C₂²×#10018 #72306474

order	1024
nilpotency class	3
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂¹¹

Aliases

direct product

C₂²×#10018

Subgroups

	subgroup	G/H
center, socle	C₂⁴	#392
commutator, Frattini, cosocle	C₂×C₄²	C₂⁵
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ C₂³×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

14621 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 28 points:

(5,6)(7,9)(8,11)(10,12)(14,19,20,26)(16,24)(21,27,22,28)(23,25);(6,9,11,10)(13,14,17,21,18,20,15,22)(16,19,25,27,24,26,23,28);(1,2);(3,4);(1,2)(3,4)(5,6)(7,10)(8,11)(9,12)(13,15)(14,19)(16,23)(17,18)(20,26)(21,28)(22,27)(24,25)

Conjugacy classes & character table

124 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂²×#246927
maximal normal subgroup of	C₂²×#246927

GAP

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

gap> G := Group((5,6)(7,9)(8,11)(10,12)(14,19,20,26)(16,24)(21,27,22,28)(23,25), (6,9,11,10)(13,14,17,21,18,20,15,22)(16,19,25,27,24,26,23,28), (1,2), (3,4), (1,2)(3,4)(5,6)(7,10)(8,11)(9,12)(13,15)(14,19)(16,23)(17,18)(20,26)(21,28)(22,27)(24,25));

C22×#10018 #72306474