C₂×#72232185 #72325778

order	2048
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁷ · Schur multiplier (H₂) = C₂¹⁸×C₄

Aliases

direct product

C₂×#72232185

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 13918 conjugacy classes of subgroups; only the 35688 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:

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

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

C2×#72232185 #72325778