C₂×#72256563 #72289189

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

Aliases

direct product

C₂×#72256563

Subgroups

	subgroup	G/H
center, socle	C₂⁵	C₂⁶
commutator, Frattini, cosocle	C₂⁴	C₂⁷
prime radical	C₂³×#2846	C₂
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁵ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

2822036 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:

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

Conjugacy classes & character table

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

Relationships

maximal subgroup of	C₂×#72268588, C₂×#72256567
maximal normal subgroup of	C₂×#72268588, C₂×#72256567

GAP

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

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

C2×#72256563 #72289189