C₂×#72293965 #72295293

order	2048
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹³×C₄²

Aliases

direct product

C₂×#72293965

Subgroups

	subgroup	G/H
center, socle	C₂⁶	C₂⁵
commutator, Frattini, cosocle	C₂⁵	C₂⁶
LCS	G ▷ C₂⁵ ▷ C₁
UCS	C₁ ◁ C₂⁶ ◁ G
LECS	G ▷ C₂⁵ ▷ C₁

At least 17144 conjugacy classes of subgroups; only the 7640 normal subgroups are shown.

Full subgroup data →

Presentation

minimal degree	μ(G) = 34
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 6 generators on 34 points:

(11,12)(13,16)(14,15)(17,18)(19,20,22,23)(21,24,25,26)(27,28,31,33)(29,30,34,32);(7,8)(9,10)(13,17)(16,18)(21,25)(24,26)(27,29,31,34)(28,32,33,30);(4,6)(8,10)(12,15)(13,17)(20,23)(24,26)(27,30,31,32)(28,29,33,34);(3,4)(5,6)(7,8)(9,10)(11,12)(13,16)(14,15)(17,18)(19,20)(21,24)(22,23)(25,26)(27,30)(28,29)(31,32)(33,34);(11,13)(12,16)(14,17)(15,18)(19,21)(20,24)(22,25)(23,26)(27,29)(28,30)(31,34)(32,33);(1,2)

Conjugacy classes & character table

248 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72293994
maximal normal subgroup of	C₂×#72293994

GAP

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

gap> G := Group((11,12)(13,16)(14,15)(17,18)(19,20,22,23)(21,24,25,26)(27,28,31,33)(29,30,34,32), (7,8)(9,10)(13,17)(16,18)(21,25)(24,26)(27,29,31,34)(28,32,33,30), (4,6)(8,10)(12,15)(13,17)(20,23)(24,26)(27,30,31,32)(28,29,33,34), (3,4)(5,6)(7,8)(9,10)(11,12)(13,16)(14,15)(17,18)(19,20)(21,24)(22,23)(25,26)(27,30)(28,29)(31,32)(33,34), (11,13)(12,16)(14,17)(15,18)(19,21)(20,24)(22,25)(23,26)(27,29)(28,30)(31,34)(32,33), (1,2));

C2×#72293965 #72295293