Abstract group 5373.15: $C_{597}:C_9$

• Label: 5373.15
• Order: \( 3^{3} \cdot 199 \)
• Exponent: \( 3^{2} \cdot 199 \)
• Nilpotent: no
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 3^{3} \)
• $\card{Z(G)}$: \( 3 \)
• $\card{\Aut(G)}$: \( 2^{2} \cdot 3^{3} \cdot 11 \cdot 199 \)
• $\card{\mathrm{Out}(G)}$: \( 2^{2} \cdot 3 \cdot 11 \)
• Perm deg.: $202$
• Trans deg.: $597$
• Rank: $2$

Group information

Description:	$C_{597}:C_9$
Order:	\(5373\)\(\medspace = 3^{3} \cdot 199 \)
Exponent:	\(1791\)\(\medspace = 3^{2} \cdot 199 \)
Automorphism group:	$C_{199}:(C_{11}:(C_{18}\times S_3))$, of order \(236412\)\(\medspace = 2^{2} \cdot 3^{3} \cdot 11 \cdot 199 \)
Composition factors:	$C_3$ x 3, $C_{199}$
Derived length:	$2$

This group is nonabelian, metacyclic (hence solvable, supersolvable, monomial, and metabelian), hyperelementary for $p = 3$, and an A-group.

Group statistics

Order	1	3	9	199	597
Elements	1	1196	3582	198	396	5373
Conjugacy classes	1	8	18	22	44	93
Divisions	1	4	3	1	1	10
Autjugacy classes	1	5	6	1	1	14

Dimension	1	2	6	9	198	396
Irr. complex chars.	27	0	0	66	0	0	93
Irr. rational chars.	1	4	3	0	1	1	10

Minimal presentations

Permutation degree:	$202$
Transitive degree:	$597$
Rank:	$2$
Inequivalent generating pairs:	$72$

Minimal degrees of faithful linear representations

	Over $\mathbb{C}$	Over $\mathbb{R}$	Over $\mathbb{Q}$
Irreducible	9	18	396
Arbitrary	9	18	200

Constructions

Presentation:	$\langle a, b \mid a^{9}=b^{597}=1, b^{a}=b^{43} \rangle$
Permutation group:	Degree $202$ $\langle(2,4,14,5,15,26,8,17,7)(3,9,29,10,30,36,12,32,11)(6,20,59,21,60,71,24,62,23) \!\cdots\! \rangle$
Matrix group:	$\left\langle \left(\begin{array}{rr} 1 & 1 \\ 0 & 1 \end{array}\right), \left(\begin{array}{rr} 1 & 0 \\ 0 & 175 \end{array}\right), \left(\begin{array}{rr} 106 & 0 \\ 0 & 92 \end{array}\right) \right\rangle \subseteq \GL_{2}(\F_{199})$
Direct product:	$C_3$ $\, \times\, $ $(C_{199}:C_9)$
Semidirect product:	$C_{597}$ $\,\rtimes\,$ $C_9$	$(C_{199}:C_3)$ $\,\rtimes\,$ $C_9$ (2)	$C_{199}$ $\,\rtimes\,$ $(C_3\times C_9)$		more information
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Non-split product:	$(C_{597}:C_3)$ . $C_3$	$(C_{199}:C_3)$ . $C_3^2$			more information

Elements of the group are displayed as words in the presentation generators from the presentation above.

Homology

Abelianization:	$C_{3} \times C_{9} $
Schur multiplier:	$C_{3}$
Commutator length:	$1$

Subgroups

There are 1604 subgroups in 20 conjugacy classes, 12 normal (7 characteristic).

Characteristic subgroups are shown in this color. Normal (but not characteristic) subgroups are shown in this color.

Special subgroups

Center:	$Z \simeq$ $C_3$	$G/Z \simeq$ $C_{199}:C_9$
Commutator:	$G' \simeq$ $C_{199}$	$G/G' \simeq$ $C_3\times C_9$
Frattini:	$\Phi \simeq$ $C_1$	$G/\Phi \simeq$ $C_{597}:C_9$
Fitting:	$\operatorname{Fit} \simeq$ $C_{597}$	$G/\operatorname{Fit} \simeq$ $C_9$
Radical:	$R \simeq$ $C_{597}:C_9$	$G/R \simeq$ $C_1$
Socle:	$\operatorname{soc} \simeq$ $C_{597}$	$G/\operatorname{soc} \simeq$ $C_9$
3-Sylow subgroup:	$P_{ 3 } \simeq$ $C_3\times C_9$
199-Sylow subgroup:	$P_{ 199 } \simeq$ $C_{199}$

Hi

Subgroup diagram and profile

Classes of subgroups up to conjugation

Classes of subgroups up to automorphism

Normal subgroups

Normal subgroups up to automorphism

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Classes of subgroups up to conjugation

Order 5373: $C_{597}:C_9$

Order 1791: $C_{199}:C_9$ x 3, $C_{597}:C_3$

Order 597: $C_{199}:C_3$ x 3, $C_{597}$

Order 199: $C_{199}$

Order 27: $C_3\times C_9$

Order 9: $C_9$ x 3, $C_3^2$

Order 3: $C_3$ x 4

Order 1: $C_1$

Classes of subgroups up to automorphism

Order 5373: $C_{597}:C_9$

Order 1791: $C_{597}:C_3$, $C_{199}:C_9$

Order 597: $C_{199}:C_3$ x 2, $C_{597}$

Order 199: $C_{199}$

Order 27: $C_3\times C_9$

Order 9: $C_3^2$, $C_9$

Order 3: $C_3$ x 3

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 5373: $C_{597}:C_9$ ($C_1$)

Order 1791: $C_{199}:C_9$ ($C_3$) x 3, $C_{597}:C_3$ ($C_3$)

Order 597: $C_{199}:C_3$ ($C_9$) x 2, $C_{597}$ ($C_9$), $C_{199}:C_3$ ($C_3^2$)

Order 199: $C_{199}$ ($C_3\times C_9$)

Order 3: $C_3$ ($C_{199}:C_9$)

Order 1: $C_1$ ($C_{597}:C_9$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 5373: $C_{597}:C_9$ ($C_1$)

Order 1791: $C_{597}:C_3$ ($C_3$), $C_{199}:C_9$ ($C_3$)

Order 597: $C_{597}$ ($C_9$), $C_{199}:C_3$ ($C_3^2$), $C_{199}:C_3$ ($C_9$)

Order 199: $C_{199}$ ($C_3\times C_9$)

Order 3: $C_3$ ($C_{199}:C_9$)

Order 1: $C_1$ ($C_{597}:C_9$)

Series

Derived series	$C_{597}:C_9$	$\rhd$	$C_{199}$	$\rhd$	$C_1$
Chief series	$C_{597}:C_9$	$\rhd$	$C_{199}:C_9$	$\rhd$	$C_{199}:C_3$	$\rhd$	$C_{199}$	$\rhd$	$C_1$
Lower central series	$C_{597}:C_9$	$\rhd$	$C_{199}$
Upper central series	$C_1$	$\lhd$	$C_3$

Supergroups

This group is a maximal subgroup of 5 larger groups in the database.

This group is a maximal quotient of 3 larger groups in the database.

Character theory

Complex character table

See the $93 \times 93$ character table. Alternatively, you may search for characters of this group with desired properties.

Rational character table

		1A	3A	3B	3C	3D	9A	9B	9C	199A	597A
	Size	1	2	398	398	398	1194	1194	1194	198	396
	3 P	1A	3A	3B	3C	3D	9A	9B	9C	199A	597A
	199 P	1A	1A	1A	1A	1A	3B	3B	3B	199A	199A
5373.15.1a		$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$
5373.15.1b		$2$	$-1$	$2$	$-1$	$2$	$-1$	$2$	$-1$	$2$	$-1$
5373.15.1c		$2$	$-1$	$2$	$-1$	$2$	$-1$	$-1$	$-1$	$2$	$-1$
5373.15.1d		$2$	$-1$	$2$	$-1$	$2$	$2$	$-1$	$2$	$2$	$-1$
5373.15.1e		$2$	$2$	$2$	$2$	$2$	$-1$	$-1$	$-1$	$2$	$2$
5373.15.1f		$6$	$-3$	$-3$	$-3$	$-3$	$0$	$0$	$0$	$6$	$-3$
5373.15.1g		$6$	$-3$	$-3$	$6$	$-3$	$0$	$0$	$0$	$6$	$-3$
5373.15.1h		$6$	$6$	$-3$	$-3$	$-3$	$0$	$0$	$0$	$6$	$6$
5373.15.9a		$198$	$198$	$0$	$0$	$0$	$0$	$0$	$0$	$-1$	$-1$
5373.15.9b		$396$	$-198$	$0$	$0$	$0$	$0$	$0$	$0$	$-2$	$1$