C₂³×#14693 #72315137

order	2048
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶×C₄ · Schur multiplier (H₂) = C₂²⁴

Aliases

direct product

C₂³×#14693

Subgroups

	subgroup	G/H
center, socle	C₂⁷	C₂⁴
commutator	C₂³	C₂⁶×C₄
Frattini, cosocle	C₂⁴	C₂⁷
prime radical	C₂⁹	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂⁷ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

8435208 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 22
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 7 generators on 22 points:

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

Conjugacy classes & character table

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

Relationships

maximal subgroup of	C₂³×C₄×#2846
maximal normal subgroup of	C₂³×C₄×#2846

GAP

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

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

C23×#14693 #72315137