#72277865

order	12000
automorphism	?
homology	Abelianization (H₁) = C₄ · Schur multiplier (H₂) = C₁

non-solvable

Aliases

No aliases for this group.

Splittings

derived

#72209598 ⋊ C₄

Subgroups

	subgroup	G/H
center, Frattini, hypercenter	C₂	#72210652
commutator, hypocenter	#72209598	C₄
Fitting, socle	C₂×C₅²	#6732
prime radical	C₂×#72209598	C₂
cosocle	C₂×C₅⋊D₅	C₂×A₅
LCS	G ▷ #72209598
UCS	C₁ ◁ C₂

146 conjugacy classes of subgroups; only the 13 normal subgroups are shown.

normal subgroups only

This group has 146 conjugacy classes of subgroups — too many to draw. Only its 13 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) = 29
transitive degree	μ_t(G) = 100

A minimal faithful permutation representation — 2 generators on 29 points:

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

Conjugacy classes & character table

44 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₄×#72210652
maximal normal subgroup of	C₄×#72210652

GAP

Past the SmallGroups library coverage — no SmallGroups id. Construct it from a minimal faithful permutation representation:

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