C₂×#7627401 #72332292

order	1024
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶×C₄ · Schur multiplier (H₂) = C₂¹⁹

Aliases

direct product

C₂×#7627401

Subgroups

	subgroup	G/H
center, socle	C₂⁴	C₂⁶
commutator	C₂²	C₂⁶×C₄
Frattini, cosocle	C₂³	C₂⁷
prime radical	C₂³×#560	C₂
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂³ ▷ C₁

75769 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 30
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 7 generators on 30 points:

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

Conjugacy classes & character table

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

Relationships

maximal subgroup of	C₂×#72323293
maximal normal subgroup of	C₂×#72323293
maximal quotient of	C₂×#72323421

GAP

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

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

C2×#7627401 #72332292