C₂×#72268550 #72296249

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

Aliases

direct product

C₂×#72268550

Subgroups

	subgroup	G/H
center, socle	C₂⁵	C₂⁶
commutator, Frattini, cosocle	C₂⁴	C₂⁷
prime radical	C₂×#6342371	C₂
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁵ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

2782790 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 26
transitive degree	μ_t(G) = 128

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

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

Conjugacy classes & character table

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

Relationships

maximal subgroup of	C₂×#72268592
maximal normal subgroup of	C₂×#72268592
maximal quotient of	C₂×#72296169

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

C2×#72268550 #72296249