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Cayley diagram

C7×#95 #6412

order224
nilpotency class2
isoclinismisoclinic class of D4
automorphismAut(G) = C2×C3×#87524 (order 2304) · Inn(G) = C22 · Out(G) = C2×C3×GL(2,ℤ/4)
homologyAbelianization (H1) = C42×C7 · Schur multiplier (H2) = C23

Aliases

direct productC7×#95

Subgroups

subgroupG/H
center, socle, prime radicalC23×C7C22
commutatorC2C42×C7
Frattini, cosocleC23C22×C7
LCSG ▷ C2 ▷ C1
UCSC1 ◁ C23×C7 ◁ G

76 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 19
transitive degreeμt(G) = 224

A minimal faithful permutation representation — 2 generators on 19 points:

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

Conjugacy classes & character table

140 conjugacy classes — view the full character table.

Relationships

derived subgroup ofC7⋊(#95⋊C3), #13052148, #13052149, …
Fitting subgroup of#89246, #89247, #89418, …
maximal subgroup of#89246, #89247, C7×#327, …
maximal normal subgroup of#89246, #89247, C7×#327, …
maximal quotient ofC7×#336, C7×H(3,ℤ/4), C7×#338, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(224, 44);