C₂×C₄⋊C₄×C₆₇ #14523920

order	2144
nilpotency class	2
isoclinism	isoclinic class of D₄
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₆₇ · Schur multiplier (H₂) = C₂³

Aliases

direct product

C₂×C₄⋊C₄×C₆₇

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂³×C₆₇	C₂²
commutator	C₂	C₂²×C₄×C₆₇
Frattini, cosocle	C₂²	C₂³×C₆₇
LCS	G ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₆₇ ◁ G

92 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 77
transitive degree	μ_t(G) = 2144

A minimal faithful permutation representation — 3 generators on 77 points:

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

Conjugacy classes & character table

1340 conjugacy classes. The character table is too large to display.

Relationships

derived subgroup of	#30743355, #30743356, #30743357, …
Fitting subgroup of	#16863122, #16863158, #16863165, …
maximal subgroup of	#324×C₆₇, #340×C₆₇, #343×C₆₇, …
maximal normal subgroup of	#324×C₆₇, #340×C₆₇, #343×C₆₇, …
maximal quotient of	C₂×#95×C₆₇, C₄×C₄⋊C₄×C₆₇, #380×C₆₇, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(2144, 23);

C2×C4⋊C4×C67 #14523920