C₂×D₄×#92788 #72229367

order	8112
automorphism	?
homology	Abelianization (H₁) = C₂³×C₃ · Schur multiplier (H₂) = C₂³

Aliases

direct product

C₂×D₄×#92788

Splittings

Fitting, Hall	C₂×D₄×C₁₃² ⋊ C₃
Sylow	C₁₃² ⋊ C₂×C₃×D₄

Subgroups

	subgroup	G/H
center	C₂²	C₂²×#92788
commutator, cosocle	C₂×C₁₃²	C₂³×C₃
Frattini	C₂	C₂³×#92788
Fitting	C₂×D₄×C₁₃²	C₃
socle	C₂²×C₁₃²	C₂²×C₃
hypercenter	C₂×D₄	#92788
hypocenter	C₁₃²	C₂×C₃×D₄
LCS	G ▷ C₂×C₁₃² ▷ C₁₃²
UCS	C₁ ◁ C₂² ◁ C₂×D₄

864 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 1352

A minimal faithful permutation representation — 3 generators on 32 points:

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

Conjugacy classes & character table

590 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	C₂×D₄×#11759815, #72259483, #72259622
maximal normal subgroup of	C₂×D₄×#11759815, #72259483, #72259622

GAP

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

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

C2×D4×#92788 #72229367