Finite Group Databasebeta

← Prev (#90472) Next (#90474) →

C₂²×C₄×Dic₇ #90473

order	448
isoclinism	isoclinic class of D₇
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

C₂²×C₄×Dic₇

Splittings

C₇ ⋊ C₂²×C₄²

Subgroups

	subgroup	G/H
center, hypercenter	C₂³×C₄	D₇
commutator, hypocenter	C₇	C₂²×C₄²
Frattini	C₂²	C₂³×D₇
Fitting	C₂³×C₄×C₇	C₂
socle, prime radical	C₂⁴×C₇	C₂²
cosocle	C₂²×C₇	C₂⁴
LCS	G ▷ C₇
UCS	C₁ ◁ C₂³×C₄

498 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 19
transitive degree	μ_t(G) = 448

A minimal faithful permutation representation — 4 generators on 19 points:

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

Conjugacy classes & character table

160 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	Dic₇×#95, #11727155, #11727156, …
maximal normal subgroup of	Dic₇×#95, #11727155, #11727156, …
maximal quotient of	C₂²×C₄×C₇⋊C₈, C₂²×#6377, C₂×C₄×#1136, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(448, 1235);