C₂×#2141105 #72325303

order	1024
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹¹×C₄²

Aliases

direct product

C₂×#2141105

Subgroups

	subgroup	G/H
center, socle	C₂⁵	C₂⁵
commutator, Frattini, cosocle	C₂⁴	C₂⁶
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁵ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

48335 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:

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

Conjugacy classes & character table

160 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72232083, C₂×#72232158, C₂×#72232124
maximal normal subgroup of	C₂×#72232083, C₂×#72232158, C₂×#72232124

GAP

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

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

C2×#2141105 #72325303