C₈×D₄×C₇₃ #17988153

order	4672
nilpotency class	2
isoclinism	isoclinic class of D₄
automorphism	?
homology	Abelianization (H₁) = C₂²×C₈×C₇₃ · Schur multiplier (H₂) = C₂³

Aliases

direct product

C₈×D₄×C₇₃

Subgroups

	subgroup	G/H
center	C₂×C₈×C₇₃	C₂²
commutator	C₂	C₂²×C₈×C₇₃
Frattini, cosocle	C₂×C₄	C₂³×C₇₃
socle	C₂²×C₇₃	C₂²×C₄
prime radical	C₂×D₄×C₇₃	C₄
LCS	G ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₈×C₇₃ ◁ G

134 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 85
transitive degree	μ_t(G) = 2336

A minimal faithful permutation representation — 3 generators on 85 points:

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

Conjugacy classes & character table

2920 conjugacy classes. The character table is too large to display.

Relationships

Fitting subgroup of	D₄⋊#14556641, #34000865, C₈×#14556655, …
maximal subgroup of	C₇₃×D₄⋊C₁₆, C₈×D₈×C₇₃, C₈×SD₁₆×C₇₃, …
maximal normal subgroup of	C₇₃×D₄⋊C₁₆, C₈×D₈×C₇₃, C₈×SD₁₆×C₇₃, …
maximal quotient of	C₈×D₈×C₇₃, C₈×SD₁₆×C₇₃, C₈×Q₁₆×C₇₃, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(4672, 115);

C8×D4×C73 #17988153