Abstract group 1194.4: $C_3\times D_{199}$

• Label: 1194.4
• Order: \( 2 \cdot 3 \cdot 199 \)
• Exponent: \( 2 \cdot 3 \cdot 199 \)
• Nilpotent: no
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2 \cdot 3 \)
• $\card{Z(G)}$: \( 3 \)
• $\card{\Aut(G)}$: \( 2^{2} \cdot 3^{2} \cdot 11 \cdot 199 \)
• $\card{\mathrm{Out}(G)}$: \( 2 \cdot 3^{2} \cdot 11 \)
• Perm deg.: $202$
• Trans deg.: $597$
• Rank: $2$

Group information

Description:	$C_3\times D_{199}$
Order:	\(1194\)\(\medspace = 2 \cdot 3 \cdot 199 \)
Exponent:	\(1194\)\(\medspace = 2 \cdot 3 \cdot 199 \)
Automorphism group:	$C_2\times F_{199}$, of order \(78804\)\(\medspace = 2^{2} \cdot 3^{2} \cdot 11 \cdot 199 \)
Composition factors:	$C_2$, $C_3$, $C_{199}$
Derived length:	$2$

This group is nonabelian, a Z-group (hence solvable, supersolvable, monomial, metacyclic, metabelian, and an A-group), and hyperelementary for $p = 2$.

Group statistics

Order	1	2	3	6	199	597
Elements	1	199	2	398	198	396	1194
Conjugacy classes	1	1	2	2	99	198	303
Divisions	1	1	1	1	1	1	6
Autjugacy classes	1	1	1	1	1	1	6

Dimension	1	2	198	396
Irr. complex chars.	6	297	0	0	303
Irr. rational chars.	2	2	1	1	6

Minimal presentations

Permutation degree:	$202$
Transitive degree:	$597$
Rank:	$2$
Inequivalent generating pairs:	$12$

Minimal degrees of faithful linear representations

	Over $\mathbb{C}$	Over $\mathbb{R}$	Over $\mathbb{Q}$
Irreducible	2	4	396
Arbitrary	2	4	200

Constructions

Presentation:	$\langle a, b \mid a^{2}=b^{597}=1, b^{a}=b^{397} \rangle$
Permutation group:	Degree $202$ $\langle(2,5)(3,9)(4,12)(6,15)(7,19)(8,21)(10,25)(11,14)(13,29)(16,33)(17,37)(18,39) \!\cdots\! \rangle$
Matrix group:	$\left\langle \left(\begin{array}{rr} 1 & 1 \\ 0 & 1 \end{array}\right), \left(\begin{array}{rr} 106 & 0 \\ 0 & 93 \end{array}\right) \right\rangle \subseteq \GL_{2}(\F_{199})$
Direct product:	$C_3$ $\, \times\, $ $D_{199}$
Semidirect product:	$C_{597}$ $\,\rtimes\,$ $C_2$	$C_{199}$ $\,\rtimes\,$ $C_6$			more information
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product

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

Homology

Abelianization:	$C_{6} \simeq C_{2} \times C_{3}$
Schur multiplier:	$C_1$
Commutator length:	$1$

Subgroups

There are 404 subgroups in 8 conjugacy classes, 6 normal, and all normal subgroups are characteristic.

Characteristic subgroups are shown in this color.

Special subgroups

Center:	$Z \simeq$ $C_3$	$G/Z \simeq$ $D_{199}$
Commutator:	$G' \simeq$ $C_{199}$	$G/G' \simeq$ $C_6$
Frattini:	$\Phi \simeq$ $C_1$	$G/\Phi \simeq$ $C_3\times D_{199}$
Fitting:	$\operatorname{Fit} \simeq$ $C_{597}$	$G/\operatorname{Fit} \simeq$ $C_2$
Radical:	$R \simeq$ $C_3\times D_{199}$	$G/R \simeq$ $C_1$
Socle:	$\operatorname{soc} \simeq$ $C_{597}$	$G/\operatorname{soc} \simeq$ $C_2$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $C_2$
3-Sylow subgroup:	$P_{ 3 } \simeq$ $C_3$
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 1194: $C_3\times D_{199}$

Order 597: $C_{597}$

Order 398: $D_{199}$

Order 199: $C_{199}$

Order 6: $C_6$

Order 3: $C_3$

Order 2: $C_2$

Order 1: $C_1$

Classes of subgroups up to automorphism

Order 1194: $C_3\times D_{199}$

Order 597: $C_{597}$

Order 398: $D_{199}$

Order 199: $C_{199}$

Order 6: $C_6$

Order 3: $C_3$

Order 2: $C_2$

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 1194: $C_3\times D_{199}$ ($C_1$)

Order 597: $C_{597}$ ($C_2$)

Order 398: $D_{199}$ ($C_3$)

Order 199: $C_{199}$ ($C_6$)

Order 3: $C_3$ ($D_{199}$)

Order 1: $C_1$ ($C_3\times D_{199}$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 1194: $C_3\times D_{199}$ ($C_1$)

Order 597: $C_{597}$ ($C_2$)

Order 398: $D_{199}$ ($C_3$)

Order 199: $C_{199}$ ($C_6$)

Order 3: $C_3$ ($D_{199}$)

Order 1: $C_1$ ($C_3\times D_{199}$)

Series

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

Supergroups

This group is a maximal subgroup of 6 larger groups in the database.

This group is a maximal quotient of 4 larger groups in the database.

Character theory

Complex character table

See the $303 \times 303$ character table (warning: may be slow to load). Alternatively, you may search for characters of this group with desired properties.

Rational character table

		1A	2A	3A	6A	199A	597A
	Size	1	199	2	398	198	396
	2 P	1A	1A	3A	3A	199A	597A
	3 P	1A	2A	1A	2A	199A	199A
	199 P	1A	2A	3A	6A	199A	597A
1194.4.1a		$1$	$1$	$1$	$1$	$1$	$1$
1194.4.1b		$1$	$-1$	$1$	$-1$	$1$	$1$
1194.4.1c		$2$	$2$	$-1$	$-1$	$2$	$-1$
1194.4.1d		$2$	$-2$	$-1$	$1$	$2$	$-1$
1194.4.2a		$198$	$0$	$198$	$0$	$-1$	$-1$
1194.4.2b		$396$	$0$	$-198$	$0$	$-2$	$1$