C₂×#72237463 #72339353

order	2048
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶×C₄ · Schur multiplier (H₂) = C₂²⁰

Aliases

direct product

C₂×#72237463

Subgroups

	subgroup	G/H
center, socle	C₂⁵	C₂⁶
commutator	C₂³	C₂⁶×C₄
Frattini, cosocle	C₂⁴	C₂⁷
prime radical	C₂⁴×D₄²	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂⁵ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

946456 conjugacy classes of subgroups.

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:

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

Conjugacy classes & character table

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

Relationships

maximal subgroup of	C₂×#72237469
maximal normal subgroup of	C₂×#72237469

GAP

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

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

C2×#72237463 #72339353