Abstract group 960.6444: $C_{60}.C_4^2$

• Label: 960.6444
• Order: \( 2^{6} \cdot 3 \cdot 5 \)
• Exponent: \( 2^{3} \cdot 3 \cdot 5 \)
• Nilpotent: no
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{6} \cdot 3 \)
• $\card{Z(G)}$: \( 2^{4} \cdot 3 \)
• $\card{\Aut(G)}$: \( 2^{10} \cdot 5 \)
• $\card{\mathrm{Out}(G)}$: \( 2^{8} \)
• Perm deg.: $18$
• Trans deg.: $240$
• Rank: $3$

Group information

Description:	$C_{60}.C_4^2$
Order:	\(960\)\(\medspace = 2^{6} \cdot 3 \cdot 5 \)
Exponent:	\(120\)\(\medspace = 2^{3} \cdot 3 \cdot 5 \)
Automorphism group:	$C_5:(C_2^5.C_2^5)$, of order \(5120\)\(\medspace = 2^{10} \cdot 5 \)
Composition factors:	$C_2$ x 6, $C_3$, $C_5$
Derived length:	$2$

This group is nonabelian, supersolvable (hence solvable and monomial), hyperelementary for $p = 2$, metabelian, and an A-group.

Group statistics

Order	1	2	3	4	5	6	8	10	12	15	20	24	30	40	60	120
Elements	1	23	2	104	4	46	128	12	208	8	16	256	24	32	32	64	960
Conjugacy classes	1	7	2	24	1	14	32	3	48	2	4	64	6	8	8	16	240
Divisions	1	7	1	12	1	7	8	3	12	1	2	8	3	2	2	2	72
Autjugacy classes	1	4	1	6	1	4	4	2	6	1	2	4	2	1	2	1	42

Dimension	1	2	4	8	16	32
Irr. complex chars.	192	0	48	0	0	0	240
Irr. rational chars.	8	20	24	14	4	2	72

Minimal presentations

Permutation degree:	$18$
Transitive degree:	$240$
Rank:	$3$
Inequivalent generating triples:	$52416$

Minimal degrees of faithful linear representations

	Over $\mathbb{C}$	Over $\mathbb{R}$	Over $\mathbb{Q}$
Irreducible	none	none	none
Arbitrary	not computed	not computed	not computed

Constructions

Presentation:	$\langle a, b, c \mid a^{4}=b^{8}=c^{30}=[a,b]=[b,c]=1, c^{a}=c^{13} \rangle$
Permutation group:	Degree $18$ $\langle(2,3,4,5)(6,7)(8,9)(10,12)(11,13)(14,15), (6,7), (6,7)(8,10,11,14,9,12,13,15) \!\cdots\! \rangle$
Matrix group:	$\left\langle \left(\begin{array}{rrrr} 0 & 1 & 2 & 3 \\ 2 & 2 & 0 & 0 \\ 1 & 0 & 2 & 0 \\ 4 & 1 & 2 & 0 \end{array}\right), \left(\begin{array}{rrrr} 4 & 0 & 0 & 0 \\ 0 & 4 & 0 & 0 \\ 0 & 0 & 4 & 0 \\ 0 & 0 & 0 & 4 \end{array}\right), \left(\begin{array}{rrrr} 1 & 0 & 0 & 0 \\ 4 & 4 & 3 & 1 \\ 1 & 1 & 0 & 4 \\ 3 & 0 & 3 & 3 \end{array}\right), \left(\begin{array}{rrrr} 1 & 0 & 0 & 0 \\ 4 & 1 & 4 & 1 \\ 2 & 1 & 1 & 3 \\ 4 & 1 & 2 & 2 \end{array}\right), \left(\begin{array}{rrrr} 4 & 2 & 4 & 1 \\ 4 & 2 & 2 & 3 \\ 2 & 3 & 2 & 4 \\ 0 & 0 & 0 & 1 \end{array}\right), \left(\begin{array}{rrrr} 3 & 1 & 2 & 3 \\ 1 & 0 & 4 & 1 \\ 3 & 1 & 0 & 3 \\ 3 & 2 & 4 & 4 \end{array}\right), \left(\begin{array}{rrrr} 1 & 0 & 0 & 0 \\ 4 & 3 & 0 & 1 \\ 3 & 1 & 2 & 2 \\ 0 & 2 & 1 & 1 \end{array}\right), \left(\begin{array}{rrrr} 4 & 0 & 0 & 0 \\ 2 & 4 & 2 & 3 \\ 1 & 3 & 4 & 4 \\ 2 & 3 & 1 & 2 \end{array}\right) \right\rangle \subseteq \GL_{4}(\F_{5})$
Direct product:	$C_2$ $\, \times\, $ $C_8$ $\, \times\, $ $C_3$ $\, \times\, $ $F_5$
Semidirect product:	$(D_5\times C_{24})$ $\,\rtimes\,$ $C_4$ (2)	$(D_5:C_{24})$ $\,\rtimes\,$ $C_4$ (2)	$(C_8\times D_5)$ $\,\rtimes\,$ $C_{12}$ (2)	$(D_5:C_8)$ $\,\rtimes\,$ $C_{12}$ (2)	all 22
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Non-split product:	$C_{60}$ . $C_4^2$	$(C_8\times D_{10})$ . $C_6$	$(D_{10}:C_8)$ . $C_6$	$(C_{12}\times F_5)$ . $C_4$ (2)	all 60
Aut. group:	$\Aut(C_5:C_{224})$	$\Aut(C_5:C_{288})$

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

Homology

Abelianization:	$C_{2} \times C_{4} \times C_{24} \simeq C_{2} \times C_{4} \times C_{8} \times C_{3}$
Schur multiplier:	$C_{2}^{2} \times C_{4}$
Commutator length:	$1$

Subgroups

There are 884 subgroups in 324 conjugacy classes, 184 normal (60 characteristic).

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

Special subgroups

Center:	$Z \simeq$ $C_2\times C_{24}$	$G/Z \simeq$ $F_5$
Commutator:	$G' \simeq$ $C_5$	$G/G' \simeq$ $C_2\times C_4\times C_{24}$
Frattini:	$\Phi \simeq$ $C_4$	$G/\Phi \simeq$ $D_{10}:C_{12}$
Fitting:	$\operatorname{Fit} \simeq$ $C_2\times C_{120}$	$G/\operatorname{Fit} \simeq$ $C_4$
Radical:	$R \simeq$ $C_{60}.C_4^2$	$G/R \simeq$ $C_1$
Socle:	$\operatorname{soc} \simeq$ $C_2\times C_{30}$	$G/\operatorname{soc} \simeq$ $C_4^2$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $C_2\times C_4\times C_8$
3-Sylow subgroup:	$P_{ 3 } \simeq$ $C_3$
5-Sylow subgroup:	$P_{ 5 } \simeq$ $C_5$

Hi

Subgroup diagram and profile

Classes of subgroups up to conjugation

No diagram available

Classes of subgroups up to automorphism

No diagram available

Normal subgroups

No diagram available

Normal subgroups up to automorphism

No diagram available

Classes of subgroups up to conjugation

Order 960: $C_{60}.C_4^2$

Order 480: $F_5\times C_{24}$ x 4, $D_{10}\times C_{24}$, $C_{30}:C_4^2$, $D_{10}:C_{24}$

Order 320: $C_{20}.C_4^2$

Order 240: $D_5\times C_{24}$ x 4, $C_{12}\times F_5$ x 4, $D_5:C_{24}$ x 4, $D_{10}:C_{12}$ x 2, $C_{10}:C_{24}$ x 2, $C_2\times C_{120}$, $C_{10}:C_{24}$, $C_{12}\times D_{10}$

Order 192: $C_2\times C_4\times C_{24}$

Order 160: $C_8\times F_5$ x 4, $C_{10}:C_4^2$, $D_{10}:C_8$, $C_8\times D_{10}$

Order 120: $C_6\times F_5$ x 12, $C_5:C_{24}$ x 4, $D_5\times C_{12}$ x 4, $C_{120}$ x 2, $C_5:C_{24}$ x 2, $C_6\times D_{10}$, $C_2\times C_{60}$, $C_{10}:C_{12}$

Order 96: $C_4\times C_{24}$ x 4, $C_2^2\times C_{24}$ x 2, $C_2\times C_4\times C_{12}$

Order 80: $C_8\times D_5$ x 4, $C_4\times F_5$ x 4, $D_5:C_8$ x 4, $C_2^2\times F_5$ x 2, $C_{10}:C_8$ x 2, $C_{10}:C_8$, $C_4\times D_{10}$, $C_2\times C_{40}$

Order 64: $C_2\times C_4\times C_8$

Order 60: $C_3\times F_5$ x 8, $C_3\times D_{10}$ x 6, $C_{60}$ x 2, $C_5:C_{12}$ x 2, $C_2\times C_{30}$

Order 48: $C_2\times C_{24}$ x 12, $C_4\times C_{12}$ x 4, $C_2^2\times C_{12}$ x 3

Order 40: $C_2\times F_5$ x 12, $C_4\times D_5$ x 4, $C_5:C_8$ x 4, $C_{40}$ x 2, $C_5:C_8$ x 2, $C_2\times C_{20}$, $C_{10}:C_4$, $C_2\times D_{10}$

Order 32: $C_4\times C_8$ x 4, $C_2^2\times C_8$ x 2, $C_2\times C_4^2$

Order 30: $C_3\times D_5$ x 4, $C_{30}$ x 3

Order 24: $C_2\times C_{12}$ x 18, $C_{24}$ x 8, $C_2^2\times C_6$

Order 20: $F_5$ x 8, $D_{10}$ x 6, $C_{20}$ x 2, $C_5:C_4$ x 2, $C_2\times C_{10}$

Order 16: $C_2\times C_8$ x 12, $C_4^2$ x 4, $C_2^2\times C_4$ x 3

Order 15: $C_{15}$

Order 12: $C_{12}$ x 12, $C_2\times C_6$ x 7

Order 10: $D_5$ x 4, $C_{10}$ x 3

Order 8: $C_2\times C_4$ x 18, $C_8$ x 8, $C_2^3$

Order 6: $C_6$ x 7

Order 5: $C_5$

Order 4: $C_4$ x 12, $C_2^2$ x 7

Order 3: $C_3$

Order 2: $C_2$ x 7

Order 1: $C_1$

Classes of subgroups up to automorphism

Order 960: $C_{60}.C_4^2$

Order 480: $D_{10}\times C_{24}$, $F_5\times C_{24}$, $C_{30}:C_4^2$, $D_{10}:C_{24}$

Order 320: $C_{20}.C_4^2$

Order 240: $D_5\times C_{24}$ x 2, $C_{12}\times F_5$ x 2, $D_5:C_{24}$ x 2, $C_2\times C_{120}$, $C_{10}:C_{24}$, $D_{10}:C_{12}$, $C_{12}\times D_{10}$, $C_{10}:C_{24}$

Order 192: $C_2\times C_4\times C_{24}$

Order 160: $C_8\times F_5$, $C_{10}:C_4^2$, $D_{10}:C_8$, $C_8\times D_{10}$

Order 120: $D_5\times C_{12}$ x 4, $C_6\times F_5$ x 2, $C_5:C_{24}$, $C_6\times D_{10}$, $C_{120}$, $C_2\times C_{60}$, $C_5:C_{24}$, $C_{10}:C_{12}$

Order 96: $C_2^2\times C_{24}$ x 2, $C_4\times C_{24}$, $C_2\times C_4\times C_{12}$

Order 80: $C_8\times D_5$ x 2, $C_4\times F_5$ x 2, $D_5:C_8$ x 2, $C_{10}:C_8$, $C_2^2\times F_5$, $C_4\times D_{10}$, $C_{10}:C_8$, $C_2\times C_{40}$

Order 64: $C_2\times C_4\times C_8$

Order 60: $C_3\times D_{10}$ x 3, $C_{60}$ x 2, $C_5:C_{12}$ x 2, $C_3\times F_5$, $C_2\times C_{30}$

Order 48: $C_2\times C_{24}$ x 7, $C_2^2\times C_{12}$ x 2, $C_4\times C_{12}$ x 2

Order 40: $C_4\times D_5$ x 4, $C_2\times F_5$ x 2, $C_2\times C_{20}$, $C_{10}:C_4$, $C_5:C_8$, $C_{40}$, $C_2\times D_{10}$, $C_5:C_8$

Order 32: $C_2^2\times C_8$ x 2, $C_4\times C_8$, $C_2\times C_4^2$

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

Order 24: $C_2\times C_{12}$ x 8, $C_{24}$ x 3, $C_2^2\times C_6$

Order 20: $D_{10}$ x 3, $C_{20}$ x 2, $C_5:C_4$ x 2, $C_2\times C_{10}$, $F_5$

Order 16: $C_2\times C_8$ x 7, $C_4^2$ x 2, $C_2^2\times C_4$ x 2

Order 15: $C_{15}$

Order 12: $C_{12}$ x 5, $C_2\times C_6$ x 4

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

Order 8: $C_2\times C_4$ x 8, $C_8$ x 3, $C_2^3$

Order 6: $C_6$ x 4

Order 5: $C_5$

Order 4: $C_4$ x 5, $C_2^2$ x 4

Order 3: $C_3$

Order 2: $C_2$ x 4

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 960: $C_{60}.C_4^2$ ($C_1$)

Order 480: $F_5\times C_{24}$ ($C_2$) x 4, $D_{10}\times C_{24}$ ($C_2$), $C_{30}:C_4^2$ ($C_2$), $D_{10}:C_{24}$ ($C_2$)

Order 320: $C_{20}.C_4^2$ ($C_3$)

Order 240: $D_5\times C_{24}$ ($C_2^2$) x 2, $D_5\times C_{24}$ ($C_4$) x 2, $D_{10}:C_{12}$ ($C_4$) x 2, $C_{10}:C_{24}$ ($C_4$) x 2, $C_{12}\times F_5$ ($C_2^2$) x 2, $C_{12}\times F_5$ ($C_4$) x 2, $D_5:C_{24}$ ($C_2^2$) x 2, $D_5:C_{24}$ ($C_4$) x 2, $C_2\times C_{120}$ ($C_4$), $C_{10}:C_{24}$ ($C_4$), $C_{12}\times D_{10}$ ($C_2^2$)

Order 160: $C_8\times F_5$ ($C_6$) x 4, $C_{10}:C_4^2$ ($C_6$), $D_{10}:C_8$ ($C_6$), $C_8\times D_{10}$ ($C_6$)

Order 120: $C_6\times F_5$ ($C_8$) x 8, $C_5:C_{24}$ ($C_2\times C_4$) x 4, $C_6\times F_5$ ($C_2\times C_4$) x 4, $D_5\times C_{12}$ ($C_2\times C_4$) x 3, $C_{120}$ ($C_2\times C_4$) x 2, $C_5:C_{24}$ ($C_2\times C_4$) x 2, $C_6\times D_{10}$ ($C_2\times C_4$), $C_2\times C_{60}$ ($C_2\times C_4$), $C_{10}:C_{12}$ ($C_2\times C_4$), $D_5\times C_{12}$ ($C_2^3$)

Order 80: $C_2^2\times F_5$ ($C_{12}$) x 2, $C_8\times D_5$ ($C_2\times C_6$) x 2, $C_8\times D_5$ ($C_{12}$) x 2, $C_{10}:C_8$ ($C_{12}$) x 2, $C_4\times F_5$ ($C_2\times C_6$) x 2, $C_4\times F_5$ ($C_{12}$) x 2, $D_5:C_8$ ($C_2\times C_6$) x 2, $D_5:C_8$ ($C_{12}$) x 2, $C_{10}:C_8$ ($C_{12}$), $C_4\times D_{10}$ ($C_2\times C_6$), $C_2\times C_{40}$ ($C_{12}$)

Order 60: $C_3\times F_5$ ($C_2\times C_8$) x 8, $C_3\times D_{10}$ ($C_2\times C_8$) x 4, $C_{60}$ ($C_4^2$), $C_{60}$ ($C_2^2\times C_4$), $C_5:C_{12}$ ($C_4^2$), $C_5:C_{12}$ ($C_2^2\times C_4$), $C_2\times C_{30}$ ($C_4^2$), $C_3\times D_{10}$ ($C_4^2$), $C_3\times D_{10}$ ($C_2^2\times C_4$)

Order 48: $C_2\times C_{24}$ ($F_5$)

Order 40: $C_2\times F_5$ ($C_{24}$) x 8, $C_5:C_8$ ($C_2\times C_{12}$) x 4, $C_2\times F_5$ ($C_2\times C_{12}$) x 4, $C_4\times D_5$ ($C_2\times C_{12}$) x 3, $C_{40}$ ($C_2\times C_{12}$) x 2, $C_5:C_8$ ($C_2\times C_{12}$) x 2, $C_2\times C_{20}$ ($C_2\times C_{12}$), $C_{10}:C_4$ ($C_2\times C_{12}$), $C_4\times D_5$ ($C_2^2\times C_6$), $C_2\times D_{10}$ ($C_2\times C_{12}$)

Order 30: $C_{30}$ ($C_4\times C_8$) x 2, $C_3\times D_5$ ($C_2^2\times C_8$) x 2, $C_3\times D_5$ ($C_4\times C_8$) x 2, $C_{30}$ ($C_2\times C_4^2$)

Order 24: $C_{24}$ ($C_2\times F_5$) x 2, $C_2\times C_{12}$ ($C_2\times F_5$)

Order 20: $F_5$ ($C_2\times C_{24}$) x 8, $D_{10}$ ($C_2\times C_{24}$) x 4, $C_2\times C_{10}$ ($C_4\times C_{12}$), $D_{10}$ ($C_2^2\times C_{12}$), $D_{10}$ ($C_4\times C_{12}$), $C_{20}$ ($C_2^2\times C_{12}$), $C_{20}$ ($C_4\times C_{12}$), $C_5:C_4$ ($C_2^2\times C_{12}$), $C_5:C_4$ ($C_4\times C_{12}$)

Order 16: $C_2\times C_8$ ($C_3\times F_5$)

Order 15: $C_{15}$ ($C_2\times C_4\times C_8$)

Order 12: $C_2\times C_6$ ($C_4\times F_5$), $C_{12}$ ($C_2^2\times F_5$), $C_{12}$ ($C_4\times F_5$)

Order 10: $C_{10}$ ($C_4\times C_{24}$) x 2, $D_5$ ($C_4\times C_{24}$) x 2, $D_5$ ($C_2^2\times C_{24}$) x 2, $C_{10}$ ($C_2\times C_4\times C_{12}$)

Order 8: $C_8$ ($C_6\times F_5$) x 2, $C_2\times C_4$ ($C_6\times F_5$)

Order 6: $C_6$ ($C_8\times F_5$) x 2, $C_6$ ($C_{10}:C_4^2$)

Order 5: $C_5$ ($C_2\times C_4\times C_{24}$)

Order 4: $C_2^2$ ($C_{12}\times F_5$), $C_4$ ($D_{10}:C_{12}$), $C_4$ ($C_{12}\times F_5$)

Order 3: $C_3$ ($C_{20}.C_4^2$)

Order 2: $C_2$ ($F_5\times C_{24}$) x 2, $C_2$ ($C_{30}:C_4^2$)

Order 1: $C_1$ ($C_{60}.C_4^2$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 960: $C_{60}.C_4^2$ ($C_1$)

Order 480: $D_{10}\times C_{24}$ ($C_2$), $F_5\times C_{24}$ ($C_2$), $C_{30}:C_4^2$ ($C_2$), $D_{10}:C_{24}$ ($C_2$)

Order 320: $C_{20}.C_4^2$ ($C_3$)

Order 240: $C_2\times C_{120}$ ($C_4$), $C_{10}:C_{24}$ ($C_4$), $D_5\times C_{24}$ ($C_2^2$), $D_5\times C_{24}$ ($C_4$), $D_{10}:C_{12}$ ($C_4$), $C_{12}\times D_{10}$ ($C_2^2$), $C_{10}:C_{24}$ ($C_4$), $C_{12}\times F_5$ ($C_2^2$), $C_{12}\times F_5$ ($C_4$), $D_5:C_{24}$ ($C_2^2$), $D_5:C_{24}$ ($C_4$)

Order 160: $C_8\times F_5$ ($C_6$), $C_{10}:C_4^2$ ($C_6$), $D_{10}:C_8$ ($C_6$), $C_8\times D_{10}$ ($C_6$)

Order 120: $D_5\times C_{12}$ ($C_2\times C_4$) x 3, $C_5:C_{24}$ ($C_2\times C_4$), $C_6\times D_{10}$ ($C_2\times C_4$), $C_6\times F_5$ ($C_2\times C_4$), $C_6\times F_5$ ($C_8$), $C_{120}$ ($C_2\times C_4$), $C_2\times C_{60}$ ($C_2\times C_4$), $C_5:C_{24}$ ($C_2\times C_4$), $C_{10}:C_{12}$ ($C_2\times C_4$), $D_5\times C_{12}$ ($C_2^3$)

Order 80: $C_{10}:C_8$ ($C_{12}$), $C_2^2\times F_5$ ($C_{12}$), $C_8\times D_5$ ($C_2\times C_6$), $C_8\times D_5$ ($C_{12}$), $C_4\times D_{10}$ ($C_2\times C_6$), $C_{10}:C_8$ ($C_{12}$), $C_4\times F_5$ ($C_2\times C_6$), $C_4\times F_5$ ($C_{12}$), $D_5:C_8$ ($C_2\times C_6$), $D_5:C_8$ ($C_{12}$), $C_2\times C_{40}$ ($C_{12}$)

Order 60: $C_3\times F_5$ ($C_2\times C_8$), $C_{60}$ ($C_4^2$), $C_{60}$ ($C_2^2\times C_4$), $C_5:C_{12}$ ($C_4^2$), $C_5:C_{12}$ ($C_2^2\times C_4$), $C_2\times C_{30}$ ($C_4^2$), $C_3\times D_{10}$ ($C_2\times C_8$), $C_3\times D_{10}$ ($C_4^2$), $C_3\times D_{10}$ ($C_2^2\times C_4$)

Order 48: $C_2\times C_{24}$ ($F_5$)

Order 40: $C_4\times D_5$ ($C_2\times C_{12}$) x 3, $C_2\times C_{20}$ ($C_2\times C_{12}$), $C_{10}:C_4$ ($C_2\times C_{12}$), $C_4\times D_5$ ($C_2^2\times C_6$), $C_5:C_8$ ($C_2\times C_{12}$), $C_{40}$ ($C_2\times C_{12}$), $C_2\times D_{10}$ ($C_2\times C_{12}$), $C_2\times F_5$ ($C_2\times C_{12}$), $C_2\times F_5$ ($C_{24}$), $C_5:C_8$ ($C_2\times C_{12}$)

Order 30: $C_{30}$ ($C_4\times C_8$), $C_{30}$ ($C_2\times C_4^2$), $C_3\times D_5$ ($C_2^2\times C_8$), $C_3\times D_5$ ($C_4\times C_8$)

Order 24: $C_2\times C_{12}$ ($C_2\times F_5$), $C_{24}$ ($C_2\times F_5$)

Order 20: $C_2\times C_{10}$ ($C_4\times C_{12}$), $D_{10}$ ($C_2^2\times C_{12}$), $D_{10}$ ($C_2\times C_{24}$), $D_{10}$ ($C_4\times C_{12}$), $F_5$ ($C_2\times C_{24}$), $C_{20}$ ($C_2^2\times C_{12}$), $C_{20}$ ($C_4\times C_{12}$), $C_5:C_4$ ($C_2^2\times C_{12}$), $C_5:C_4$ ($C_4\times C_{12}$)

Order 16: $C_2\times C_8$ ($C_3\times F_5$)

Order 15: $C_{15}$ ($C_2\times C_4\times C_8$)

Order 12: $C_2\times C_6$ ($C_4\times F_5$), $C_{12}$ ($C_2^2\times F_5$), $C_{12}$ ($C_4\times F_5$)

Order 10: $C_{10}$ ($C_4\times C_{24}$), $C_{10}$ ($C_2\times C_4\times C_{12}$), $D_5$ ($C_4\times C_{24}$), $D_5$ ($C_2^2\times C_{24}$)

Order 8: $C_2\times C_4$ ($C_6\times F_5$), $C_8$ ($C_6\times F_5$)

Order 6: $C_6$ ($C_8\times F_5$), $C_6$ ($C_{10}:C_4^2$)

Order 5: $C_5$ ($C_2\times C_4\times C_{24}$)

Order 4: $C_2^2$ ($C_{12}\times F_5$), $C_4$ ($D_{10}:C_{12}$), $C_4$ ($C_{12}\times F_5$)

Order 3: $C_3$ ($C_{20}.C_4^2$)

Order 2: $C_2$ ($F_5\times C_{24}$), $C_2$ ($C_{30}:C_4^2$)

Order 1: $C_1$ ($C_{60}.C_4^2$)

Series

Derived series

$C_{60}.C_4^2$

$\rhd$

$C_5$

$\rhd$

$C_1$

Chief series

$C_{60}.C_4^2$

$\rhd$

$F_5\times C_{24}$

$\rhd$

$C_{12}\times F_5$

$\rhd$

$D_5\times C_{12}$

$\rhd$

$C_3\times D_{10}$

$\rhd$

$C_{30}$

$\rhd$

$C_{15}$

$\rhd$

$C_5$

$\rhd$

$C_1$

Lower central series

$C_{60}.C_4^2$

$\rhd$

$C_5$

Upper central series

$C_1$

$\lhd$

$C_2\times C_{24}$

Character theory

Complex character table

See the $240 \times 240$ 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 $72 \times 72$ rational character table.