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C₁₆×Dic₇ #89295

order	448
isoclinism	isoclinic class of D₇
automorphism	?
homology	Abelianization (H₁) = C₄×C₁₆ · Schur multiplier (H₂) = C₄

Aliases

C₁₆×Dic₇

Splittings

C₇ ⋊ C₄×C₁₆

Subgroups

	subgroup	G/H
center, hypercenter	C₂×C₁₆	D₇
commutator, hypocenter	C₇	C₄×C₁₆
Frattini	C₂×C₈	C₂×D₇
Fitting	C₂×C₇×C₁₆	C₂
socle, prime radical	C₂²×C₇	C₂×C₈
cosocle	C₂×C₇×C₈	C₂²
LCS	G ▷ C₇
UCS	C₁ ◁ C₂×C₁₆

58 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 27
transitive degree	μ_t(G) = 448

A minimal faithful permutation representation — 2 generators on 27 points:

(2,3)(4,5)(6,7)(8,9,10,11);(1,2,4,6,7,5,3)(12,13,14,17,15,18,20,23,16,19,21,24,22,25,26,27)

Conjugacy classes & character table

160 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	Dic₇×C₃₂, #11725395, Dic₇⋊C₃₂, …
maximal normal subgroup of	Dic₇×C₃₂, #11725395, Dic₇⋊C₃₂, …
maximal quotient of	C₁₆×C₇⋊C₈, #11725305, Dic₇×C₃₂, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(448, 57);