C₂×#72232095 #72325680

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

Aliases

direct product

C₂×#72232095

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 22532 conjugacy classes of subgroups; only the 35496 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:

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

Conjugacy classes & character table

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

Relationships

maximal subgroup of	C₂×#72232191
maximal normal subgroup of	C₂×#72232191

GAP

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

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

C2×#72232095 #72325680