C₂×#11714178 #72268280

order	1536
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁷

Aliases

direct product

C₂×#11714178

Splittings

Fitting

C₂×#3027 ⋊ D₃

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#1020⋊C₂
commutator, cosocle, hypocenter	#997	C₂⁴
Frattini	C₂³	C₂³×S₄
Fitting	C₂×#3027	D₃
hypercenter	C₂×C₄	#1020⋊C₂
LCS	G ▷ #997
UCS	C₁ ◁ C₂² ◁ C₂×C₄

3289 conjugacy classes of subgroups; only the 107 normal subgroups are shown.

normal subgroups only

This group has 3289 conjugacy classes of subgroups — too many to draw. Only its 107 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) = 34
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 34 points:

(3,6,28)(4,22,32,10,11,24)(5,25,33,13,14,27)(7,18)(8,16,17)(9,19,34)(20,31)(21,29,30);(3,5)(4,9)(6,15)(7,14)(8,13)(10,21)(11,20)(12,19)(16,26)(17,27)(18,25)(22,31)(23,29)(24,30)(28,33)(32,34);(3,4,8,10)(5,9,13,21)(6,11,16,22)(7,12,18,23)(14,19,25,29)(15,20,26,31)(17,24,28,32)(27,30,33,34);(1,2)

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72233471, C₂×#72236895, C₂×#72236853, …
maximal normal subgroup of	C₂×#72233471, C₂×#72236895, C₂×#72236853, …

GAP

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

gap> G := Group((3,6,28)(4,22,32,10,11,24)(5,25,33,13,14,27)(7,18)(8,16,17)(9,19,34)(20,31)(21,29,30), (3,5)(4,9)(6,15)(7,14)(8,13)(10,21)(11,20)(12,19)(16,26)(17,27)(18,25)(22,31)(23,29)(24,30)(28,33)(32,34), (3,4,8,10)(5,9,13,21)(6,11,16,22)(7,12,18,23)(14,19,25,29)(15,20,26,31)(17,24,28,32)(27,30,33,34), (1,2));