C₂×#151367 #72340367

order	1024
nilpotency class	5
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁷

Aliases

direct product

C₂×#151367

Subgroups

	subgroup	G/H
center, socle	C₂²	#12202
commutator, Frattini, cosocle	#379	C₂⁴
prime radical	C₂×#23895	C₂
LCS	G ▷ #379 ▷ C₂⁴ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴ ◁ C₂⁵ ◁ C₂×#379 ◁ G
LECS	G ▷ #379 ▷ C₂⁴ ▷ C₂² ▷ C₂ ▷ C₁

4221 conjugacy classes of subgroups; only the 149 normal subgroups are shown.

normal subgroups only

This group has 4221 conjugacy classes of subgroups — too many to draw. Only its 149 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) = 34
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 34 points:

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

Conjugacy classes & character table

64 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72213804, #72340379, #72340388
maximal normal subgroup of	C₂×#72213804, #72340379, #72340388

GAP

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

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