D₄×#92788 #72229009

order	4056
automorphism	?
homology	Abelianization (H₁) = C₂²×C₃ · Schur multiplier (H₂) = C₂

Aliases

direct product

D₄×#92788

Splittings

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

Subgroups

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

256 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 30
transitive degree	μ_t(G) = 676

A minimal faithful permutation representation — 3 generators on 30 points:

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

Conjugacy classes & character table

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

Relationships

maximal subgroup of	D₄×#11759815, C₂×D₄×#92788, #72229416, …
maximal normal subgroup of	D₄×#11759815, C₂×D₄×#92788, #72229416, …
maximal quotient of	C₂×D₄×#92788

GAP

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

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

D4×#92788 #72229009