Abstract group 39402.n: $C_{199}:C_{198}$

• Label: 39402.n
• Order: \( 2 \cdot 3^{2} \cdot 11 \cdot 199 \)
• Exponent: \( 2 \cdot 3^{2} \cdot 11 \cdot 199 \)
• Nilpotent: no
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2 \cdot 3^{2} \cdot 11 \)
• $\card{Z(G)}$: \( 2 \cdot 11 \)
• $\card{\Aut(G)}$: \( 2^{2} \cdot 3^{2} \cdot 5 \cdot 11 \cdot 199 \)
• $\card{\mathrm{Out}(G)}$: \( 2^{2} \cdot 5 \cdot 11 \)
• Perm deg.: $212$
• Trans deg.: $4378$
• Rank: $2$

Group information

Description:	$C_{199}:C_{198}$
Order:	\(39402\)\(\medspace = 2 \cdot 3^{2} \cdot 11 \cdot 199 \)
Exponent:	\(39402\)\(\medspace = 2 \cdot 3^{2} \cdot 11 \cdot 199 \)
Automorphism group:	$C_{199}:(C_2\times C_{990})$, of order \(394020\)\(\medspace = 2^{2} \cdot 3^{2} \cdot 5 \cdot 11 \cdot 199 \)
Composition factors:	$C_2$, $C_3$ x 2, $C_{11}$, $C_{199}$
Derived length:	$2$

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

Group statistics

Order	1	2	3	6	9	11	18	22	33	66	99	198	199	398	2189	4378
Elements	1	1	398	398	1194	10	1194	10	3980	3980	11940	11940	198	198	1980	1980	39402
Conjugacy classes	1	1	2	2	6	10	6	10	20	20	60	60	22	22	220	220	682
Divisions	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	16
Autjugacy classes	1	1	2	2	6	1	6	1	2	2	6	6	1	1	1	1	40

Minimal presentations

Permutation degree:	$212$
Transitive degree:	$4378$
Rank:	$2$
Inequivalent generating pairs:	$2592$

Minimal degrees of faithful linear representations

	Over $\mathbb{C}$	Over $\mathbb{R}$	Over $\mathbb{Q}$
Irreducible	9	not computed	not computed
Arbitrary	not computed	not computed	not computed

Constructions

Presentation:	$\langle a, b \mid a^{198}=b^{199}=1, b^{a}=b^{180} \rangle$
Permutation group:	Degree $212$ $\langle(1,2,4,10,26,56,106,160,196,182,199,175,121,64,119,101,90,44,20,7,18,9,24,52,83,140,163,153,126,181,193,179,122,177,143,85,42,19,21,45,86,50,23,17,39,78,133,185,132,77,124,108,162,190,146,88,43,47,81,138,187,141,97,49,38,68,125,180,129,73,98,79,134,92,46,89,94,116,170,137,149,183,128,152,96,70,32,69,59,112,136,80,40,41,82,139,188,195,157,191,167,135,186,161,107,76,36,55,105,158,150,93,84,51,100,58,110,144,168,113,159,164,109,57,65,29,63,117,104,62,87,142,172,154,99,53,28,61,115,169,114,60,27,34,66,123,176,192,151,145,102,120,173,156,103,54,25,12,31,67,30,11,14,35,74,130,184,147,189,166,174,155,111,165,131,75,37,16,6,15,5,13,33,72,71,127,91,148,178,198,194,197,171,118,95,48,22,8,3) \!\cdots\! \rangle$
Matrix group:	$\left\langle \left(\begin{array}{rr} 107 & 0 \\ 0 & 93 \end{array}\right), \left(\begin{array}{rr} 1 & 1 \\ 0 & 1 \end{array}\right), \left(\begin{array}{rr} 148 & 0 \\ 0 & 173 \end{array}\right) \right\rangle \subseteq \GL_{2}(\F_{199})$
Direct product:	$C_2$ $\, \times\, $ $C_{11}$ $\, \times\, $ $(C_{199}:C_9)$
Semidirect product:	$C_{4378}$ $\,\rtimes\,$ $C_9$	$C_{398}$ $\,\rtimes\,$ $C_{99}$	$C_{2189}$ $\,\rtimes\,$ $C_{18}$	$C_{199}$ $\,\rtimes\,$ $C_{198}$	more information
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

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 1608 subgroups in 24 conjugacy classes, 16 normal, and all normal subgroups are characteristic.

Characteristic subgroups are shown in this color.

Special subgroups

Center:	$Z \simeq$ $C_{22}$	$G/Z \simeq$ $C_{199}:C_9$
Commutator:	$G' \simeq$ $C_{199}$	$G/G' \simeq$ $C_{198}$
Frattini:	$\Phi \simeq$ $C_1$	$G/\Phi \simeq$ $C_{199}:C_{198}$
Fitting:	$\operatorname{Fit} \simeq$ $C_{4378}$	$G/\operatorname{Fit} \simeq$ $C_9$
Radical:	$R \simeq$ $C_{199}:C_{198}$	$G/R \simeq$ $C_1$
Socle:	$\operatorname{soc} \simeq$ $C_{4378}$	$G/\operatorname{soc} \simeq$ $C_9$
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

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Classes of subgroups up to conjugation

Order 39402: $C_{199}:C_{198}$

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

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

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

Order 4378: $C_{4378}$

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

Order 2189: $C_{2189}$

Order 1791: $C_{199}:C_9$

Order 1194: $C_{199}:C_6$

Order 597: $C_{199}:C_3$

Order 398: $C_{398}$

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: $C_{199}:C_{198}$

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

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

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

Order 4378: $C_{4378}$

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

Order 2189: $C_{2189}$

Order 1791: $C_{199}:C_9$

Order 1194: $C_{199}:C_6$

Order 597: $C_{199}:C_3$

Order 398: $C_{398}$

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: $C_{199}:C_{198}$ ($C_1$)

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: $C_{4378}$ ($C_9$)

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

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

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: $C_{398}$ ($C_{99}$)

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

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

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

Order 2: $C_2$ ($C_{199}:C_{99}$)

Order 1: $C_1$ ($C_{199}:C_{198}$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 39402: $C_{199}:C_{198}$ ($C_1$)

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: $C_{4378}$ ($C_9$)

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

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

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: $C_{398}$ ($C_{99}$)

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

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

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

Order 2: $C_2$ ($C_{199}:C_{99}$)

Order 1: $C_1$ ($C_{199}:C_{198}$)

Series

Derived series

$C_{199}:C_{198}$

$\rhd$

$C_{199}$

$\rhd$

$C_1$

Chief series

$C_{199}:C_{198}$

$\rhd$

$C_{199}:C_{99}$

$\rhd$

$C_{199}:C_{33}$

$\rhd$

$C_{2189}$

$\rhd$

$C_{199}$

$\rhd$

$C_1$

Lower central series

$C_{199}:C_{198}$

$\rhd$

$C_{199}$

Upper central series

$C_1$

$\lhd$

$C_{22}$

Supergroups

This group is a maximal subgroup of 1 larger groups in the database.

This group is a maximal quotient of 0 larger groups in the database.

Character theory

Complex character table

The $682 \times 682$ character table is not available for this group.

Rational character table

The $16 \times 16$ rational character table is not available for this group.