Finite Group Databasebeta

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C₇×C₈×C₂₂₃ #72190830

order	12488
nilpotency class	1
isoclinism	isoclinic class of C₁
automorphism	Aut(G) = C₂⁴×C₃²×C₃₇ (order 5328) · Inn(G) = C₁ · Out(G) = C₂⁴×C₃²×C₃₇
homology	Abelianization (H₁) = G · Schur multiplier (H₂) = C₁

Aliases

cyclic	C₁₂₄₈₈
abelian	C₇×C₈×C₂₂₃

Subgroups

	subgroup	G/H
center	G	C₁
commutator	C₁	G
Frattini, cosocle	C₄	C₂×C₇×C₂₂₃
socle, prime radical	C₂×C₇×C₂₂₃	C₄
LCS	G ▷ C₁
UCS	C₁ ◁ G

16 conjugacy classes of subgroups.

Full subgroup data →

Conjugacy classes & character table

12488 conjugacy classes. The character table is too large to display.

Relationships

derived subgroup of	Dic₁₂₄₈₈, D₂₄₉₇₆
maximal subgroup of	C₇×C₁₆×C₂₂₃, C₂×C₇×C₈×C₂₂₃, Dic₆₂₄₄, …
maximal normal subgroup of	C₇×C₁₆×C₂₂₃, C₂×C₇×C₈×C₂₂₃, Dic₆₂₄₄, …
maximal quotient of	C₇×C₁₆×C₂₂₃, C₂×C₇×C₈×C₂₂₃

GAP

Past the SmallGroups library coverage — no SmallGroups id.