C₂×#11714181 #72343917

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

Aliases

direct product

C₂×#11714181

Splittings

Fitting

C₂×#3025 ⋊ D₃

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#1020⋊C₂
commutator, cosocle, hypocenter	#997	C₂⁴
Frattini	C₂³	C₂³×S₄
Fitting	C₂×#3025	D₃
hypercenter	C₂×C₄	#1020⋊C₂
LCS	G ▷ #997
UCS	C₁ ◁ C₂² ◁ C₂×C₄

3253 conjugacy classes of subgroups; only the 107 normal subgroups are shown.

normal subgroups only

This group has 3253 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:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72236803, C₂×#72236843, C₂×#72236831, …
maximal normal subgroup of	C₂×#72236803, C₂×#72236843, C₂×#72236831, …

GAP

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

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