C₂×#637112 #72339960

order	1024
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹²×C₄²

Aliases

direct product

C₂×#637112

Subgroups

	subgroup	G/H
center, socle	C₂⁵	C₂⁵
commutator, Frattini, cosocle	C₂⁴	C₂⁶
prime radical	C₂⁴×#121	C₂
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁵ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

69165 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:

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

Conjugacy classes & character table

184 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72337224, #72339988, #72339993, …
maximal normal subgroup of	C₂×#72337224, #72339988, #72339993, …
maximal quotient of	#72339993, C₂×#72337366

GAP

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

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

C2×#637112 #72339960