₂) = C₂¹⁹×C₄

Aliases

direct product

C₂×#72297844

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 14592 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:

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

Conjugacy classes & character table

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

Relationships

maximal subgroup of	C₂×#72323275
maximal normal subgroup of	C₂×#72323275

GAP

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

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

C2×#72297844 #72324147