C₂×C₃₁×C₂₉⋊C₇ #27160404

order	12586
isoclinism	isoclinic class of C₂₉⋊C₇
automorphism	?
homology	Abelianization (H₁) = C₂×C₇×C₃₁ · Schur multiplier (H₂) = C₁

Aliases

direct product	C₂×C₃₁×C₂₉⋊C₇
unambiguous semidirect product	(C₂×C₂₉)⋊(C₇×C₃₁)
unambiguous semidirect product	(C₂×C₂₉×C₃₁)⋊C₇

Splittings

Fitting, socle, Hall	C₂×C₂₉×C₃₁ ⋊ C₇
derived, Sylow	C₂₉ ⋊ C₂×C₇×C₃₁
Hall	C₂₉×C₃₁ ⋊ C₂×C₇
Hall	C₂×C₂₉ ⋊ C₇×C₃₁

Subgroups

	subgroup	G/H
center, hypercenter	C₂×C₃₁	C₂₉⋊C₇
commutator, cosocle, hypocenter	C₂₉	C₂×C₇×C₃₁
Frattini	C₁	G
Fitting, socle	C₂×C₂₉×C₃₁	C₇
LCS	G ▷ C₂₉
UCS	C₁ ◁ C₂×C₃₁

16 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 62
transitive degree	μ_t(G) = 1798

A minimal faithful permutation representation — 2 generators on 62 points:

(2,3,5,8,9,15,20)(4,7,12,19,27,6,10)(11,18,16,24,28,14,17)(13,21,23,26,25,22,29)(30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60)(61,62);(1,2,4,7,13,22,21,8,14,23,24,10,17,25,19,28,3,6,11,12,20,27,29,15,5,9,16,18,26)

Conjugacy classes & character table

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

Relationships

maximal subgroup of	C₄×C₃₁×C₂₉⋊C₇, Dic₃₁×C₂₉⋊C₇, C₃₁×Dic₂₉⋊C₇, …
maximal normal subgroup of	C₄×C₃₁×C₂₉⋊C₇, Dic₃₁×C₂₉⋊C₇, C₃₁×Dic₂₉⋊C₇, …
maximal quotient of	C₄×C₃₁×C₂₉⋊C₇, C₂²×C₃₁×C₂₉⋊C₇, C₂×C₃×C₃₁×C₂₉⋊C₇, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(12586, 2);

C2×C31×C29⋊C7 #27160404