H(5,𝔽₂)×C₆₇ #14523946

order	2144
nilpotency class	2
isoclinism	isoclinic class of H(5,𝔽₂)
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₆₇ · Schur multiplier (H₂) = C₂⁵

Aliases

direct product

H(5,𝔽₂)×C₆₇

Subgroups

	subgroup	G/H
center, socle	C₂×C₆₇	C₂⁴
commutator, Frattini, cosocle	C₂	C₂⁴×C₆₇
LCS	G ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₆₇ ◁ G

166 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 75
transitive degree	μ_t(G) = 536

A minimal faithful permutation representation — 4 generators on 75 points:

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

Conjugacy classes & character table

1139 conjugacy classes. The character table is too large to display.

Relationships

Fitting subgroup of	#16863564, #16863574, #16863583, …
maximal subgroup of	#453×C₆₇, #455×C₆₇, B(4,𝔽₂)×C₆₇, …
maximal normal subgroup of	#453×C₆₇, #455×C₆₇, B(4,𝔽₂)×C₆₇, …
maximal quotient of	#518×C₆₇, #520×C₆₇, #534×C₆₇, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(2144, 49);

H(5,𝔽2)×C67 #14523946