Abstract group 67950.h: $C_{2265}:C_{30}$

• Label: 67950.h
• Order: \( 2 \cdot 3^{2} \cdot 5^{2} \cdot 151 \)
• Exponent: \( 2 \cdot 3 \cdot 5 \cdot 151 \)
• Nilpotent: no
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2 \cdot 3^{2} \cdot 5^{2} \)
• $\card{Z(G)}$: \( 2 \cdot 3 \cdot 5 \)
• $\card{\Aut(G)}$: \( 2^{4} \cdot 3^{2} \cdot 5^{3} \cdot 151 \)
• $\card{\mathrm{Out}(G)}$: \( 2^{4} \cdot 3 \cdot 5^{2} \)
• Perm deg.: $161$
• Trans deg.: $4530$
• Rank: $2$

Group information

Description:	$C_{2265}:C_{30}$
Order:	\(67950\)\(\medspace = 2 \cdot 3^{2} \cdot 5^{2} \cdot 151 \)
Exponent:	\(4530\)\(\medspace = 2 \cdot 3 \cdot 5 \cdot 151 \)
Automorphism group:	$C_{2265}.C_{150}.C_2^3$, of order \(2718000\)\(\medspace = 2^{4} \cdot 3^{2} \cdot 5^{3} \cdot 151 \)
Composition factors:	$C_2$, $C_3$ x 2, $C_5$ x 2, $C_{151}$
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	151	302	453	755	906	1510	2265	4530
Elements	1	1	908	3024	908	3024	27792	27792	150	150	300	600	300	600	1200	1200	67950
Conjugacy classes	1	1	8	24	8	24	192	192	10	10	20	40	20	40	80	80	750
Divisions	1	1	4	6	4	6	24	24	1	1	1	1	1	1	1	1	78
Autjugacy classes	1	1	3	5	3	5	15	15	1	1	1	1	1	1	1	1	56

Minimal presentations

Permutation degree:	$161$
Transitive degree:	$4530$
Rank:	$2$
Inequivalent generating pairs:	$576$

Minimal degrees of faithful linear representations

	Over $\mathbb{C}$	Over $\mathbb{R}$	Over $\mathbb{Q}$
Irreducible	15	not computed	not computed
Arbitrary	not computed	not computed	not computed

Constructions

Presentation:	$\langle a, b \mid a^{30}=b^{2265}=1, b^{a}=b^{1246} \rangle$
Permutation group:	Degree $161$ $\langle(2,4,18,27,25,8,7,20,6,9,10,22,29,23,5)(3,11,35,48,42,14,13,37,15,16,17,39,45,40,12) \!\cdots\! \rangle$
Matrix group:	$\left\langle \left(\begin{array}{rr} 1 & 1 \\ 0 & 1 \end{array}\right), \left(\begin{array}{rr} 75 & 0 \\ 0 & 149 \end{array}\right), \left(\begin{array}{rr} 1 & 0 \\ 0 & 38 \end{array}\right) \right\rangle \subseteq \GL_{2}(\F_{151})$
Direct product:	$C_2$ $\, \times\, $ $C_3$ $\, \times\, $ $C_5$ $\, \times\, $ $(C_{151}:C_{15})$
Semidirect product:	$C_{4530}$ $\,\rtimes\,$ $C_{15}$	$C_{2265}$ $\,\rtimes\,$ $C_{30}$	$C_{302}$ $\,\rtimes\,$ $C_{15}^2$	$(C_{2265}:C_5)$ $\,\rtimes\,$ $C_6$	all 22
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 13392 subgroups in 192 conjugacy classes, 104 normal (26 characteristic).

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

Special subgroups

Center:	$Z \simeq$ $C_{30}$	$G/Z \simeq$ $C_{151}:C_{15}$
Commutator:	$G' \simeq$ $C_{151}$	$G/G' \simeq$ $C_{15}\times C_{30}$
Frattini:	$\Phi \simeq$ $C_1$	$G/\Phi \simeq$ $C_{2265}:C_{30}$
Fitting:	$\operatorname{Fit} \simeq$ $C_{4530}$	$G/\operatorname{Fit} \simeq$ $C_{15}$
Radical:	$R \simeq$ $C_{2265}:C_{30}$	$G/R \simeq$ $C_1$
Socle:	$\operatorname{soc} \simeq$ $C_{4530}$	$G/\operatorname{soc} \simeq$ $C_{15}$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $C_2$
3-Sylow subgroup:	$P_{ 3 } \simeq$ $C_3^2$
5-Sylow subgroup:	$P_{ 5 } \simeq$ $C_5^2$
151-Sylow subgroup:	$P_{ 151 } \simeq$ $C_{151}$

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 67950: $C_{2265}:C_{30}$

Order 33975: $C_{15}\times C_{151}:C_{15}$

Order 22650: $C_{10}\times C_{151}:C_{15}$ x 3, $C_{30}\times C_{151}:C_5$

Order 13590: $C_6\times C_{151}:C_{15}$ x 5, $C_{30}\times C_{151}:C_3$

Order 11325: $C_{755}:C_{15}$ x 3, $C_{2265}:C_5$

Order 7550: $C_{1510}:C_5$

Order 6795: $C_{453}:C_{15}$ x 5, $C_{2265}:C_3$

Order 4530: $C_{151}:C_{30}$ x 15, $C_{151}:C_{30}$ x 5, $C_{1510}:C_3$ x 3, $C_{4530}$

Order 3775: $C_{755}:C_5$

Order 2718: $C_{453}:C_6$

Order 2265: $C_{151}:C_{15}$ x 15, $C_{151}:C_{15}$ x 5, $C_{151}:C_{15}$ x 3, $C_{2265}$

Order 1510: $C_{151}:C_{10}$ x 5, $C_{1510}$

Order 1359: $C_{453}:C_3$

Order 906: $C_{151}:C_6$ x 3, $C_{906}$

Order 755: $C_{151}:C_5$ x 5, $C_{755}$

Order 453: $C_{151}:C_3$ x 3, $C_{453}$

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

Order 302: $C_{302}$

Order 225: $C_{15}^2$

Order 151: $C_{151}$

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 67950: $C_{2265}:C_{30}$

Order 33975: $C_{15}\times C_{151}:C_{15}$

Order 22650: $C_{10}\times C_{151}:C_{15}$, $C_{30}\times C_{151}:C_5$

Order 13590: $C_{30}\times C_{151}:C_3$, $C_6\times C_{151}:C_{15}$

Order 11325: $C_{2265}:C_5$, $C_{755}:C_{15}$

Order 7550: $C_{1510}:C_5$

Order 6795: $C_{2265}:C_3$, $C_{453}:C_{15}$

Order 4530: $C_{151}:C_{30}$, $C_{1510}:C_3$, $C_{151}:C_{30}$, $C_{4530}$

Order 3775: $C_{755}:C_5$

Order 2718: $C_{453}:C_6$

Order 2265: $C_{2265}$, $C_{151}:C_{15}$, $C_{151}:C_{15}$, $C_{151}:C_{15}$

Order 1510: $C_{1510}$, $C_{151}:C_{10}$

Order 1359: $C_{453}:C_3$

Order 906: $C_{906}$, $C_{151}:C_6$

Order 755: $C_{755}$, $C_{151}:C_5$

Order 453: $C_{453}$, $C_{151}:C_3$

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

Order 302: $C_{302}$

Order 225: $C_{15}^2$

Order 151: $C_{151}$

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 67950: $C_{2265}:C_{30}$ ($C_1$)

Order 33975: $C_{15}\times C_{151}:C_{15}$ ($C_2$)

Order 22650: $C_{10}\times C_{151}:C_{15}$ ($C_3$) x 3, $C_{30}\times C_{151}:C_5$ ($C_3$)

Order 13590: $C_6\times C_{151}:C_{15}$ ($C_5$) x 5, $C_{30}\times C_{151}:C_3$ ($C_5$)

Order 11325: $C_{755}:C_{15}$ ($C_6$) x 3, $C_{2265}:C_5$ ($C_6$)

Order 7550: $C_{1510}:C_5$ ($C_3^2$)

Order 6795: $C_{453}:C_{15}$ ($C_{10}$) x 5, $C_{2265}:C_3$ ($C_{10}$)

Order 4530: $C_{151}:C_{30}$ ($C_{15}$) x 15, $C_{151}:C_{30}$ ($C_{15}$) x 5, $C_{1510}:C_3$ ($C_{15}$) x 3, $C_{4530}$ ($C_{15}$)

Order 3775: $C_{755}:C_5$ ($C_3\times C_6$)

Order 2718: $C_{453}:C_6$ ($C_5^2$)

Order 2265: $C_{151}:C_{15}$ ($C_{30}$) x 15, $C_{151}:C_{15}$ ($C_{30}$) x 5, $C_{151}:C_{15}$ ($C_{30}$) x 3, $C_{2265}$ ($C_{30}$)

Order 1510: $C_{151}:C_{10}$ ($C_3\times C_{15}$) x 5, $C_{1510}$ ($C_3\times C_{15}$)

Order 1359: $C_{453}:C_3$ ($C_5\times C_{10}$)

Order 906: $C_{151}:C_6$ ($C_5\times C_{15}$) x 3, $C_{906}$ ($C_5\times C_{15}$)

Order 755: $C_{151}:C_5$ ($C_3\times C_{30}$) x 5, $C_{755}$ ($C_3\times C_{30}$)

Order 453: $C_{151}:C_3$ ($C_5\times C_{30}$) x 3, $C_{453}$ ($C_5\times C_{30}$)

Order 302: $C_{302}$ ($C_{15}^2$)

Order 151: $C_{151}$ ($C_{15}\times C_{30}$)

Order 30: $C_{30}$ ($C_{151}:C_{15}$)

Order 15: $C_{15}$ ($C_{151}:C_{30}$)

Order 10: $C_{10}$ ($C_{453}:C_{15}$)

Order 6: $C_6$ ($C_{755}:C_{15}$)

Order 5: $C_5$ ($C_{151}:(C_3\times C_{30})$)

Order 3: $C_3$ ($C_{151}:(C_5\times C_{30})$)

Order 2: $C_2$ ($C_{151}:C_{15}^2$)

Order 1: $C_1$ ($C_{2265}:C_{30}$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 67950: $C_{2265}:C_{30}$ ($C_1$)

Order 33975: $C_{15}\times C_{151}:C_{15}$ ($C_2$)

Order 22650: $C_{10}\times C_{151}:C_{15}$ ($C_3$), $C_{30}\times C_{151}:C_5$ ($C_3$)

Order 13590: $C_{30}\times C_{151}:C_3$ ($C_5$), $C_6\times C_{151}:C_{15}$ ($C_5$)

Order 11325: $C_{2265}:C_5$ ($C_6$), $C_{755}:C_{15}$ ($C_6$)

Order 7550: $C_{1510}:C_5$ ($C_3^2$)

Order 6795: $C_{2265}:C_3$ ($C_{10}$), $C_{453}:C_{15}$ ($C_{10}$)

Order 4530: $C_{151}:C_{30}$ ($C_{15}$), $C_{1510}:C_3$ ($C_{15}$), $C_{151}:C_{30}$ ($C_{15}$), $C_{4530}$ ($C_{15}$)

Order 3775: $C_{755}:C_5$ ($C_3\times C_6$)

Order 2718: $C_{453}:C_6$ ($C_5^2$)

Order 2265: $C_{2265}$ ($C_{30}$), $C_{151}:C_{15}$ ($C_{30}$), $C_{151}:C_{15}$ ($C_{30}$), $C_{151}:C_{15}$ ($C_{30}$)

Order 1510: $C_{1510}$ ($C_3\times C_{15}$), $C_{151}:C_{10}$ ($C_3\times C_{15}$)

Order 1359: $C_{453}:C_3$ ($C_5\times C_{10}$)

Order 906: $C_{906}$ ($C_5\times C_{15}$), $C_{151}:C_6$ ($C_5\times C_{15}$)

Order 755: $C_{755}$ ($C_3\times C_{30}$), $C_{151}:C_5$ ($C_3\times C_{30}$)

Order 453: $C_{453}$ ($C_5\times C_{30}$), $C_{151}:C_3$ ($C_5\times C_{30}$)

Order 302: $C_{302}$ ($C_{15}^2$)

Order 151: $C_{151}$ ($C_{15}\times C_{30}$)

Order 30: $C_{30}$ ($C_{151}:C_{15}$)

Order 15: $C_{15}$ ($C_{151}:C_{30}$)

Order 10: $C_{10}$ ($C_{453}:C_{15}$)

Order 6: $C_6$ ($C_{755}:C_{15}$)

Order 5: $C_5$ ($C_{151}:(C_3\times C_{30})$)

Order 3: $C_3$ ($C_{151}:(C_5\times C_{30})$)

Order 2: $C_2$ ($C_{151}:C_{15}^2$)

Order 1: $C_1$ ($C_{2265}:C_{30}$)

Series

Derived series

$C_{2265}:C_{30}$

$\rhd$

$C_{2265}:C_{30}$

$\rhd$

$C_{151}$

$\rhd$

$C_{151}$

$\rhd$

$C_1$

$\rhd$

$C_1$

Chief series

$C_{2265}:C_{30}$

$\rhd$

$C_{2265}:C_{30}$

$\rhd$

$C_{15}\times C_{151}:C_{15}$

$\rhd$

$C_{15}\times C_{151}:C_{15}$

$\rhd$

$C_{2265}:C_5$

$\rhd$

$C_{2265}:C_5$

$\rhd$

$C_{2265}$

$\rhd$

$C_{2265}$

$\rhd$

$C_{755}$

$\rhd$

$C_{755}$

$\rhd$

$C_{151}$

$\rhd$

$C_{151}$

$\rhd$

$C_1$

$\rhd$

$C_1$

Lower central series

$C_{2265}:C_{30}$

$\rhd$

$C_{2265}:C_{30}$

$\rhd$

$C_{151}$

$\rhd$

$C_{151}$

Upper central series

$C_1$

$\lhd$

$C_1$

$\lhd$

$C_{30}$

$\lhd$

$C_{30}$

Supergroups

This group is a maximal subgroup of 3 larger groups in the database.

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

Character theory

Complex character table

The $750 \times 750$ character table is not available for this group.

Rational character table

The $78 \times 78$ rational character table is not available for this group.