H(3,𝔽₄)×C₇₁ #17985692

order	4544
nilpotency class	2
isoclinism	isoclinic class of #560
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₇₁ · Schur multiplier (H₂) = C₂²×C₄²

Aliases

direct product

H(3,𝔽₄)×C₇₁

Subgroups

	subgroup	G/H
center, socle	C₂²×C₇₁	C₂⁴
commutator, Frattini, cosocle	C₂²	C₂⁴×C₇₁
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₇₁ ◁ G

260 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 87
transitive degree	μ_t(G) = 1136

A minimal faithful permutation representation — 4 generators on 87 points:

(1,2,5,8)(3,10,11,7)(4,12,14,16)(6,9,15,13)(17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87);(1,3,5,11)(2,7,8,10)(4,13,14,9)(6,12,15,16);(1,4)(2,3)(5,14)(6,7)(8,11)(9,12)(10,15)(13,16);(1,3)(2,4)(5,11)(6,12)(7,9)(8,14)(10,13)(15,16)

Conjugacy classes & character table

1349 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	C₇₁×#1655, C₇₁×#1659, C₇₁×#2199, …
maximal normal subgroup of	C₇₁×#1655, C₇₁×#1659, C₇₁×#2199, …
maximal quotient of	C₇₁×#2375, C₇₁×#2379, C₇₁×#2670, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(4544, 242);

H(3,𝔽4)×C71 #17985692