C₇×#4700 #13055735

order	1344
isoclinism	isoclinic class of #204
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₇ · Schur multiplier (H₂) = C₂²

Aliases

direct product

C₇×#4700

Splittings

Fitting	C₃×C₄×C₇×C₈ ⋊ C₂
Hall	C₃×C₇ ⋊ #443
Sylow	C₃ ⋊ C₇×#443

Subgroups

	subgroup	G/H
center	C₇×C₈	D₁₂
commutator	C₃×C₄	C₂²×C₄×C₇
Frattini	C₂×C₄	C₂²×D₃×C₇
Fitting	C₃×C₄×C₇×C₈	C₂
socle	C₂×C₃×C₇	C₄×D₄
prime radical	C₇×#904	C₂
cosocle	C₂×C₃×C₄	C₂³×C₇
hypercenter	C₄×C₇×C₈	D₃
hypocenter	C₃	C₇×#443
LCS	G ▷ C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₇×C₈ ◁ C₂×C₇×C₈ ◁ C₄×C₇×C₈

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

normal subgroups only

This group has 212 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) = 26
transitive degree	μ_t(G) = 336

A minimal faithful permutation representation — 3 generators on 26 points:

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

Conjugacy classes & character table

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

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1344, 5024);