C₂×#72302231 #72339388

order	2048
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁵×C₄ · Schur multiplier (H₂) = C₂²⁰×C₄

Aliases

direct product

C₂×#72302231

Subgroups

	subgroup	G/H
center, socle	C₂⁶	C₂⁵
commutator	C₂⁴	C₂⁵×C₄
Frattini, cosocle	C₂⁵	C₂⁶
prime radical	C₂⁷×D₄	C₂
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁶ ◁ G
LECS	G ▷ C₂⁵ ▷ C₁

8174472 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

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

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

Conjugacy classes & character table

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

Relationships

maximal subgroup of	C₂×#72334969, C₂×#72337391
maximal normal subgroup of	C₂×#72334969, C₂×#72337391

GAP

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

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

C2×#72302231 #72339388