C₂⁴×H(3,𝔽₄) #72295509

order	1024
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁸ · Schur multiplier (H₂) = C₂²⁴×C₄²

Aliases

direct product

C₂⁴×H(3,𝔽₄)

Subgroups

	subgroup	G/H
center, socle	C₂⁶	C₂⁴
commutator, Frattini, cosocle	C₂²	C₂⁸
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂⁶ ◁ G
LECS	G ▷ C₂² ▷ C₁

704005 conjugacy classes of subgroups; only the 419957 normal subgroups are shown.

normal subgroups only

This group has 704005 conjugacy classes of subgroups — too many to draw. Only its 419957 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) = 24
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 8 generators on 24 points:

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

Conjugacy classes & character table

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

Relationships

maximal subgroup of	C₂⁴×#3532, C₂⁴×#3549, C₂³×#51068, …
maximal normal subgroup of	C₂⁴×#3532, C₂⁴×#3549, C₂³×#51068, …
maximal quotient of	C₂³×#52301, C₂⁴×#2684, C₂³×#45439

GAP

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

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