C₂×#11714179 #72343828

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

Aliases

direct product

C₂×#11714179

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

3293 conjugacy classes of subgroups; only the 107 normal subgroups are shown.

normal subgroups only

This group has 3293 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:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72236801, C₂×#72236804, C₂×#72236829, …
maximal normal subgroup of	C₂×#72236801, C₂×#72236804, C₂×#72236829, …

GAP

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

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