Groups of normal p-rank one

A non-trivial p-group G always has a normal subgroup C_p◃G. One says that G has normal p-rank one if has no C_p²◃G. For odd p, only cyclic p-groups have this property. For p=2 dihedral (except D₄), quaternion and semidihedral groups have normal p-rank one as well.

Groups of order 2

		d	ρ	Label	ID
C₂	Cyclic group	2	1+	C2	2,1

Groups of order 3

		d	ρ	Label	ID
C₃	Cyclic group; = A₃ = triangle rotations	3	1	C3	3,1

Groups of order 4

		d	ρ	Label	ID
C₄	Cyclic group; = square rotations	4	1	C4	4,1

Groups of order 5

		d	ρ	Label	ID
C₅	Cyclic group; = pentagon rotations	5	1	C5	5,1

Groups of order 7

		d	ρ	Label	ID
C₇	Cyclic group	7	1	C7	7,1

Groups of order 8

		d	ρ	Label	ID
C₈	Cyclic group	8	1	C8	8,1
Q₈	Quaternion group; = C₄ .C₂ = Dic₂ = 2_-¹⁺²	8	2-	Q8	8,4

Groups of order 9

		d	ρ	Label	ID
C₉	Cyclic group	9	1	C9	9,1

Groups of order 11

		d	ρ	Label	ID
C₁₁	Cyclic group	11	1	C11	11,1

Groups of order 13

		d	ρ	Label	ID
C₁₃	Cyclic group	13	1	C13	13,1

Groups of order 16

		d	ρ	Label	ID
C₁₆	Cyclic group	16	1	C16	16,1
D₈	Dihedral group	8	2+	D8	16,7
Q₁₆	Generalised quaternion group; = C₈ .C₂ = Dic₄	16	2-	Q16	16,9
SD₁₆	Semidihedral group; = Q₈ ⋊C₂ = QD₁₆	8	2	SD16	16,8

Groups of order 17

		d	ρ	Label	ID
C₁₇	Cyclic group	17	1	C17	17,1

Groups of order 19

		d	ρ	Label	ID
C₁₉	Cyclic group	19	1	C19	19,1

Groups of order 23

		d	ρ	Label	ID
C₂₃	Cyclic group	23	1	C23	23,1

Groups of order 25

		d	ρ	Label	ID
C₂₅	Cyclic group	25	1	C25	25,1

Groups of order 27

		d	ρ	Label	ID
C₂₇	Cyclic group	27	1	C27	27,1

Groups of order 29

		d	ρ	Label	ID
C₂₉	Cyclic group	29	1	C29	29,1

Groups of order 31

		d	ρ	Label	ID
C₃₁	Cyclic group	31	1	C31	31,1

Groups of order 32

		d	ρ	Label	ID
C₃₂	Cyclic group	32	1	C32	32,1
D₁₆	Dihedral group	16	2+	D16	32,18
Q₃₂	Generalised quaternion group; = C₁₆ .C₂ = Dic₈	32	2-	Q32	32,20
SD₃₂	Semidihedral group; = C₁₆ ⋊₂ C₂ = QD₃₂	16	2	SD32	32,19

Groups of order 37

		d	ρ	Label	ID
C₃₇	Cyclic group	37	1	C37	37,1

Groups of order 41

		d	ρ	Label	ID
C₄₁	Cyclic group	41	1	C41	41,1

Groups of order 43

		d	ρ	Label	ID
C₄₃	Cyclic group	43	1	C43	43,1

Groups of order 47

		d	ρ	Label	ID
C₄₇	Cyclic group	47	1	C47	47,1

Groups of order 49

		d	ρ	Label	ID
C₄₉	Cyclic group	49	1	C49	49,1

Groups of order 53

		d	ρ	Label	ID
C₅₃	Cyclic group	53	1	C53	53,1

Groups of order 59

		d	ρ	Label	ID
C₅₉	Cyclic group	59	1	C59	59,1

Groups of order 61

		d	ρ	Label	ID
C₆₁	Cyclic group	61	1	C61	61,1

Groups of order 64

		d	ρ	Label	ID
C₆₄	Cyclic group	64	1	C64	64,1
D₃₂	Dihedral group	32	2+	D32	64,52
Q₆₄	Generalised quaternion group; = C₃₂ .C₂ = Dic₁₆	64	2-	Q64	64,54
SD₆₄	Semidihedral group; = C₃₂ ⋊₂ C₂ = QD₆₄	32	2	SD64	64,53

Groups of order 67

		d	ρ	Label	ID
C₆₇	Cyclic group	67	1	C67	67,1

Groups of order 71

		d	ρ	Label	ID
C₇₁	Cyclic group	71	1	C71	71,1

Groups of order 73

		d	ρ	Label	ID
C₇₃	Cyclic group	73	1	C73	73,1

Groups of order 79

		d	ρ	Label	ID
C₇₉	Cyclic group	79	1	C79	79,1

Groups of order 81

		d	ρ	Label	ID
C₈₁	Cyclic group	81	1	C81	81,1

Groups of order 83

		d	ρ	Label	ID
C₈₃	Cyclic group	83	1	C83	83,1

Groups of order 89

		d	ρ	Label	ID
C₈₉	Cyclic group	89	1	C89	89,1

Groups of order 97

		d	ρ	Label	ID
C₉₇	Cyclic group	97	1	C97	97,1

Groups of order 101

		d	ρ	Label	ID
C₁₀₁	Cyclic group	101	1	C101	101,1

Groups of order 103

		d	ρ	Label	ID
C₁₀₃	Cyclic group	103	1	C103	103,1

Groups of order 107

		d	ρ	Label	ID
C₁₀₇	Cyclic group	107	1	C107	107,1

Groups of order 109

		d	ρ	Label	ID
C₁₀₉	Cyclic group	109	1	C109	109,1

Groups of order 113

		d	ρ	Label	ID
C₁₁₃	Cyclic group	113	1	C113	113,1

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