C₄×#12778 #72282175

order	1024
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂⁶×C₄²

Aliases

direct product

C₄×#12778

Subgroups

	subgroup	G/H
center	C₂×C₄	#1790
commutator	C₂×Q₈	C₂²×C₄²
Frattini, cosocle	C₂²×#41	C₂⁴
socle	C₂²	C₂×#1790
prime radical	C₂×#3549	C₂²
LCS	G ▷ C₂×Q₈ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×C₄×#41 ◁ G
LECS	G ▷ C₂²×#41 ▷ C₂² ▷ C₂ ▷ C₁

5598 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 20
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 20 points:

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

Conjugacy classes & character table

136 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₄×#511819, #72283126, C₄×#11714278, …
maximal normal subgroup of	C₄×#511819, #72283126, #72311898, …

GAP

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

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

C4×#12778 #72282175