D₄×#2613 #72334293

order	1024
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹⁴×C₄

Aliases

direct product

D₄×#2613

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂⁴	C₂⁶
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

74557 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 20
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 20 points:

(6,8)(10,12)(13,14)(15,18)(16,17)(19,20);(2,4)(6,8)(9,10)(11,12)(14,17)(18,19);(10,12);(9,10)(11,12);(1,2)(3,4)(5,6,7,8)(13,14)(15,19)(16,17)(18,20);(13,15)(14,18)(16,20)(17,19)

Conjugacy classes & character table

160 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	D₄×#34980, C₂×D₄×#2613
maximal normal subgroup of	D₄×#34980, C₂×D₄×#2613
maximal quotient of	D₄×#14715, C₂×D₄×#2613, #72334979, …

GAP

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

gap> G := Group((6,8)(10,12)(13,14)(15,18)(16,17)(19,20), (2,4)(6,8)(9,10)(11,12)(14,17)(18,19), (10,12), (9,10)(11,12), (1,2)(3,4)(5,6,7,8)(13,14)(15,19)(16,17)(18,20), (13,15)(14,18)(16,20)(17,19));

D4×#2613 #72334293