C₂²×#63011 #72232063

order	1024
nilpotency class	2
isoclinism	isoclinic class of #63011
automorphism	?
homology	Abelianization (H₁) = C₂⁸ · Schur multiplier (H₂) = C₂²⁶

Aliases

direct product

C₂²×#63011

Subgroups

	subgroup	G/H
center, socle	C₂⁴	C₂⁶
commutator, Frattini, cosocle	C₂²	C₂⁸
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂² ▷ C₁

557505 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 20
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 8 generators on 20 points:

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

Conjugacy classes & character table

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

Relationships

Fitting subgroup of	#72262047, #72262232, C₂²×#63011⋊C₃, …
maximal subgroup of	#72232064, C₂²×#7622880, #72252039, …
maximal normal subgroup of	#72232064, C₂²×#7622880, #72252039, …
maximal quotient of	#72233731, #72234898, C₂²×#7684106, …

GAP

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

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

C22×#63011 #72232063