C₂×#72241009 #72339338

order	2048
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁷ · Schur multiplier (H₂) = C₂¹⁸×C₄²

Aliases

direct product

C₂×#72241009

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

2752180 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)(19,20)(21,23)(22,24)(25,26);(4,6)(8,10)(12,15)(13,17)(19,20,22,24)(21,25,26,23);(4,6)(7,8,9,10)(13,17)(16,18)(19,20)(21,25)(22,24)(23,26);(3,4)(5,6)(7,8)(9,10)(11,12)(13,16)(14,15)(17,18);(1,2);(19,21)(20,23)(22,26)(24,25);(11,13)(12,16)(14,17)(15,18)(19,21)(20,23)(22,26)(24,25)

Conjugacy classes & character table

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

Relationships

maximal subgroup of	C₂×#72339320, C₂×#72256567
maximal normal subgroup of	C₂×#72339320, C₂×#72256567

GAP

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

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

C2×#72241009 #72339338