C₂×C₂²⋊C₄×#95 #72320459

order	1024
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄³ · Schur multiplier (H₂) = C₂¹¹×C₄²

Aliases

direct product

C₂×C₂²⋊C₄×#95

Subgroups

	subgroup	G/H
center, socle	C₂⁶	C₂⁴
commutator	C₂²	C₂²×C₄³
Frattini, cosocle	C₂⁵	C₂⁵
prime radical	C₂⁷	C₂³
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂⁶ ◁ G
LECS	G ▷ C₂⁵ ▷ C₁

64109 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 22
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 5 generators on 22 points:

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

Conjugacy classes & character table

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

Relationships

maximal subgroup of	C₂×C₂²⋊C₄×#392, C₂×#72278310, C₂×#95×#120, …
maximal normal subgroup of	C₂×C₂²⋊C₄×#392, C₂×#72278310, C₂×#95×#120, …
maximal quotient of	C₂²×C₂²⋊C₄×#95

GAP

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

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

C2×C22⋊C4×#95 #72320459