C₂×#72233696 #72334663

order	2048
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁷ · Schur multiplier (H₂) = C₂²⁰

Aliases

direct product

C₂×#72233696

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

1105620 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:

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

Conjugacy classes & character table

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

Relationships

maximal subgroup of	C₂×#72233717, C₂×#72334382, #72337273
maximal normal subgroup of	C₂×#72233717, C₂×#72334382, #72337273

GAP

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

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

C2×#72233696 #72334663