Abstract group 297.2: $C_3\times C_{99}$

• Label: 297.2
• Order: \( 3^{3} \cdot 11 \)
• Exponent: \( 3^{2} \cdot 11 \)
• Abelian: yes
• $\card{\Aut(G)}$: \( 2^{3} \cdot 3^{3} \cdot 5 \)
• Perm deg.: $23$
• Trans deg.: $297$
• Rank: $2$

Group information

Description:	$C_3\times C_{99}$
Order:	\(297\)\(\medspace = 3^{3} \cdot 11 \)
Exponent:	\(99\)\(\medspace = 3^{2} \cdot 11 \)
Automorphism group:	$C_5\times C_6^2:S_3$, of order \(1080\)\(\medspace = 2^{3} \cdot 3^{3} \cdot 5 \)
Composition factors:	$C_3$ x 3, $C_{11}$
Nilpotency class:	$1$
Derived length:	$1$

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

Group statistics

Order	1	3	9	11	33	99
Elements	1	8	18	10	80	180	297
Conjugacy classes	1	8	18	10	80	180	297
Divisions	1	4	3	1	4	3	16
Autjugacy classes	1	2	1	1	2	1	8

Dimension	1	2	6	10	20	60
Irr. complex chars.	297	0	0	0	0	0	297
Irr. rational chars.	1	4	3	1	4	3	16

Minimal presentations

Permutation degree:	$23$
Transitive degree:	$297$
Rank:	$2$
Inequivalent generating pairs:	$48$

Minimal degrees of faithful linear representations

	Over $\mathbb{C}$	Over $\mathbb{R}$	Over $\mathbb{Q}$
Irreducible	none	none	none
Arbitrary	2	4	18

Constructions

Presentation:	Abelian group $\langle a, b \mid a^{3}=b^{99}=1 \rangle$
Permutation group:	Degree $23$ $\langle(4,12,9,6,11,8,5,10,7), (1,3,2), (13,23,22,21,20,19,18,17,16,15,14), (4,6,5)(7,9,8)(10,12,11)\rangle$
Matrix group:	$\left\langle \left(\begin{array}{rr} 16 & 0 \\ 0 & 13 \end{array}\right), \left(\begin{array}{rr} 106 & 0 \\ 0 & 92 \end{array}\right) \right\rangle \subseteq \GL_{2}(\F_{199})$
Direct product:	$C_3$ $\, \times\, $ $C_9$ $\, \times\, $ $C_{11}$
Semidirect product:	not isomorphic to a non-trivial semidirect product
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Non-split product:	$C_{33}$ . $C_3^2$	$C_3^2$ . $C_{33}$	$(C_3\times C_{33})$ . $C_3$	$C_3$ . $(C_3\times C_{33})$	more information

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

Homology

Primary decomposition:	$C_{3} \times C_{9} \times C_{11}$
Schur multiplier:	$C_{3}$
Commutator length:	$0$

Subgroups

There are 20 subgroups, all normal (8 characteristic).

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

Special subgroups

Center:	$Z \simeq$ $C_3\times C_{99}$	$G/Z \simeq$ $C_1$
Commutator:	$G' \simeq$ $C_1$	$G/G' \simeq$ $C_3\times C_{99}$
Frattini:	$\Phi \simeq$ $C_3$	$G/\Phi \simeq$ $C_3\times C_{33}$
Fitting:	$\operatorname{Fit} \simeq$ $C_3\times C_{99}$	$G/\operatorname{Fit} \simeq$ $C_1$
Radical:	$R \simeq$ $C_3\times C_{99}$	$G/R \simeq$ $C_1$
Socle:	$\operatorname{soc} \simeq$ $C_3\times C_{33}$	$G/\operatorname{soc} \simeq$ $C_3$
3-Sylow subgroup:	$P_{ 3 } \simeq$ $C_3\times C_9$
11-Sylow subgroup:	$P_{ 11 } \simeq$ $C_{11}$

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 297: $C_3\times C_{99}$

Order 99: $C_{99}$ x 3, $C_3\times C_{33}$

Order 33: $C_{33}$ x 4

Order 27: $C_3\times C_9$

Order 11: $C_{11}$

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 297: $C_3\times C_{99}$

Order 99: $C_3\times C_{33}$, $C_{99}$

Order 33: $C_{33}$ x 2

Order 27: $C_3\times C_9$

Order 11: $C_{11}$

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

Order 3: $C_3$ x 2

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 297: $C_3\times C_{99}$ ($C_1$)

Order 99: $C_{99}$ ($C_3$) x 3, $C_3\times C_{33}$ ($C_3$)

Order 33: $C_{33}$ ($C_9$) x 3, $C_{33}$ ($C_3^2$)

Order 27: $C_3\times C_9$ ($C_{11}$)

Order 11: $C_{11}$ ($C_3\times C_9$)

Order 9: $C_9$ ($C_{33}$) x 3, $C_3^2$ ($C_{33}$)

Order 3: $C_3$ ($C_{99}$) x 3, $C_3$ ($C_3\times C_{33}$)

Order 1: $C_1$ ($C_3\times C_{99}$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 297: $C_3\times C_{99}$ ($C_1$)

Order 99: $C_3\times C_{33}$ ($C_3$), $C_{99}$ ($C_3$)

Order 33: $C_{33}$ ($C_3^2$), $C_{33}$ ($C_9$)

Order 27: $C_3\times C_9$ ($C_{11}$)

Order 11: $C_{11}$ ($C_3\times C_9$)

Order 9: $C_3^2$ ($C_{33}$), $C_9$ ($C_{33}$)

Order 3: $C_3$ ($C_3\times C_{33}$), $C_3$ ($C_{99}$)

Order 1: $C_1$ ($C_3\times C_{99}$)

Series

Derived series	$C_3\times C_{99}$	$\rhd$	$C_1$
Chief series	$C_3\times C_{99}$	$\rhd$	$C_3\times C_{33}$	$\rhd$	$C_{33}$	$\rhd$	$C_{11}$	$\rhd$	$C_1$
Lower central series	$C_3\times C_{99}$	$\rhd$	$C_1$
Upper central series	$C_1$	$\lhd$	$C_3\times C_{99}$

Supergroups

This group is a maximal subgroup of 32 larger groups in the database.

This group is a maximal quotient of 19 larger groups in the database.

Character theory

Complex character table

See the $297 \times 297$ character table (warning: may be slow to load). Alternatively, you may search for characters of this group with desired properties.

Rational character table

		1A	3A	3B	3C	3D	9A	9B	9C	11A	33A	33B	33C	33D	99A	99B	99C
	Size	1	2	2	2	2	6	6	6	10	20	20	20	20	60	60	60
	3 P	1A	3A	3B	3C	3D	9A	9B	9C	11A	33A	33B	33C	33D	99A	99B	99C
	11 P	1A	1A	1A	1A	1A	3D	3D	3D	11A	11A	11A	11A	11A	33D	33D	33D
297.2.1a		$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$
297.2.1b		$2$	$-1$	$2$	$-1$	$-1$	$-1$	$-1$	$2$	$2$	$2$	$-1$	$2$	$-1$	$-1$	$-1$	$2$
297.2.1c		$2$	$-1$	$2$	$-1$	$-1$	$-1$	$-1$	$-1$	$2$	$2$	$-1$	$2$	$-1$	$2$	$2$	$-1$
297.2.1d		$2$	$-1$	$2$	$-1$	$-1$	$2$	$2$	$-1$	$2$	$2$	$-1$	$2$	$-1$	$-1$	$-1$	$-1$
297.2.1e		$2$	$2$	$2$	$2$	$2$	$-1$	$-1$	$-1$	$2$	$2$	$2$	$2$	$2$	$-1$	$-1$	$-1$
297.2.1f		$6$	$-3$	$-3$	$-3$	$-3$	$0$	$0$	$0$	$6$	$-3$	$6$	$-3$	$-3$	$0$	$0$	$0$
297.2.1g		$6$	$-3$	$-3$	$-3$	$6$	$0$	$0$	$0$	$6$	$-3$	$-3$	$-3$	$-3$	$0$	$0$	$0$
297.2.1h		$6$	$6$	$-3$	$6$	$-3$	$0$	$0$	$0$	$6$	$-3$	$-3$	$-3$	$6$	$0$	$0$	$0$
297.2.1i		$10$	$10$	$10$	$10$	$10$	$10$	$10$	$10$	$-1$	$-1$	$-1$	$-1$	$-1$	$-1$	$-1$	$-1$
297.2.1j		$20$	$-10$	$20$	$-10$	$-10$	$-10$	$-10$	$20$	$-2$	$-2$	$1$	$-2$	$1$	$1$	$1$	$-2$
297.2.1k		$20$	$-10$	$20$	$-10$	$-10$	$-10$	$-10$	$-10$	$-2$	$-2$	$1$	$-2$	$1$	$-2$	$-2$	$1$
297.2.1l		$20$	$-10$	$20$	$-10$	$-10$	$20$	$20$	$-10$	$-2$	$-2$	$1$	$-2$	$1$	$1$	$1$	$1$
297.2.1m		$20$	$20$	$20$	$20$	$20$	$-10$	$-10$	$-10$	$-2$	$-2$	$-2$	$-2$	$-2$	$1$	$1$	$1$
297.2.1n		$60$	$-30$	$-30$	$-30$	$-30$	$0$	$0$	$0$	$-6$	$3$	$-6$	$3$	$3$	$0$	$0$	$0$
297.2.1o		$60$	$-30$	$-30$	$-30$	$60$	$0$	$0$	$0$	$-6$	$3$	$3$	$3$	$3$	$0$	$0$	$0$
297.2.1p		$60$	$60$	$-30$	$60$	$-30$	$0$	$0$	$0$	$-6$	$3$	$3$	$3$	$-6$	$0$	$0$	$0$