This quiz is about alternating group:A4, which we take as the alternating group on the set of size four.
Elements
See element structure of alternating group:A4 for background information.
Element order and conjugacy class structure
Review the conjugacy class structure:
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For a symmetric group, cycle type determines conjugacy class. The statement is almost true for the alternating group, except for the fact that some conjugacy classes of even permutations in the symmetric group split into two in the alternating group, as per the splitting criterion for conjugacy classes in the alternating group, which says that a conjugacy class of even permutations splits in the alternating group if and only if its cycle decomposition comprises odd cycles of distinct length.
Here are the unsplit conjugacy classes:
Partition 
Verbal description of cycle type 
Elements with the cycle type 
Size of conjugacy class 
Formula for size 
Element order

1 + 1 + 1 + 1 
four cycles of size one each, i.e., four fixed points 
 the identity element 
1 

1

2 + 2 
double transposition: two cycles of size two 
, , 
3 

2

Total 
 
, , and 
4 
NA 
NA

In this case, the union of the unsplit conjugacy classes is a proper normal subgroup isomorphic to the Klein fourgroup. Note that this phenomenon is unique to the case .
Here is the split conjugacy class:
Partition 
Verbal description of cycle type 
Elements with the cycle type 
Combined size of conjugacy classes 
Formula for combined size 
Size of each half 
First split half 
Second split half 
Real? 
Rational? 
Element order

3 + 1 
one 3cycle, one fixed point 
, , , , , , , 
8 

4 
, , , 
, , , 
No 
No 
3

Subgroups
See subgroup structure of alternating group:A4 for background information.
Basic stuff
Summary table on the structure of subgroups:
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Quick summary
Table classifying subgroups up to automorphism
View the lattice of subgroups as a picture:
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