C₂²×#122963 #72286352

order	2048
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₁₆ · Schur multiplier (H₂) = C₂¹⁰

Aliases

direct product

C₂²×#122963

Subgroups

	subgroup	G/H
center	C₂³×C₄	#442
commutator	C₈	C₂⁴×C₁₆
Frattini, cosocle	C₈²	C₂⁵
socle	C₂⁴	#1580
prime radical	C₂³×D₁₆	C₈
LCS	G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₄ ◁ C₂²×C₄×C₈ ◁ C₂²×#1312 ◁ G
LECS	G ▷ C₈² ▷ C₄² ▷ C₂² ▷ C₁

4720 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 1024

A minimal faithful permutation representation — 5 generators on 36 points:

(1,2);(3,4);(5,6,8,12,9,13,15,19,10,14,18,7,11,16,20,17)(21,22,24,28,26,30,33,31)(23,27,34,36,25,35,32,29);(22,27)(23,32)(24,33)(25,34)(28,29)(30,35)(31,36);(5,7,15,6,11,19,8,16,10,12,20,14,9,17,18,13)(21,23,33,32,26,25,24,34)(22,27,31,29,30,35,28,36)

Conjugacy classes & character table

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

Relationships

maximal subgroup of	C₂²×#72224639, #72287331, #72322318, …
maximal normal subgroup of	C₂²×#72224639, #72287331, #72322318, …
maximal quotient of	C₂²×#72224639, #72323120

GAP

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

gap> G := Group((1,2), (3,4), (5,6,8,12,9,13,15,19,10,14,18,7,11,16,20,17)(21,22,24,28,26,30,33,31)(23,27,34,36,25,35,32,29), (22,27)(23,32)(24,33)(25,34)(28,29)(30,35)(31,36), (5,7,15,6,11,19,8,16,10,12,20,14,9,17,18,13)(21,23,33,32,26,25,24,34)(22,27,31,29,30,35,28,36));

C22×#122963 #72286352