C₂²×#12609 #72307278

order	1024
nilpotency class	3
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄² · Schur multiplier (H₂) = C₂¹⁰

Aliases

direct product

C₂²×#12609

Subgroups

	subgroup	G/H
center	C₂²×C₄	C₄×C₂²⋊C₄
commutator	C₂×C₄	C₂³×C₄²
Frattini, cosocle	C₂²×C₈	C₂⁵
socle	C₂³	#1749
prime radical	C₂³×#41	C₂×C₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂³×#97 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

8913 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 36 points:

(1,2);(3,4);(1,2)(3,4)(7,15,19,12)(9,24)(14,25)(16,18,28,22)(17,32)(20,27,35,30)(26,29)(31,33,36,34);(5,6,11,10)(7,16)(8,21,23,13)(9,14,24,25)(12,22)(15,18)(17,29,32,26)(19,28)(20,31)(27,33)(30,34)(35,36);(5,7,17,33,10,15,29,31,11,19,32,34,6,12,26,36)(8,22,9,20,21,16,25,30,23,18,24,35,13,28,14,27)

Conjugacy classes & character table

232 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72280389, C₂×#72249772, C₂²×#495739, …
maximal normal subgroup of	C₂×#72280389, C₂×#72249772, C₂²×#495739, …
maximal quotient of	C₂×#72319950

GAP

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

gap> G := Group((1,2), (3,4), (1,2)(3,4)(7,15,19,12)(9,24)(14,25)(16,18,28,22)(17,32)(20,27,35,30)(26,29)(31,33,36,34), (5,6,11,10)(7,16)(8,21,23,13)(9,14,24,25)(12,22)(15,18)(17,29,32,26)(19,28)(20,31)(27,33)(30,34)(35,36), (5,7,17,33,10,15,29,31,11,19,32,34,6,12,26,36)(8,22,9,20,21,16,25,30,23,18,24,35,13,28,14,27));

C22×#12609 #72307278