C₂×#72278198 #72296477

order	2048
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹⁷

Aliases

direct product

C₂×#72278198

Subgroups

	subgroup	G/H
center, socle	C₂⁶	C₂⁵
commutator, Frattini, cosocle	C₂⁵	C₂⁶
LCS	G ▷ C₂⁵ ▷ C₁
UCS	C₁ ◁ C₂⁶ ◁ G
LECS	G ▷ C₂⁵ ▷ C₁

At least 15858 conjugacy classes of subgroups; only the 8584 normal subgroups are shown.

Full subgroup data →

Presentation

minimal degree	μ(G) = 26
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 6 generators on 26 points:

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

Conjugacy classes & character table

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

Relationships

maximal subgroup of	C₂×#72278229
maximal normal subgroup of	C₂×#72278229

GAP

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

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

C2×#72278198 #72296477