C₄×#362 #12645

order	256
nilpotency class	4
isoclinism	isoclinic class of #360
automorphism	Aut(G) order 8192
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂²×C₄

Aliases

direct product

C₄×#362

Subgroups

	subgroup	G/H
center	C₂×C₄	#102
commutator	C₈	C₂×C₄²
Frattini, cosocle	C₂²×C₈	C₂³
socle	C₂²	C₂×#102
prime radical	C₂³	C₂×SD₁₆
LCS	G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

217 conjugacy classes of subgroups; only the 97 normal subgroups are shown.

normal subgroups only

This group has 217 conjugacy classes of subgroups — too many to draw. Only its 97 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) = 36
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 3 generators on 36 points:

(1,2,3,4);(5,6,9,13,10,15,25,14)(7,16,18,29,19,33,24,32)(8,20,22,12,23,30,36,31)(11,21,26,17,28,34,27,35);(5,7,10,19)(6,11,15,28)(8,21,23,34)(9,24,25,18)(12,29,31,32)(13,27,14,26)(16,20,33,30)(17,22,35,36)

Conjugacy classes & character table

64 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₈×#362, #128693, #128712, …
maximal normal subgroup of	C₈×#362, #128693, #128712, …
maximal quotient of	C₄×#1334, #128591, #128604, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(256, 5633);