Abstract group 360.91: $C_{60}:C_6$

• Label: 360.91
• Order: \( 2^{3} \cdot 3^{2} \cdot 5 \)
• Exponent: \( 2^{2} \cdot 3 \cdot 5 \)
• Nilpotent: no
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{3} \cdot 3^{2} \)
• $\card{Z(G)}$: \( 2^{2} \cdot 3^{2} \)
• $\card{\Aut(G)}$: \( 2^{8} \cdot 3 \cdot 5 \)
• $\card{\mathrm{Out}(G)}$: \( 2^{7} \cdot 3 \)
• Perm deg.: $15$
• Trans deg.: $180$
• Rank: $2$

Group information

Description:	$C_{60}:C_6$
Order:	\(360\)\(\medspace = 2^{3} \cdot 3^{2} \cdot 5 \)
Exponent:	\(60\)\(\medspace = 2^{2} \cdot 3 \cdot 5 \)
Automorphism group:	$C_2^2\times \GL(2,3)\times F_5$, of order \(3840\)\(\medspace = 2^{8} \cdot 3 \cdot 5 \)
Composition factors:	$C_2$ x 3, $C_3$ x 2, $C_5$
Derived length:	$2$

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

Group statistics

Order	1	2	3	4	5	6	10	12	15	20	30	60
Elements	1	11	8	12	4	88	4	96	32	8	32	64	360
Conjugacy classes	1	3	8	4	2	24	2	32	16	4	16	32	144
Divisions	1	3	4	2	1	12	1	8	4	1	4	4	45
Autjugacy classes	1	2	1	2	1	2	1	2	1	1	1	1	16

Dimension	1	2	4	8	16
Irr. complex chars.	72	72	0	0	0	144
Irr. rational chars.	4	18	10	9	4	45

Minimal presentations

Permutation degree:	$15$
Transitive degree:	$180$
Rank:	$2$
Inequivalent generating pairs:	$6$

Minimal degrees of faithful linear representations

	Over $\mathbb{C}$	Over $\mathbb{R}$	Over $\mathbb{Q}$
Irreducible	none	none	none
Arbitrary	3	6	10

Constructions

Presentation:	$\langle a, b \mid a^{6}=b^{60}=1, b^{a}=b^{49} \rangle$
Permutation group:	Degree $15$ $\langle(2,3)(4,5), (9,10,11,12), (13,14,15), (6,7,8)(13,14,15), (9,11)(10,12), (1,2,4,5,3)\rangle$
Matrix group:	$\left\langle \left(\begin{array}{rr} 1 & 10 \\ 0 & 1 \end{array}\right), \left(\begin{array}{rr} 7 & 0 \\ 0 & 7 \end{array}\right), \left(\begin{array}{rr} 19 & 0 \\ 0 & 1 \end{array}\right), \left(\begin{array}{rr} 16 & 15 \\ 15 & 1 \end{array}\right), \left(\begin{array}{rr} 1 & 6 \\ 0 & 1 \end{array}\right), \left(\begin{array}{rr} 19 & 0 \\ 0 & 19 \end{array}\right) \right\rangle \subseteq \GL_{2}(\Z/30\Z)$
Direct product:	$C_4$ $\, \times\, $ $C_3$ ${}^2$ $\, \times\, $ $D_5$
Semidirect product:	$C_{60}$ $\,\rtimes\,$ $C_6$ (4)	$(C_3\times C_{60})$ $\,\rtimes\,$ $C_2$	$(C_5:C_{12})$ $\,\rtimes\,$ $C_6$ (4)	$C_{20}$ $\,\rtimes\,$ $(C_3\times C_6)$	all 9
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Non-split product:	$C_{10}$ . $C_6^2$	$(C_3\times D_{10})$ . $C_6$ (4)	$D_{10}$ . $(C_3\times C_6)$	$(C_3\times C_6)$ . $D_{10}$	all 9

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

Homology

Abelianization:	$C_{6} \times C_{12} \simeq C_{2} \times C_{4} \times C_{3}^{2}$
Schur multiplier:	$C_{6}$
Commutator length:	$1$

Subgroups

There are 216 subgroups in 96 conjugacy classes, 66 normal (18 characteristic).

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

Special subgroups

Center:	$Z \simeq$ $C_3\times C_{12}$	$G/Z \simeq$ $D_5$
Commutator:	$G' \simeq$ $C_5$	$G/G' \simeq$ $C_6\times C_{12}$
Frattini:	$\Phi \simeq$ $C_2$	$G/\Phi \simeq$ $C_3^2\times D_{10}$
Fitting:	$\operatorname{Fit} \simeq$ $C_3\times C_{60}$	$G/\operatorname{Fit} \simeq$ $C_2$
Radical:	$R \simeq$ $C_{60}:C_6$	$G/R \simeq$ $C_1$
Socle:	$\operatorname{soc} \simeq$ $C_3\times C_{30}$	$G/\operatorname{soc} \simeq$ $C_2^2$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $C_2\times C_4$
3-Sylow subgroup:	$P_{ 3 } \simeq$ $C_3^2$
5-Sylow subgroup:	$P_{ 5 } \simeq$ $C_5$

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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Subgroup information

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Classes of subgroups up to conjugation

Order 360: $C_{60}:C_6$

Order 180: $C_3^2\times D_{10}$, $C_3\times C_{60}$, $C_{15}:C_{12}$

Order 120: $D_5\times C_{12}$ x 4

Order 90: $C_3^2\times D_5$ x 2, $C_3\times C_{30}$

Order 72: $C_6\times C_{12}$

Order 60: $C_{60}$ x 4, $C_5:C_{12}$ x 4, $C_3\times D_{10}$ x 4

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

Order 40: $C_4\times D_5$

Order 36: $C_3\times C_{12}$ x 2, $C_6^2$

Order 30: $C_3\times D_5$ x 8, $C_{30}$ x 4

Order 24: $C_2\times C_{12}$ x 4

Order 20: $D_{10}$, $C_{20}$, $C_5:C_4$

Order 18: $C_3\times C_6$ x 3

Order 15: $C_{15}$ x 4

Order 12: $C_{12}$ x 8, $C_2\times C_6$ x 4

Order 10: $D_5$ x 2, $C_{10}$

Order 9: $C_3^2$

Order 8: $C_2\times C_4$

Order 6: $C_6$ x 12

Order 5: $C_5$

Order 4: $C_4$ x 2, $C_2^2$

Order 3: $C_3$ x 4

Order 2: $C_2$ x 3

Order 1: $C_1$

Classes of subgroups up to automorphism

Order 360: $C_{60}:C_6$

Order 180: $C_3^2\times D_{10}$, $C_3\times C_{60}$, $C_{15}:C_{12}$

Order 120: $D_5\times C_{12}$

Order 90: $C_3^2\times D_5$, $C_3\times C_{30}$

Order 72: $C_6\times C_{12}$

Order 60: $C_{60}$, $C_5:C_{12}$, $C_3\times D_{10}$

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

Order 40: $C_4\times D_5$

Order 36: $C_3\times C_{12}$ x 2, $C_6^2$

Order 30: $C_{30}$, $C_3\times D_5$

Order 24: $C_2\times C_{12}$

Order 20: $D_{10}$, $C_{20}$, $C_5:C_4$

Order 18: $C_3\times C_6$ x 2

Order 15: $C_{15}$

Order 12: $C_{12}$ x 2, $C_2\times C_6$

Order 10: $C_{10}$, $D_5$

Order 9: $C_3^2$

Order 8: $C_2\times C_4$

Order 6: $C_6$ x 2

Order 5: $C_5$

Order 4: $C_4$ x 2, $C_2^2$

Order 3: $C_3$

Order 2: $C_2$ x 2

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 360: $C_{60}:C_6$ ($C_1$)

Order 180: $C_3^2\times D_{10}$ ($C_2$), $C_3\times C_{60}$ ($C_2$), $C_{15}:C_{12}$ ($C_2$)

Order 120: $D_5\times C_{12}$ ($C_3$) x 4

Order 90: $C_3^2\times D_5$ ($C_4$) x 2, $C_3\times C_{30}$ ($C_2^2$)

Order 60: $C_{60}$ ($C_6$) x 4, $C_5:C_{12}$ ($C_6$) x 4, $C_3\times D_{10}$ ($C_6$) x 4

Order 45: $C_3\times C_{15}$ ($C_2\times C_4$)

Order 40: $C_4\times D_5$ ($C_3^2$)

Order 36: $C_3\times C_{12}$ ($D_5$)

Order 30: $C_3\times D_5$ ($C_{12}$) x 8, $C_{30}$ ($C_2\times C_6$) x 4

Order 20: $D_{10}$ ($C_3\times C_6$), $C_{20}$ ($C_3\times C_6$), $C_5:C_4$ ($C_3\times C_6$)

Order 18: $C_3\times C_6$ ($D_{10}$)

Order 15: $C_{15}$ ($C_2\times C_{12}$) x 4

Order 12: $C_{12}$ ($C_3\times D_5$) x 4

Order 10: $D_5$ ($C_3\times C_{12}$) x 2, $C_{10}$ ($C_6^2$)

Order 9: $C_3^2$ ($C_4\times D_5$)

Order 6: $C_6$ ($C_3\times D_{10}$) x 4

Order 5: $C_5$ ($C_6\times C_{12}$)

Order 4: $C_4$ ($C_3^2\times D_5$)

Order 3: $C_3$ ($D_5\times C_{12}$) x 4

Order 2: $C_2$ ($C_3^2\times D_{10}$)

Order 1: $C_1$ ($C_{60}:C_6$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 360: $C_{60}:C_6$ ($C_1$)

Order 180: $C_3^2\times D_{10}$ ($C_2$), $C_3\times C_{60}$ ($C_2$), $C_{15}:C_{12}$ ($C_2$)

Order 120: $D_5\times C_{12}$ ($C_3$)

Order 90: $C_3^2\times D_5$ ($C_4$), $C_3\times C_{30}$ ($C_2^2$)

Order 60: $C_{60}$ ($C_6$), $C_5:C_{12}$ ($C_6$), $C_3\times D_{10}$ ($C_6$)

Order 45: $C_3\times C_{15}$ ($C_2\times C_4$)

Order 40: $C_4\times D_5$ ($C_3^2$)

Order 36: $C_3\times C_{12}$ ($D_5$)

Order 30: $C_{30}$ ($C_2\times C_6$), $C_3\times D_5$ ($C_{12}$)

Order 20: $D_{10}$ ($C_3\times C_6$), $C_{20}$ ($C_3\times C_6$), $C_5:C_4$ ($C_3\times C_6$)

Order 18: $C_3\times C_6$ ($D_{10}$)

Order 15: $C_{15}$ ($C_2\times C_{12}$)

Order 12: $C_{12}$ ($C_3\times D_5$)

Order 10: $C_{10}$ ($C_6^2$), $D_5$ ($C_3\times C_{12}$)

Order 9: $C_3^2$ ($C_4\times D_5$)

Order 6: $C_6$ ($C_3\times D_{10}$)

Order 5: $C_5$ ($C_6\times C_{12}$)

Order 4: $C_4$ ($C_3^2\times D_5$)

Order 3: $C_3$ ($D_5\times C_{12}$)

Order 2: $C_2$ ($C_3^2\times D_{10}$)

Order 1: $C_1$ ($C_{60}:C_6$)

Series

Derived series

$C_{60}:C_6$

$\rhd$

$C_5$

$\rhd$

$C_1$

Chief series

$C_{60}:C_6$

$\rhd$

$C_3\times C_{60}$

$\rhd$

$C_{60}$

$\rhd$

$C_{30}$

$\rhd$

$C_{15}$

$\rhd$

$C_5$

$\rhd$

$C_1$

Lower central series

$C_{60}:C_6$

$\rhd$

$C_5$

Upper central series

$C_1$

$\lhd$

$C_3\times C_{12}$

Supergroups

This group is a maximal subgroup of 52 larger groups in the database.

This group is a maximal quotient of 20 larger groups in the database.

Character theory

Complex character table

See the $144 \times 144$ character table (warning: may be slow to load). Alternatively, you may search for characters of this group with desired properties.

Rational character table

See the $45 \times 45$ rational character table.