C₂×#72232159 #72325741

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

Aliases

direct product

C₂×#72232159

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 22156 conjugacy classes of subgroups; only the 36728 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:

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

Conjugacy classes & character table

332 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((14,18)(16,17)(20,25)(23,26), (4,6)(14,18)(16,17)(19,20)(21,26)(22,23)(24,25), (7,8)(9,10)(11,12,15,13)(14,17,18,16)(19,21,24,22)(20,26,25,23), (7,8,9,10)(12,13)(16,17)(21,22)(23,26), (1,2), (4,6)(7,8)(9,10)(11,13,15,12)(14,17,18,16)(19,22,24,21)(20,23,25,26), (3,4)(5,6)(11,14)(12,16)(13,17)(15,18)(19,23)(20,22)(21,25)(24,26));

C2×#72232159 #72325741