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C₂³×C₄×D₇ #90604

order	448
isoclinism	isoclinic class of D₇
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂¹⁰

Aliases

C₂³×C₄×D₇

Splittings

Fitting	C₂³×C₄×C₇ ⋊ C₂
derived, Sylow	C₇ ⋊ C₂⁴×C₄

Subgroups

	subgroup	G/H
center, hypercenter	C₂³×C₄	D₇
commutator, hypocenter	C₇	C₂⁴×C₄
Frattini	C₂	C₂⁴×D₇
Fitting	C₂³×C₄×C₇	C₂
socle	C₂⁴×C₇	C₂²
prime radical	C₂⁴×D₇	C₂
cosocle	C₂×C₇	C₂⁵
LCS	G ▷ C₇
UCS	C₁ ◁ C₂³×C₄

1362 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 17
transitive degree	μ_t(G) = 224

A minimal faithful permutation representation — 5 generators on 17 points:

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

Conjugacy classes & character table

160 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	(C₂×D₇)⋊#95, #11727709, C₂×#89438, …
maximal normal subgroup of	(C₂×D₇)⋊#95, #11727709, C₂×#89438, …
maximal quotient of	C₂²×C₄×Dic₁₄, C₂×C₄×#1157, C₂²×C₄²×D₇, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(448, 1366);