C₂×#11714186 #72343134

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

Aliases

direct product

C₂×#11714186

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

3237 conjugacy classes of subgroups; only the 107 normal subgroups are shown.

normal subgroups only

This group has 3237 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:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72236920, C₂×#72236866, C₂×#72236902, …
maximal normal subgroup of	C₂×#72236920, C₂×#72236866, C₂×#72236902, …

GAP

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

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