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C₇²×#100 #13096161

order	1568
nilpotency class	3
isoclinism	isoclinic class of #99
automorphism	?
homology	Abelianization (H₁) = C₂×C₄×C₇² · Schur multiplier (H₂) = C₂×C₇

Aliases

C₇²×#100

Subgroups

	subgroup	G/H
center, socle	C₂×C₇²	C₂²⋊C₄
commutator	C₂²	C₂×C₄×C₇²
Frattini, cosocle	C₂×C₄	C₂²×C₇²
prime radical	C₂×C₇²×D₄	C₂
LCS	G ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₇² ◁ C₂×C₄×C₇² ◁ G

230 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 22
transitive degree	μ_t(G) = 392

A minimal faithful permutation representation — 2 generators on 22 points:

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

Conjugacy classes & character table

539 conjugacy classes — view the full character table.

Relationships

Fitting subgroup of	#100×C₇⋊D₇, #72246218, #100×#3861, …
maximal subgroup of	#100×C₇⋊D₇, C₇²×#453, #72246218, …
maximal normal subgroup of	#100×C₇⋊D₇, C₇²×#453, #72246218, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1568, 327);