C₂×#72221868 #72244208

order	17496
automorphism	?
homology	Abelianization (H₁) = C₂³×C₃ · Schur multiplier (H₂) = C₂³

Aliases

direct product

C₂×#72221868

Splittings

Fitting	C₂×#14555333 ⋊ C₂²
Sylow	#14555333 ⋊ C₂³

Subgroups

	subgroup	G/H
center, hypercenter	C₂×C₃	#72221867
commutator, cosocle, hypocenter	#10624869	C₂³×C₃
Frattini	C₃²	C₂×C₃×C₃⋊#1113
Fitting	C₂×#14555333	C₂²
socle	C₂×C₃²	C₃×C₃⋊#1113
prime radical	C₂×#72221851	C₃
LCS	G ▷ #10624869
UCS	C₁ ◁ C₂×C₃

4333 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 56
transitive degree	μ_t(G) = 108

A minimal faithful permutation representation — 3 generators on 56 points:

(2,14,20,5,7,15)(3,10,11,8,9,22)(4,12)(6,16)(13,27,23,21,25,18)(17,24)(19,26)(28,34,50,31,44,32,38,29,40)(30,52,48,46,42,53,36,51,49,45,43,47,37,41,39,35,33,54);(1,11,8,4,9,10,12,3,22)(2,25,21,15,17,24,20,13,27,5,19,16,7,23,18,14,6,26)(30,35)(33,41)(36,46)(37,45)(39,47)(42,52)(43,51)(48,54)(49,53);(1,2,5)(3,6,13)(4,7,15)(8,21,16)(9,17,23)(10,18,24)(11,19,25)(12,20,14)(22,27,26)(28,30,35)(29,33,41)(31,37,45)(32,39,47)(34,43,51)(36,46,38)(40,49,53)(42,52,44)(48,54,50)(55,56)

Conjugacy classes & character table

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

Relationships

maximal subgroup of

C₂×#72221873

GAP

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

gap> G := Group((2,14,20,5,7,15)(3,10,11,8,9,22)(4,12)(6,16)(13,27,23,21,25,18)(17,24)(19,26)(28,34,50,31,44,32,38,29,40)(30,52,48,46,42,53,36,51,49,45,43,47,37,41,39,35,33,54), (1,11,8,4,9,10,12,3,22)(2,25,21,15,17,24,20,13,27,5,19,16,7,23,18,14,6,26)(30,35)(33,41)(36,46)(37,45)(39,47)(42,52)(43,51)(48,54)(49,53), (1,2,5)(3,6,13)(4,7,15)(8,21,16)(9,17,23)(10,18,24)(11,19,25)(12,20,14)(22,27,26)(28,30,35)(29,33,41)(31,37,45)(32,39,47)(34,43,51)(36,46,38)(40,49,53)(42,52,44)(48,54,50)(55,56));

C2×#72221868 #72244208