C₂²×#14684 #72339135

order	1024
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁵×C₄ · Schur multiplier (H₂) = C₂¹⁵×C₄

Aliases

direct product

C₂²×#14684

Subgroups

	subgroup	G/H
center, socle	C₂⁶	C₂⁴
commutator	C₂³	C₂⁵×C₄
Frattini, cosocle	C₂⁴	C₂⁶
prime radical	C₂⁷	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂⁶ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

82869 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 6 generators on 28 points:

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

Conjugacy classes & character table

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

Relationships

maximal subgroup of	C₂²×#555905, C₂²×#613325, C₂²×#613192, …
maximal normal subgroup of	C₂²×#555905, C₂²×#613325, C₂²×#613192, …
maximal quotient of	C₂×#72339322

GAP

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

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

C22×#14684 #72339135