C₇×#4696 #13055731

order	1344
automorphism	?
homology	Abelianization (H₁) = C₂³×C₇ · Schur multiplier (H₂) = C₂×C₄

Aliases

direct product

C₇×#4696

Splittings

Fitting	C₃×C₄×C₇×C₈ ⋊ C₂
Hall	C₃×C₇ ⋊ #495
Sylow	C₃ ⋊ C₇×#495

Subgroups

	subgroup	G/H
center	C₂²×C₇	C₂×D₁₂
commutator, cosocle	C₂×C₃×C₄	C₂³×C₇
Frattini	C₂×C₄	C₂²×D₃×C₇
Fitting	C₃×C₄×C₇×C₈	C₂
socle	C₂²×C₃×C₇	C₂×D₄
prime radical	C₂×C₇×D₂₄	C₂
hypercenter	C₄×C₇×C₈	D₃
hypocenter	C₃	C₇×#495
LCS	G ▷ C₂×C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂²×C₇ ◁ C₄²×C₇ ◁ C₄×C₇×C₈

260 conjugacy classes of subgroups; only the 94 normal subgroups are shown.

normal subgroups only

This group has 260 conjugacy classes of subgroups — too many to draw. Only its 94 normal subgroups are shown.

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

Full subgroup data →

Presentation

minimal degree	μ(G) = 30
transitive degree	μ_t(G) = 672

A minimal faithful permutation representation — 3 generators on 30 points:

(1,2,3,4,5,6,7)(8,9,10)(15,17,18,24,19,25,27,30)(16,21,22,23,20,28,26,29);(11,12,13,14)(16,20)(21,28)(22,26)(23,29);(9,10)(12,14)(15,16)(17,23)(18,26)(19,20)(21,24)(22,27)(25,29)(28,30)

Conjugacy classes & character table

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

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1344, 5020);