#72210376

order	2352
automorphism	?
homology	Abelianization (H₁) = C₂³×C₃ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Splittings

Fitting	C₂²×C₇² ⋊ C₂²×C₃
Hall	#11716115 ⋊ C₃
Hall	#3861 ⋊ C₂×D₄
Sylow	C₇² ⋊ C₂×C₃×D₄

Subgroups

	subgroup	G/H
center, Frattini	C₂	C₂×#64423⋊C₂
commutator, socle, cosocle	C₂×C₇²	C₂³×C₃
Fitting	C₂²×C₇²	C₂²×C₃
hypercenter	C₂²	#64423⋊C₂
hypocenter	C₇²	C₂×C₃×D₄
LCS	G ▷ C₂×C₇² ▷ C₇²
UCS	C₁ ◁ C₂ ◁ C₂²

312 conjugacy classes of subgroups; only the 53 normal subgroups are shown.

normal subgroups only

This group has 312 conjugacy classes of subgroups — too many to draw. Only its 53 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Full subgroup data →

Presentation

Polycyclic presentation:

2^3,3,2,7^2|a^2=e,b^a=be,f^a=f^6,f^c=f^6,f^d=f^4,g^a=f^2g,g^c=g^6,g^d=g^4

Conjugacy classes & character table

52 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72210376, D₄×#64423⋊C₂, #72214230, …
maximal normal subgroup of	C₂×#72210376, D₄×#64423⋊C₂, #72214230, …
maximal quotient of	C₂×#72210376, #72214489, #72214510, …
holomorph of	Dic₇, C₂×D₇

GAP

Past the SmallGroups library coverage — no SmallGroups id. Construct it from its polycyclic presentation:

gap> F := FreeGroup(IsSyllableWordsFamily, "a", "b", "c", "d", "e", "f", "g");;
gap> coll := SingleCollector(F, [2, 2, 2, 3, 2, 7, 7]);;
gap> SetPower(coll, 1, F.5);;
gap> SetConjugate(coll, 2, 1, F.2*F.5);;
gap> SetConjugate(coll, 6, 1, F.6^6);;
gap> SetConjugate(coll, 6, 3, F.6^6);;
gap> SetConjugate(coll, 6, 4, F.6^4);;
gap> SetConjugate(coll, 7, 1, F.6^2*F.7);;
gap> SetConjugate(coll, 7, 3, F.7^6);;
gap> SetConjugate(coll, 7, 4, F.7^4);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);