C₄×D₄×C₆₇ #14523922

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

Aliases

direct product

C₄×D₄×C₆₇

Subgroups

	subgroup	G/H
center	C₂×C₄×C₆₇	C₂²
commutator	C₂	C₂²×C₄×C₆₇
Frattini, cosocle	C₂²	C₂³×C₆₇
socle	C₂²×C₆₇	C₂³
prime radical	C₂×D₄×C₆₇	C₂
LCS	G ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₆₇ ◁ G

94 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 75
transitive degree	μ_t(G) = 1072

A minimal faithful permutation representation — 3 generators on 75 points:

(1,2,3,4)(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);(5,6,7,8);(6,8)

Conjugacy classes & character table

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

Relationships

Fitting subgroup of	D₄⋊(C₆₇⋊C₈), D₄×C₆₇⋊C₈, #16863468, …
maximal subgroup of	D₄⋊C₈×C₆₇, C₈×D₄×C₆₇, #435×C₆₇, …
maximal normal subgroup of	D₄⋊C₈×C₆₇, C₈×D₄×C₆₇, #435×C₆₇, …
maximal quotient of	C₄×C₂²⋊C₄×C₆₇, C₄×C₄⋊C₄×C₆₇, #385×C₆₇, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(2144, 25);

C4×D4×C67 #14523922