D₄×#13091120 #72228581

order	11664
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂³×C₃

Aliases

direct product

D₄×#13091120

Splittings

Fitting	D₄×#10624749 ⋊ C₂
Sylow	#10624749 ⋊ C₂×D₄

Subgroups

	subgroup	G/H
center	C₂	C₂²×#13091120
commutator, cosocle	C₂×#10624749	C₂³
Frattini	C₂×C₃³	C₂²×C₃⋊(C₃⋊D₃)
Fitting	D₄×#10624749	C₂
socle	C₂×C₃	C₂²×#92338
hypercenter	D₄	#13091120
hypocenter	#10624749	C₂×D₄
LCS	G ▷ C₂×#10624749 ▷ #10624749
UCS	C₁ ◁ C₂ ◁ D₄

4472 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 31
transitive degree	μ_t(G) = 108

A minimal faithful permutation representation — 4 generators on 31 points:

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

Conjugacy classes & character table

210 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	D₄×#72210336
maximal normal subgroup of	D₄×#72210336
holomorph of	C₄×3₋¹⁺²

GAP

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

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

D4×#13091120 #72228581