C₂×#72232497 #72299501

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

Aliases

direct product

C₂×#72232497

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 7500 conjugacy classes of subgroups; only the 35400 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:

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

Conjugacy classes & character table

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

Relationships

maximal subgroup of	C₂×#72297602
maximal normal subgroup of	C₂×#72297602

GAP

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

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

C2×#72232497 #72299501