C₂×C₂²⋊C₄×H(5,𝔽₂) #72298635

order	1024
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶×C₄ · Schur multiplier (H₂) = C₂²¹

Aliases

direct product

C₂×C₂²⋊C₄×H(5,𝔽₂)

Subgroups

	subgroup	G/H
center, socle	C₂⁴	C₂⁶
commutator	C₂²	C₂⁶×C₄
Frattini, cosocle	C₂³	C₂⁷
prime radical	C₂⁴×H(5,𝔽₂)	C₂
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂³ ▷ C₁

318325 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 18
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 7 generators on 18 points:

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

Conjugacy classes & character table

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

Relationships

maximal subgroup of	C₂×#120×H(5,𝔽₂), #72316646, C₂²×C₂²⋊C₄×H(5,𝔽₂), …
maximal normal subgroup of	C₂×#120×H(5,𝔽₂), #72316646, C₂²×C₂²⋊C₄×H(5,𝔽₂), …
maximal quotient of	C₂×#72233563, C₂×D₄²×C₂²⋊C₄, C₂²×C₂²⋊C₄×H(5,𝔽₂), …

GAP

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

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

C2×C22⋊C4×H(5,𝔽2) #72298635