C₄⋊C₈×C₆₇ #14523909

order	2144
nilpotency class	2
isoclinism	isoclinic class of D₄
automorphism	?
homology	Abelianization (H₁) = C₂×C₈×C₆₇ · Schur multiplier (H₂) = C₂

Aliases

direct product	C₄⋊C₈×C₆₇
unambiguous semidirect product	C₄⋊(C₈×C₆₇)

Subgroups

	subgroup	G/H
center	C₂×C₄×C₆₇	C₂²
commutator	C₂	C₂×C₈×C₆₇
Frattini, cosocle	C₂×C₄	C₂²×C₆₇
socle, prime radical	C₂²×C₆₇	C₂×C₄
LCS	G ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₆₇ ◁ G

38 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 79
transitive degree	μ_t(G) = 2144

A minimal faithful permutation representation — 2 generators on 79 points:

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

Conjugacy classes & character table

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

Relationships

Fitting subgroup of	#16863123, #16863126, #16863139, …
maximal subgroup of	D₄⋊C₈×C₆₇, Q₈⋊C₈×C₆₇, #331×C₆₇, …
maximal normal subgroup of	D₄⋊C₈×C₆₇, Q₈⋊C₈×C₆₇, #331×C₆₇, …
maximal quotient of	#334×C₆₇, #335×C₆₇, #336×C₆₇, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(2144, 12);

C4⋊C8×C67 #14523909