C₂²×#555905 #72339147

order	2048
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶×C₄ · Schur multiplier (H₂) = C₂²¹×C₄

Aliases

direct product

C₂²×#555905

Subgroups

	subgroup	G/H
center, socle	C₂⁶	C₂⁵
commutator	C₂³	C₂⁶×C₄
Frattini, cosocle	C₂⁴	C₂⁷
prime radical	C₂⁵×#127	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂⁶ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

1156392 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 7 generators on 28 points:

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

Conjugacy classes & character table

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

Relationships

maximal subgroup of	C₂²×#72225737
maximal normal subgroup of	C₂²×#72225737

GAP

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

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

C22×#555905 #72339147