Abstract group 94950.b: $C_{3165}:C_{30}$

• Label: 94950.b
• Order: \( 2 \cdot 3^{2} \cdot 5^{2} \cdot 211 \)
• Exponent: \( 2 \cdot 3 \cdot 5 \cdot 211 \)
• Nilpotent: no
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2 \cdot 3^{2} \cdot 5^{2} \)
• $\card{Z(G)}$: \( 3 \cdot 5 \)
• $\card{\Aut(G)}$: \( 2^{4} \cdot 3^{2} \cdot 5^{2} \cdot 7 \cdot 211 \)
• $\card{\mathrm{Out}(G)}$: \( 2^{3} \cdot 3 \cdot 5 \cdot 7 \)
• Perm deg.: $219$
• Trans deg.: $3165$
• Rank: $2$

Group information

Description:	$C_{3165}:C_{30}$
Order:	\(94950\)\(\medspace = 2 \cdot 3^{2} \cdot 5^{2} \cdot 211 \)
Exponent:	\(6330\)\(\medspace = 2 \cdot 3 \cdot 5 \cdot 211 \)
Automorphism group:	$C_{3165}.C_{210}.C_2^3$, of order \(5317200\)\(\medspace = 2^{4} \cdot 3^{2} \cdot 5^{2} \cdot 7 \cdot 211 \)
Composition factors:	$C_2$, $C_3$ x 2, $C_5$ x 2, $C_{211}$
Derived length:	$2$

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

Group statistics

Order	1	2	3	5	6	10	15	30	211	633	1055	3165
Elements	1	211	1268	4224	1688	5064	38832	40512	210	420	840	1680	94950
Conjugacy classes	1	1	8	24	8	24	192	192	7	14	28	56	555
Divisions	1	1	4	6	4	6	24	24	1	1	1	1	74
Autjugacy classes	1	1	3	5	3	5	15	15	1	1	1	1	52

Minimal presentations

Permutation degree:	$219$
Transitive degree:	$3165$
Rank:	$2$
Inequivalent generating pairs:	$576$

Minimal degrees of faithful linear representations

	Over $\mathbb{C}$	Over $\mathbb{R}$	Over $\mathbb{Q}$
Irreducible	30	not computed	not computed
Arbitrary	not computed	not computed	not computed

Constructions

Presentation:	$\langle a, b \mid a^{30}=b^{3165}=1, b^{a}=b^{1816} \rangle$
Permutation group:	Degree $219$ $\langle(2,4,14,45,12,37,66,19,33,62,41,10,36,29,7)(3,9,35,30,8,17,54,39,48,73,21,5,15,44,11) \!\cdots\! \rangle$
Matrix group:	$\left\langle \left(\begin{array}{rr} 137 & 0 \\ 0 & 134 \end{array}\right), \left(\begin{array}{rr} 1 & 1 \\ 0 & 1 \end{array}\right), \left(\begin{array}{rr} 1 & 0 \\ 0 & 128 \end{array}\right) \right\rangle \subseteq \GL_{2}(\F_{211})$
Direct product:	$C_3$ $\, \times\, $ $C_5$ $\, \times\, $ $(C_{211}:C_{30})$
Semidirect product:	$C_{3165}$ $\,\rtimes\,$ $C_{30}$	$D_{211}$ $\,\rtimes\,$ $C_{15}^2$	$(C_{3165}:C_6)$ $\,\rtimes\,$ $C_5$	$C_{633}$ $\,\rtimes\,$ $(C_5\times C_{30})$	all 28
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:	$C_{15} \times C_{30} \simeq C_{2} \times C_{3}^{2} \times C_{5}^{2}$
Schur multiplier:	$C_{15}$
Commutator length:	$1$

Subgroups

There are 19512 subgroups in 192 conjugacy classes, 100 normal (22 characteristic).

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

Special subgroups

Center:	$Z \simeq$ $C_{15}$	$G/Z \simeq$ $C_{211}:C_{30}$
Commutator:	$G' \simeq$ $C_{211}$	$G/G' \simeq$ $C_{15}\times C_{30}$
Frattini:	$\Phi \simeq$ $C_1$	$G/\Phi \simeq$ $C_{3165}:C_{30}$
Fitting:	$\operatorname{Fit} \simeq$ $C_{3165}$	$G/\operatorname{Fit} \simeq$ $C_{30}$
Radical:	$R \simeq$ $C_{3165}:C_{30}$	$G/R \simeq$ $C_1$
Socle:	$\operatorname{soc} \simeq$ $C_{3165}$	$G/\operatorname{soc} \simeq$ $C_{30}$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $C_2$
3-Sylow subgroup:	$P_{ 3 } \simeq$ $C_3^2$
5-Sylow subgroup:	$P_{ 5 } \simeq$ $C_5^2$
211-Sylow subgroup:	$P_{ 211 } \simeq$ $C_{211}$

Hi

Subgroup diagram and profile

Classes of subgroups up to conjugation

No diagram available: subgroups only stored up to automorphism

Classes of subgroups up to automorphism

Normal subgroups

No diagram available: subgroups only stored up to automorphism

Normal subgroups up to automorphism

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Subgroup information

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Classes of subgroups up to conjugation

Order 94950: $C_{3165}:C_{30}$

Order 47475: $C_{3165}:C_{15}$

Order 31650: $C_{1055}:C_{30}$ x 3, $C_{1055}:C_{30}$

Order 18990: $C_{633}:C_{30}$ x 5, $C_{3165}:C_6$

Order 15825: $C_{1055}:C_{15}$ x 3, $C_{1055}:C_{15}$

Order 10550: $C_{1055}:C_{10}$

Order 9495: $C_{633}:C_{15}$ x 5, $C_{3165}:C_3$

Order 6330: $C_{211}:C_{30}$ x 15, $C_{211}:C_{30}$ x 5, $C_{1055}:C_6$ x 3, $C_{15}\times D_{211}$

Order 5275: $C_{1055}:C_5$

Order 3798: $C_{633}:C_6$

Order 3165: $C_{211}:C_{15}$ x 15, $C_{211}:C_{15}$ x 5, $C_{1055}:C_3$ x 3, $C_{3165}$

Order 2110: $C_{211}:C_{10}$ x 5, $C_5\times D_{211}$

Order 1899: $C_{633}:C_3$

Order 1266: $C_{211}:C_6$ x 3, $C_3\times D_{211}$

Order 1055: $C_{211}:C_5$ x 5, $C_{1055}$

Order 633: $C_{211}:C_3$ x 3, $C_{633}$

Order 450: $C_{15}\times C_{30}$

Order 422: $D_{211}$

Order 225: $C_{15}^2$

Order 211: $C_{211}$

Order 150: $C_5\times C_{30}$ x 4

Order 90: $C_3\times C_{30}$ x 6

Order 75: $C_5\times C_{15}$ x 4

Order 50: $C_5\times C_{10}$

Order 45: $C_3\times C_{15}$ x 6

Order 30: $C_{30}$ x 24

Order 25: $C_5^2$

Order 18: $C_3\times C_6$

Order 15: $C_{15}$ x 24

Order 10: $C_{10}$ x 6

Order 9: $C_3^2$

Order 6: $C_6$ x 4

Order 5: $C_5$ x 6

Order 3: $C_3$ x 4

Order 2: $C_2$

Order 1: $C_1$

Classes of subgroups up to automorphism

Order 94950: $C_{3165}:C_{30}$

Order 47475: $C_{3165}:C_{15}$

Order 31650: $C_{1055}:C_{30}$, $C_{1055}:C_{30}$

Order 18990: $C_{3165}:C_6$, $C_{633}:C_{30}$

Order 15825: $C_{1055}:C_{15}$, $C_{1055}:C_{15}$

Order 10550: $C_{1055}:C_{10}$

Order 9495: $C_{3165}:C_3$, $C_{633}:C_{15}$

Order 6330: $C_{211}:C_{30}$, $C_{1055}:C_6$, $C_{211}:C_{30}$, $C_{15}\times D_{211}$

Order 5275: $C_{1055}:C_5$

Order 3798: $C_{633}:C_6$

Order 3165: $C_{3165}$, $C_{1055}:C_3$, $C_{211}:C_{15}$, $C_{211}:C_{15}$

Order 2110: $C_5\times D_{211}$, $C_{211}:C_{10}$

Order 1899: $C_{633}:C_3$

Order 1266: $C_3\times D_{211}$, $C_{211}:C_6$

Order 1055: $C_{1055}$, $C_{211}:C_5$

Order 633: $C_{633}$, $C_{211}:C_3$

Order 450: $C_{15}\times C_{30}$

Order 422: $D_{211}$

Order 225: $C_{15}^2$

Order 211: $C_{211}$

Order 150: $C_5\times C_{30}$ x 2

Order 90: $C_3\times C_{30}$ x 2

Order 75: $C_5\times C_{15}$ x 2

Order 50: $C_5\times C_{10}$

Order 45: $C_3\times C_{15}$ x 2

Order 30: $C_{30}$ x 4

Order 25: $C_5^2$

Order 18: $C_3\times C_6$

Order 15: $C_{15}$ x 4

Order 10: $C_{10}$ x 2

Order 9: $C_3^2$

Order 6: $C_6$ x 2

Order 5: $C_5$ x 2

Order 3: $C_3$ x 2

Order 2: $C_2$

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 94950: $C_{3165}:C_{30}$ ($C_1$)

Order 47475: $C_{3165}:C_{15}$ ($C_2$)

Order 31650: $C_{1055}:C_{30}$ ($C_3$) x 3, $C_{1055}:C_{30}$ ($C_3$)

Order 18990: $C_{633}:C_{30}$ ($C_5$) x 5, $C_{3165}:C_6$ ($C_5$)

Order 15825: $C_{1055}:C_{15}$ ($C_6$) x 3, $C_{1055}:C_{15}$ ($C_6$)

Order 10550: $C_{1055}:C_{10}$ ($C_3^2$)

Order 9495: $C_{633}:C_{15}$ ($C_{10}$) x 5, $C_{3165}:C_3$ ($C_{10}$)

Order 6330: $C_{211}:C_{30}$ ($C_{15}$) x 15, $C_{211}:C_{30}$ ($C_{15}$) x 5, $C_{1055}:C_6$ ($C_{15}$) x 3, $C_{15}\times D_{211}$ ($C_{15}$)

Order 5275: $C_{1055}:C_5$ ($C_3\times C_6$)

Order 3798: $C_{633}:C_6$ ($C_5^2$)

Order 3165: $C_{211}:C_{15}$ ($C_{30}$) x 15, $C_{211}:C_{15}$ ($C_{30}$) x 5, $C_{1055}:C_3$ ($C_{30}$) x 3, $C_{3165}$ ($C_{30}$)

Order 2110: $C_{211}:C_{10}$ ($C_3\times C_{15}$) x 5, $C_5\times D_{211}$ ($C_3\times C_{15}$)

Order 1899: $C_{633}:C_3$ ($C_5\times C_{10}$)

Order 1266: $C_{211}:C_6$ ($C_5\times C_{15}$) x 3, $C_3\times D_{211}$ ($C_5\times C_{15}$)

Order 1055: $C_{211}:C_5$ ($C_3\times C_{30}$) x 5, $C_{1055}$ ($C_3\times C_{30}$)

Order 633: $C_{211}:C_3$ ($C_5\times C_{30}$) x 3, $C_{633}$ ($C_5\times C_{30}$)

Order 422: $D_{211}$ ($C_{15}^2$)

Order 211: $C_{211}$ ($C_{15}\times C_{30}$)

Order 15: $C_{15}$ ($C_{211}:C_{30}$)

Order 5: $C_5$ ($C_{633}:C_{30}$)

Order 3: $C_3$ ($C_{1055}:C_{30}$)

Order 1: $C_1$ ($C_{3165}:C_{30}$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 94950: $C_{3165}:C_{30}$ ($C_1$)

Order 47475: $C_{3165}:C_{15}$ ($C_2$)

Order 31650: $C_{1055}:C_{30}$ ($C_3$), $C_{1055}:C_{30}$ ($C_3$)

Order 18990: $C_{3165}:C_6$ ($C_5$), $C_{633}:C_{30}$ ($C_5$)

Order 15825: $C_{1055}:C_{15}$ ($C_6$), $C_{1055}:C_{15}$ ($C_6$)

Order 10550: $C_{1055}:C_{10}$ ($C_3^2$)

Order 9495: $C_{3165}:C_3$ ($C_{10}$), $C_{633}:C_{15}$ ($C_{10}$)

Order 6330: $C_{211}:C_{30}$ ($C_{15}$), $C_{1055}:C_6$ ($C_{15}$), $C_{211}:C_{30}$ ($C_{15}$), $C_{15}\times D_{211}$ ($C_{15}$)

Order 5275: $C_{1055}:C_5$ ($C_3\times C_6$)

Order 3798: $C_{633}:C_6$ ($C_5^2$)

Order 3165: $C_{3165}$ ($C_{30}$), $C_{1055}:C_3$ ($C_{30}$), $C_{211}:C_{15}$ ($C_{30}$), $C_{211}:C_{15}$ ($C_{30}$)

Order 2110: $C_5\times D_{211}$ ($C_3\times C_{15}$), $C_{211}:C_{10}$ ($C_3\times C_{15}$)

Order 1899: $C_{633}:C_3$ ($C_5\times C_{10}$)

Order 1266: $C_3\times D_{211}$ ($C_5\times C_{15}$), $C_{211}:C_6$ ($C_5\times C_{15}$)

Order 1055: $C_{1055}$ ($C_3\times C_{30}$), $C_{211}:C_5$ ($C_3\times C_{30}$)

Order 633: $C_{633}$ ($C_5\times C_{30}$), $C_{211}:C_3$ ($C_5\times C_{30}$)

Order 422: $D_{211}$ ($C_{15}^2$)

Order 211: $C_{211}$ ($C_{15}\times C_{30}$)

Order 15: $C_{15}$ ($C_{211}:C_{30}$)

Order 5: $C_5$ ($C_{633}:C_{30}$)

Order 3: $C_3$ ($C_{1055}:C_{30}$)

Order 1: $C_1$ ($C_{3165}:C_{30}$)

Series

Derived series

$C_{3165}:C_{30}$

$\rhd$

$C_{3165}:C_{30}$

$\rhd$

$C_{211}$

$\rhd$

$C_{211}$

$\rhd$

$C_1$

$\rhd$

$C_1$

Chief series

$C_{3165}:C_{30}$

$\rhd$

$C_{3165}:C_{30}$

$\rhd$

$C_{3165}:C_{15}$

$\rhd$

$C_{3165}:C_{15}$

$\rhd$

$C_{1055}:C_{15}$

$\rhd$

$C_{1055}:C_{15}$

$\rhd$

$C_{3165}$

$\rhd$

$C_{3165}$

$\rhd$

$C_{1055}$

$\rhd$

$C_{1055}$

$\rhd$

$C_{211}$

$\rhd$

$C_{211}$

$\rhd$

$C_1$

$\rhd$

$C_1$

Lower central series

$C_{3165}:C_{30}$

$\rhd$

$C_{3165}:C_{30}$

$\rhd$

$C_{211}$

$\rhd$

$C_{211}$

Upper central series

$C_1$

$\lhd$

$C_1$

$\lhd$

$C_{15}$

$\lhd$

$C_{15}$

Supergroups

This group is a maximal subgroup of 2 larger groups in the database.

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

Character theory

Complex character table

The $555 \times 555$ character table is not available for this group.

Rational character table

The $74 \times 74$ rational character table is not available for this group.