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C₇²×#97 #13096158

order	1568
nilpotency class	2
isoclinism	isoclinic class of D₄
automorphism	?
homology	Abelianization (H₁) = C₄²×C₇² · Schur multiplier (H₂) = C₂×C₇

Aliases

C₇²×#97

Subgroups

	subgroup	G/H
center	C₂×C₄×C₇²	C₂²
commutator	C₂	C₄²×C₇²
Frattini, cosocle	C₂×C₄	C₂²×C₇²
socle, prime radical	C₂²×C₇²	C₂×C₄
LCS	G ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₇² ◁ G

200 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 26
transitive degree	μ_t(G) = 1568

A minimal faithful permutation representation — 2 generators on 26 points:

(1,2,3,4,5,6,7)(19,20,21,24,22,23,25,26);(8,9,10,11,12,13,14)(15,16,17,18)(20,23)(24,26)

Conjugacy classes & character table

980 conjugacy classes — view the full character table.

Relationships

Fitting subgroup of	#97×#3861, #97×C₇⋊D₇, #97×#64424, …
maximal subgroup of	#97×#3861, #97×C₇⋊D₇, C₇²×#442, …
maximal normal subgroup of	#97×#3861, #97×C₇⋊D₇, C₇²×#442, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1568, 324);