Abstract group 39402.d: $F_{199}$

• Label: 39402.d
• Order: \( 2 \cdot 3^{2} \cdot 11 \cdot 199 \)
• Exponent: \( 2 \cdot 3^{2} \cdot 11 \cdot 199 \)
• Nilpotent: no
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2 \cdot 3^{2} \cdot 11 \)
• $\card{Z(G)}$: \( 1 \)
• $\card{\Aut(G)}$: \( 2 \cdot 3^{2} \cdot 11 \cdot 199 \)
• $\card{\mathrm{Out}(G)}$: \( 1 \)
• Perm deg.: $199$
• Trans deg.: $199$
• Rank: $2$

Group information

Description:	$F_{199}$
Order:	\(39402\)\(\medspace = 2 \cdot 3^{2} \cdot 11 \cdot 199 \)
Exponent:	\(39402\)\(\medspace = 2 \cdot 3^{2} \cdot 11 \cdot 199 \)
Automorphism group:	$F_{199}$, of order \(39402\)\(\medspace = 2 \cdot 3^{2} \cdot 11 \cdot 199 \)
Composition factors:	$C_2$, $C_3$ x 2, $C_{11}$, $C_{199}$
Derived length:	$2$

This group is nonabelian and a Z-group (hence solvable, supersolvable, monomial, metacyclic, metabelian, and an A-group).

Group statistics

Order	1	2	3	6	9	11	18	22	33	66	99	198	199
Elements	1	199	398	398	1194	1990	1194	1990	3980	3980	11940	11940	198	39402
Conjugacy classes	1	1	2	2	6	10	6	10	20	20	60	60	1	199
Divisions	1	1	1	1	1	1	1	1	1	1	1	1	1	13
Autjugacy classes	1	1	2	2	6	10	6	10	20	20	60	60	1	199

Dimension	1	2	6	10	20	60	198
Irr. complex chars.	198	0	0	0	0	0	1	199
Irr. rational chars.	2	2	2	2	2	2	1	13

Minimal presentations

Permutation degree:	$199$
Transitive degree:	$199$
Rank:	$2$
Inequivalent generating pairs:	$25920$

Minimal degrees of faithful linear representations

	Over $\mathbb{C}$	Over $\mathbb{R}$	Over $\mathbb{Q}$
Irreducible	198	198	198
Arbitrary	198	198	198

Constructions

Groups of Lie type:	$\AGL(1,199)$
Presentation:	$\langle a, b \mid a^{198}=b^{199}=1, b^{a}=b^{15} \rangle$
Permutation group:	Degree $199$ $\langle(2,3)(4,8)(5,7)(6,9)(10,23)(11,22)(12,24)(13,21)(14,20)(15,19)(16,26)(17,25) \!\cdots\! \rangle$
Matrix group:	$\left\langle \left(\begin{array}{rr} 175 & 0 \\ 0 & 173 \end{array}\right), \left(\begin{array}{rr} 1 & 1 \\ 0 & 1 \end{array}\right), \left(\begin{array}{rr} 106 & 0 \\ 0 & 92 \end{array}\right) \right\rangle \subseteq \GL_{2}(\F_{199})$
Direct product:	not isomorphic to a non-trivial direct product
Semidirect product:	$D_{199}$ $\,\rtimes\,$ $C_{99}$	$C_{199}$ $\,\rtimes\,$ $C_{198}$	$(C_{199}:C_{99})$ $\,\rtimes\,$ $C_2$	$(C_{199}:C_{22})$ $\,\rtimes\,$ $C_9$	all 7
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Non-split product:	$(C_{199}:C_{66})$ . $C_3$	$(C_{199}:C_{33})$ . $C_6$	$(C_{199}:C_6)$ . $C_{33}$	$(C_{199}:C_3)$ . $C_{66}$	more information
Aut. group:	$\Aut(D_{199})$	$\Aut(C_{199}:C_3)$	$\Aut(C_{199}:C_6)$	$\Aut(C_{199}:C_6)$	all 15

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

Homology

Abelianization:	$C_{198} \simeq C_{2} \times C_{9} \times C_{11}$
Schur multiplier:	$C_1$
Commutator length:	$1$

Subgroups

There are 2202 subgroups in 24 conjugacy classes, 13 normal, and all normal subgroups are characteristic.

Characteristic subgroups are shown in this color.

Special subgroups

Center:	$Z \simeq$ $C_1$	$G/Z \simeq$ $F_{199}$
Commutator:	$G' \simeq$ $C_{199}$	$G/G' \simeq$ $C_{198}$
Frattini:	$\Phi \simeq$ $C_1$	$G/\Phi \simeq$ $F_{199}$
Fitting:	$\operatorname{Fit} \simeq$ $C_{199}$	$G/\operatorname{Fit} \simeq$ $C_{198}$
Radical:	$R \simeq$ $F_{199}$	$G/R \simeq$ $C_1$
Socle:	$\operatorname{soc} \simeq$ $C_{199}$	$G/\operatorname{soc} \simeq$ $C_{198}$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $C_2$
3-Sylow subgroup:	$P_{ 3 } \simeq$ $C_9$
11-Sylow subgroup:	$P_{ 11 } \simeq$ $C_{11}$
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

For the default diagram, subgroups are sorted vertically by the number of prime divisors (counted with multiplicity) in their orders.
To see subgroups sorted vertically by order instead, check this box.

Sorry, your browser does not support the subgroup diagram.

Subgroup information Click on a subgroup in the diagram to see information about it.

See a full page version of the diagram

See a full page version of the diagram

See a full page version of the diagram

See a full page version of the diagram

Classes of subgroups up to conjugation

Order 39402: $F_{199}$

Order 19701: $C_{199}:C_{99}$

Order 13134: $C_{199}:C_{66}$

Order 6567: $C_{199}:C_{33}$

Order 4378: $C_{199}:C_{22}$

Order 3582: $C_{199}:C_{18}$

Order 2189: $C_{199}:C_{11}$

Order 1791: $C_{199}:C_9$

Order 1194: $C_{199}:C_6$

Order 597: $C_{199}:C_3$

Order 398: $D_{199}$

Order 199: $C_{199}$

Order 198: $C_{198}$

Order 99: $C_{99}$

Order 66: $C_{66}$

Order 33: $C_{33}$

Order 22: $C_{22}$

Order 18: $C_{18}$

Order 11: $C_{11}$

Order 9: $C_9$

Order 6: $C_6$

Order 3: $C_3$

Order 2: $C_2$

Order 1: $C_1$

Classes of subgroups up to automorphism

Order 39402: $F_{199}$

Order 19701: $C_{199}:C_{99}$

Order 13134: $C_{199}:C_{66}$

Order 6567: $C_{199}:C_{33}$

Order 4378: $C_{199}:C_{22}$

Order 3582: $C_{199}:C_{18}$

Order 2189: $C_{199}:C_{11}$

Order 1791: $C_{199}:C_9$

Order 1194: $C_{199}:C_6$

Order 597: $C_{199}:C_3$

Order 398: $D_{199}$

Order 199: $C_{199}$

Order 198: $C_{198}$

Order 99: $C_{99}$

Order 66: $C_{66}$

Order 33: $C_{33}$

Order 22: $C_{22}$

Order 18: $C_{18}$

Order 11: $C_{11}$

Order 9: $C_9$

Order 6: $C_6$

Order 3: $C_3$

Order 2: $C_2$

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 39402: $F_{199}$ ($C_1$)

Order 19701: $C_{199}:C_{99}$ ($C_2$)

Order 13134: $C_{199}:C_{66}$ ($C_3$)

Order 6567: $C_{199}:C_{33}$ ($C_6$)

Order 4378: $C_{199}:C_{22}$ ($C_9$)

Order 3582: $C_{199}:C_{18}$ ($C_{11}$)

Order 2189: $C_{199}:C_{11}$ ($C_{18}$)

Order 1791: $C_{199}:C_9$ ($C_{22}$)

Order 1194: $C_{199}:C_6$ ($C_{33}$)

Order 597: $C_{199}:C_3$ ($C_{66}$)

Order 398: $D_{199}$ ($C_{99}$)

Order 199: $C_{199}$ ($C_{198}$)

Order 1: $C_1$ ($F_{199}$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 39402: $F_{199}$ ($C_1$)

Order 19701: $C_{199}:C_{99}$ ($C_2$)

Order 13134: $C_{199}:C_{66}$ ($C_3$)

Order 6567: $C_{199}:C_{33}$ ($C_6$)

Order 4378: $C_{199}:C_{22}$ ($C_9$)

Order 3582: $C_{199}:C_{18}$ ($C_{11}$)

Order 2189: $C_{199}:C_{11}$ ($C_{18}$)

Order 1791: $C_{199}:C_9$ ($C_{22}$)

Order 1194: $C_{199}:C_6$ ($C_{33}$)

Order 597: $C_{199}:C_3$ ($C_{66}$)

Order 398: $D_{199}$ ($C_{99}$)

Order 199: $C_{199}$ ($C_{198}$)

Order 1: $C_1$ ($F_{199}$)

Series

Derived series

$F_{199}$

$\rhd$

$C_{199}$

$\rhd$

$C_1$

Chief series

$F_{199}$

$\rhd$

$C_{199}:C_{99}$

$\rhd$

$C_{199}:C_{33}$

$\rhd$

$C_{199}:C_{11}$

$\rhd$

$C_{199}$

$\rhd$

$C_1$

Lower central series

$F_{199}$

$\rhd$

$C_{199}$

Upper central series

$C_1$

Supergroups

This group is a maximal subgroup of 4 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 $199 \times 199$ 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	9A	11A	18A	22A	33A	66A	99A	198A	199A
	Size	1	199	398	398	1194	1990	1194	1990	3980	3980	11940	11940	198
	2 P	1A	1A	3A	3A	9A	11A	9A	11A	33A	33A	99A	99A	199A
	3 P	1A	2A	1A	2A	3A	11A	6A	22A	11A	22A	33A	66A	199A
	11 P	1A	2A	3A	6A	9A	11A	18A	22A	33A	66A	99A	198A	199A
	199 P	1A	2A	3A	6A	9A	1A	18A	2A	3A	6A	9A	18A	199A
39402.d.1a		$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$
39402.d.1b		$1$	$-1$	$1$	$-1$	$1$	$1$	$-1$	$-1$	$1$	$-1$	$1$	$-1$	$1$
39402.d.1c		$2$	$2$	$2$	$2$	$-1$	$2$	$-1$	$2$	$2$	$2$	$-1$	$-1$	$2$
39402.d.1d		$2$	$-2$	$2$	$-2$	$-1$	$2$	$1$	$-2$	$2$	$-2$	$-1$	$1$	$2$
39402.d.1e		$6$	$6$	$-3$	$-3$	$0$	$6$	$0$	$6$	$-3$	$-3$	$0$	$0$	$6$
39402.d.1f		$6$	$-6$	$-3$	$3$	$0$	$6$	$0$	$-6$	$-3$	$3$	$0$	$0$	$6$
39402.d.1g		$10$	$10$	$10$	$10$	$10$	$-1$	$10$	$-1$	$-1$	$-1$	$-1$	$-1$	$10$
39402.d.1h		$10$	$-10$	$10$	$-10$	$10$	$-1$	$-10$	$1$	$-1$	$1$	$-1$	$1$	$10$
39402.d.1i		$20$	$20$	$20$	$20$	$-10$	$-2$	$-10$	$-2$	$-2$	$-2$	$1$	$1$	$20$
39402.d.1j		$20$	$-20$	$20$	$-20$	$-10$	$-2$	$10$	$2$	$-2$	$2$	$1$	$-1$	$20$
39402.d.1k		$60$	$60$	$-30$	$-30$	$0$	$-6$	$0$	$-6$	$3$	$3$	$0$	$0$	$60$
39402.d.1l		$60$	$-60$	$-30$	$30$	$0$	$-6$	$0$	$6$	$3$	$-3$	$0$	$0$	$60$
39402.d.198a		$198$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$-1$