C₂×#72237105 #72334482

order	2048
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁷ · Schur multiplier (H₂) = C₂¹⁸×C₄²

Aliases

direct product

C₂×#72237105

Subgroups

	subgroup	G/H
center, socle	C₂⁵	C₂⁶
commutator, Frattini, cosocle	C₂⁴	C₂⁷
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁵ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

977608 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 26
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 7 generators on 26 points:

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

Conjugacy classes & character table

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

Relationships

maximal subgroup of	C₂×#72237114
maximal normal subgroup of	C₂×#72237114

GAP

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

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

C2×#72237105 #72334482