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C₄×C₇×C₈ #6413

order	224
nilpotency class	1
isoclinism	isoclinic class of C₁
automorphism	Aut(G) = C₂×C₃×#2021 (order 768) · Inn(G) = C₁ · Out(G) = C₂×C₃×#2021
homology	Abelianization (H₁) = G · Schur multiplier (H₂) = C₄

Aliases

C₄×C₇×C₈

Subgroups

	subgroup	G/H
center	G	C₁
commutator	C₁	G
Frattini, cosocle	C₂×C₄	C₂²×C₇
socle, prime radical	C₂²×C₇	C₂×C₄
LCS	G ▷ C₁
UCS	C₁ ◁ G

44 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 19
transitive degree	μ_t(G) = 224

A minimal faithful permutation representation — 2 generators on 19 points:

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

Conjugacy classes & character table

224 conjugacy classes — view the full character table.

Relationships

center of	C₇×#1311, C₇×#1312, C₇×C₈×M₁₆, …
derived subgroup of	#13240516, #13240517, #13240518, …
Fitting subgroup of	C₈×C₇⋊C₈, #89249, #89250, …
maximal subgroup of	C₈×C₇⋊C₈, #89249, #89250, …
maximal normal subgroup of	C₈×C₇⋊C₈, #89249, #89250, …
maximal quotient of	C₇×C₈², C₇×#322, C₇×#336, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(224, 45);