C₂×#14554854 #72244167

order	4374
nilpotency class	3
automorphism	?
homology	Abelianization (H₁) = C₂×C₃⁴ · Schur multiplier (H₂) = C₃⁷

Aliases

direct product

C₂×#14554854

Subgroups

	subgroup	G/H
center, socle	C₂×C₃	H(3,𝔽₃)²
commutator, Frattini, cosocle	C₃³	C₂×C₃⁴
LCS	G ▷ C₃³ ▷ C₃ ▷ C₁
UCS	C₁ ◁ C₂×C₃ ◁ C₂×C₃³ ◁ G

2800 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 29
transitive degree	μ_t(G) = 54

A minimal faithful permutation representation — 4 generators on 29 points:

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

Conjugacy classes & character table

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

Relationships

Fitting subgroup of	C₂×#72221885, C₂×#72221865, C₂×#72221901
maximal subgroup of	C₂×#72221849, C₂×#72221885, C₂×#72221865, …
maximal normal subgroup of	C₂×#72221849, C₂×#72221885, C₂×#72221865, …

GAP

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

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

C2×#14554854 #72244167