C₂×#72297575 #72335194

order	2048
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁷ · Schur multiplier (H₂) = C₂¹⁹×C₄

Aliases

direct product

C₂×#72297575

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

At least 10150 conjugacy classes of subgroups; only the 40120 normal subgroups are shown.

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

Conjugacy classes & character table

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

Relationships

maximal subgroup of	C₂×#72335129
maximal normal subgroup of	C₂×#72335129

GAP

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

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

C2×#72297575 #72335194