C₂×#229726 #72321397

order	1024
nilpotency class	3
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁹

Aliases

direct product

C₂×#229726

Subgroups

	subgroup	G/H
center	C₂³×C₄	C₂²×D₄
commutator	C₂²×C₄	C₂⁴×C₄
Frattini, cosocle	C₂×C₄²	C₂⁵
socle	C₂⁴	C₂×C₄×D₄
prime radical	C₂×#2011	C₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₄ ◁ C₂×C₄×C₄⋊C₈ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

4807 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 30
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 5 generators on 30 points:

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

Conjugacy classes & character table

184 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72318330, C₂×#72318344, C₂×#72318348
maximal normal subgroup of	C₂×#72318330, C₂×#72318344, C₂×#72318348

GAP

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

gap> G := Group((4,6)(7,8,10,11)(9,12,14,13)(15,16,20,23,21,24,19,17)(18,22,26,29,27,30,28,25), (1,2), (7,9,10,14)(8,12,11,13)(15,17,19,24,21,23,20,16)(18,25,28,30,27,29,26,22), (8,11)(9,13)(12,14)(15,18)(16,22)(17,25)(19,28)(20,26)(21,27)(23,29)(24,30), (3,4,5,6)(15,19,21,20)(16,17,24,23)(18,26,27,28)(22,29,30,25));

C2×#229726 #72321397