C₂×#72232322 #72338059

order	2048
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁷ · Schur multiplier (H₂) = C₂¹⁹

Aliases

direct product

C₂×#72232322

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 15534 conjugacy classes of subgroups; only the 35984 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:

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

Conjugacy classes & character table

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

Relationships

maximal subgroup of	C₂×#72232371
maximal normal subgroup of	C₂×#72232371

GAP

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

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

C2×#72232322 #72338059