Abstract group 480.923: $D_4\times C_{60}$

• Label: 480.923
• Order: \( 2^{5} \cdot 3 \cdot 5 \)
• Exponent: \( 2^{2} \cdot 3 \cdot 5 \)
• Nilpotent: yes
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{4} \cdot 3 \cdot 5 \)
• $\card{Z(G)}$: \( 2^{3} \cdot 3 \cdot 5 \)
• $\card{\Aut(G)}$: \( 2^{10} \)
• $\card{\mathrm{Out}(G)}$: \( 2^{8} \)
• Perm deg.: $16$
• Trans deg.: $240$
• Rank: $3$

Group information

Description:	$D_4\times C_{60}$
Order:	\(480\)\(\medspace = 2^{5} \cdot 3 \cdot 5 \)
Exponent:	\(60\)\(\medspace = 2^{2} \cdot 3 \cdot 5 \)
Automorphism group:	$C_2^5.C_2^5$, of order \(1024\)\(\medspace = 2^{10} \)
Composition factors:	$C_2$ x 5, $C_3$, $C_5$
Nilpotency class:	$2$
Derived length:	$2$

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

Group statistics

Order	1	2	3	4	5	6	10	12	15	20	30	60
Elements	1	11	2	20	4	22	44	40	8	80	88	160	480
Conjugacy classes	1	7	2	12	4	14	28	24	8	48	56	96	300
Divisions	1	7	1	7	1	7	7	7	1	7	7	7	60
Autjugacy classes	1	4	1	4	1	4	4	4	1	4	4	4	36

Dimension	1	2	4	8	16	32
Irr. complex chars.	240	60	0	0	0	0	300
Irr. rational chars.	8	14	15	15	7	1	60

Minimal presentations

Permutation degree:	$16$
Transitive degree:	$240$
Rank:	$3$
Inequivalent generating triples:	$33852$

Minimal degrees of faithful linear representations

	Over $\mathbb{C}$	Over $\mathbb{R}$	Over $\mathbb{Q}$
Irreducible	none	none	none
Arbitrary	3	4	10

Constructions

Presentation:	$\langle a, b, c \mid a^{2}=b^{4}=c^{60}=[a,b]=[b,c]=1, c^{a}=c^{31} \rangle$
Permutation group:	Degree $16$ $\langle(2,4)(5,6,7,8), (1,2)(3,4)(5,7)(6,8), (5,6,7,8), (9,11,10), (12,16,15,14,13), (1,3)(2,4), (5,7)(6,8)\rangle$
Matrix group:	$\left\langle \left(\begin{array}{rr} 19 & 0 \\ 10 & 29 \end{array}\right), \left(\begin{array}{rr} 7 & 0 \\ 0 & 7 \end{array}\right), \left(\begin{array}{rr} 16 & 15 \\ 15 & 1 \end{array}\right), \left(\begin{array}{rr} 11 & 0 \\ 0 & 11 \end{array}\right), \left(\begin{array}{rr} 1 & 6 \\ 0 & 1 \end{array}\right), \left(\begin{array}{rr} 13 & 10 \\ 10 & 23 \end{array}\right), \left(\begin{array}{rr} 19 & 0 \\ 0 & 19 \end{array}\right) \right\rangle \subseteq \GL_{2}(\Z/30\Z)$
Direct product:	$C_4$ $\, \times\, $ $C_3$ $\, \times\, $ $C_5$ $\, \times\, $ $D_4$
Semidirect product:	$C_4^2$ $\,\rtimes\,$ $C_{30}$	$(C_4\times C_{60})$ $\,\rtimes\,$ $C_2$	$(C_4:C_{60})$ $\,\rtimes\,$ $C_2$	$(C_4\times C_{20})$ $\,\rtimes\,$ $C_6$	all 24
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Non-split product:	$(D_4\times C_{30})$ . $C_2$	$(D_4\times C_{10})$ . $C_6$	$(C_6\times D_4)$ . $C_{10}$	$C_{30}$ . $(C_2\times D_4)$	all 28
Aut. group:	$\Aut(C_2\times C_{244})$	$\Aut(D_4\times C_{61})$

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

Homology

Abelianization:	$C_{2}^{2} \times C_{60} \simeq C_{2}^{2} \times C_{4} \times C_{3} \times C_{5}$
Schur multiplier:	$C_{2}^{3}$
Commutator length:	$1$

Subgroups

There are 248 subgroups in 188 conjugacy classes, 128 normal (48 characteristic).

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

Special subgroups

Center:	$Z \simeq$ $C_2\times C_{60}$	$G/Z \simeq$ $C_2^2$
Commutator:	$G' \simeq$ $C_2$	$G/G' \simeq$ $C_2^2\times C_{60}$
Frattini:	$\Phi \simeq$ $C_2^2$	$G/\Phi \simeq$ $C_2^2\times C_{30}$
Fitting:	$\operatorname{Fit} \simeq$ $D_4\times C_{60}$	$G/\operatorname{Fit} \simeq$ $C_1$
Radical:	$R \simeq$ $D_4\times C_{60}$	$G/R \simeq$ $C_1$
Socle:	$\operatorname{soc} \simeq$ $C_2\times C_{30}$	$G/\operatorname{soc} \simeq$ $C_2^3$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $C_4\times D_4$
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

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Classes of subgroups up to automorphism

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Normal subgroups

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Normal subgroups up to automorphism

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Subgroup information

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Classes of subgroups up to conjugation

Order 480: $D_4\times C_{60}$

Order 240: $C_2^2:C_{60}$ x 2, $C_2^2\times C_{60}$ x 2, $C_4:C_{60}$, $C_4\times C_{60}$, $D_4\times C_{30}$

Order 160: $D_4\times C_{20}$

Order 120: $C_2\times C_{60}$ x 9, $D_4\times C_{15}$ x 4, $C_2^2\times C_{30}$ x 2

Order 96: $D_4\times C_{12}$

Order 80: $C_2^2\times C_{20}$ x 2, $C_2^2:C_{20}$ x 2, $D_4\times C_{10}$, $C_4:C_{20}$, $C_4\times C_{20}$

Order 60: $C_2\times C_{30}$ x 9, $C_{60}$ x 7

Order 48: $C_2^2\times C_{12}$ x 2, $C_2^2:C_{12}$ x 2, $C_6\times D_4$, $C_4:C_{12}$, $C_4\times C_{12}$

Order 40: $C_2\times C_{20}$ x 9, $C_5\times D_4$ x 4, $C_2^2\times C_{10}$ x 2

Order 32: $C_4\times D_4$

Order 30: $C_{30}$ x 7

Order 24: $C_2\times C_{12}$ x 9, $C_3\times D_4$ x 4, $C_2^2\times C_6$ x 2

Order 20: $C_2\times C_{10}$ x 9, $C_{20}$ x 7

Order 16: $C_2^2:C_4$ x 2, $C_2^2\times C_4$ x 2, $C_4:C_4$, $C_4^2$, $C_2\times D_4$

Order 15: $C_{15}$

Order 12: $C_2\times C_6$ x 9, $C_{12}$ x 7

Order 10: $C_{10}$ x 7

Order 8: $C_2\times C_4$ x 9, $D_4$ x 4, $C_2^3$ x 2

Order 6: $C_6$ x 7

Order 5: $C_5$

Order 4: $C_2^2$ x 9, $C_4$ x 7

Order 3: $C_3$

Order 2: $C_2$ x 7

Order 1: $C_1$

Classes of subgroups up to automorphism

Order 480: $D_4\times C_{60}$

Order 240: $C_4:C_{60}$, $C_2^2:C_{60}$, $C_4\times C_{60}$, $D_4\times C_{30}$, $C_2^2\times C_{60}$

Order 160: $D_4\times C_{20}$

Order 120: $C_2\times C_{60}$ x 5, $C_2^2\times C_{30}$, $D_4\times C_{15}$

Order 96: $D_4\times C_{12}$

Order 80: $D_4\times C_{10}$, $C_2^2\times C_{20}$, $C_4:C_{20}$, $C_2^2:C_{20}$, $C_4\times C_{20}$

Order 60: $C_{60}$ x 4, $C_2\times C_{30}$ x 4

Order 48: $C_6\times D_4$, $C_2^2\times C_{12}$, $C_4:C_{12}$, $C_2^2:C_{12}$, $C_4\times C_{12}$

Order 40: $C_2\times C_{20}$ x 5, $C_2^2\times C_{10}$, $C_5\times D_4$

Order 32: $C_4\times D_4$

Order 30: $C_{30}$ x 4

Order 24: $C_2\times C_{12}$ x 5, $C_2^2\times C_6$, $C_3\times D_4$

Order 20: $C_2\times C_{10}$ x 4, $C_{20}$ x 4

Order 16: $C_4:C_4$, $C_2^2:C_4$, $C_4^2$, $C_2\times D_4$, $C_2^2\times C_4$

Order 15: $C_{15}$

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

Order 10: $C_{10}$ x 4

Order 8: $C_2\times C_4$ x 5, $C_2^3$, $D_4$

Order 6: $C_6$ x 4

Order 5: $C_5$

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

Order 3: $C_3$

Order 2: $C_2$ x 4

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 480: $D_4\times C_{60}$ ($C_1$)

Order 240: $C_2^2:C_{60}$ ($C_2$) x 2, $C_2^2\times C_{60}$ ($C_2$) x 2, $C_4:C_{60}$ ($C_2$), $C_4\times C_{60}$ ($C_2$), $D_4\times C_{30}$ ($C_2$)

Order 160: $D_4\times C_{20}$ ($C_3$)

Order 120: $C_2\times C_{60}$ ($C_2^2$) x 5, $D_4\times C_{15}$ ($C_4$) x 4, $C_2^2\times C_{30}$ ($C_2^2$) x 2

Order 96: $D_4\times C_{12}$ ($C_5$)

Order 80: $C_2^2\times C_{20}$ ($C_6$) x 2, $C_2^2:C_{20}$ ($C_6$) x 2, $D_4\times C_{10}$ ($C_6$), $C_4:C_{20}$ ($C_6$), $C_4\times C_{20}$ ($C_6$)

Order 60: $C_2\times C_{30}$ ($C_2\times C_4$) x 4, $C_{60}$ ($D_4$) x 2, $C_{60}$ ($C_2\times C_4$) x 2, $C_2\times C_{30}$ ($C_2^3$)

Order 48: $C_2^2\times C_{12}$ ($C_{10}$) x 2, $C_2^2:C_{12}$ ($C_{10}$) x 2, $C_6\times D_4$ ($C_{10}$), $C_4:C_{12}$ ($C_{10}$), $C_4\times C_{12}$ ($C_{10}$)

Order 40: $C_2\times C_{20}$ ($C_2\times C_6$) x 5, $C_5\times D_4$ ($C_{12}$) x 4, $C_2^2\times C_{10}$ ($C_2\times C_6$) x 2

Order 32: $C_4\times D_4$ ($C_{15}$)

Order 30: $C_{30}$ ($D_4:C_2$), $C_{30}$ ($C_2\times D_4$), $C_{30}$ ($C_2^2\times C_4$)

Order 24: $C_2\times C_{12}$ ($C_2\times C_{10}$) x 5, $C_3\times D_4$ ($C_{20}$) x 4, $C_2^2\times C_6$ ($C_2\times C_{10}$) x 2

Order 20: $C_2\times C_{10}$ ($C_2\times C_{12}$) x 4, $C_{20}$ ($C_2\times C_{12}$) x 2, $C_{20}$ ($C_3\times D_4$) x 2, $C_2\times C_{10}$ ($C_2^2\times C_6$)

Order 16: $C_2^2:C_4$ ($C_{30}$) x 2, $C_2^2\times C_4$ ($C_{30}$) x 2, $C_4:C_4$ ($C_{30}$), $C_4^2$ ($C_{30}$), $C_2\times D_4$ ($C_{30}$)

Order 15: $C_{15}$ ($C_4\times D_4$)

Order 12: $C_2\times C_6$ ($C_2\times C_{20}$) x 4, $C_{12}$ ($C_2\times C_{20}$) x 2, $C_{12}$ ($C_5\times D_4$) x 2, $C_2\times C_6$ ($C_2^2\times C_{10}$)

Order 10: $C_{10}$ ($D_4:C_6$), $C_{10}$ ($C_6\times D_4$), $C_{10}$ ($C_2^2\times C_{12}$)

Order 8: $C_2\times C_4$ ($C_2\times C_{30}$) x 5, $D_4$ ($C_{60}$) x 4, $C_2^3$ ($C_2\times C_{30}$) x 2

Order 6: $C_6$ ($D_4:C_{10}$), $C_6$ ($D_4\times C_{10}$), $C_6$ ($C_2^2\times C_{20}$)

Order 5: $C_5$ ($D_4\times C_{12}$)

Order 4: $C_2^2$ ($C_2\times C_{60}$) x 4, $C_4$ ($D_4\times C_{15}$) x 2, $C_4$ ($C_2\times C_{60}$) x 2, $C_2^2$ ($C_2^2\times C_{30}$)

Order 3: $C_3$ ($D_4\times C_{20}$)

Order 2: $C_2$ ($D_4:C_{30}$), $C_2$ ($D_4\times C_{30}$), $C_2$ ($C_2^2\times C_{60}$)

Order 1: $C_1$ ($D_4\times C_{60}$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 480: $D_4\times C_{60}$ ($C_1$)

Order 240: $C_4:C_{60}$ ($C_2$), $C_2^2:C_{60}$ ($C_2$), $C_4\times C_{60}$ ($C_2$), $D_4\times C_{30}$ ($C_2$), $C_2^2\times C_{60}$ ($C_2$)

Order 160: $D_4\times C_{20}$ ($C_3$)

Order 120: $C_2\times C_{60}$ ($C_2^2$) x 4, $C_2^2\times C_{30}$ ($C_2^2$), $D_4\times C_{15}$ ($C_4$)

Order 96: $D_4\times C_{12}$ ($C_5$)

Order 80: $D_4\times C_{10}$ ($C_6$), $C_2^2\times C_{20}$ ($C_6$), $C_4:C_{20}$ ($C_6$), $C_2^2:C_{20}$ ($C_6$), $C_4\times C_{20}$ ($C_6$)

Order 60: $C_{60}$ ($D_4$), $C_{60}$ ($C_2\times C_4$), $C_2\times C_{30}$ ($C_2^3$), $C_2\times C_{30}$ ($C_2\times C_4$)

Order 48: $C_6\times D_4$ ($C_{10}$), $C_2^2\times C_{12}$ ($C_{10}$), $C_4:C_{12}$ ($C_{10}$), $C_2^2:C_{12}$ ($C_{10}$), $C_4\times C_{12}$ ($C_{10}$)

Order 40: $C_2\times C_{20}$ ($C_2\times C_6$) x 4, $C_2^2\times C_{10}$ ($C_2\times C_6$), $C_5\times D_4$ ($C_{12}$)

Order 32: $C_4\times D_4$ ($C_{15}$)

Order 30: $C_{30}$ ($D_4:C_2$), $C_{30}$ ($C_2\times D_4$), $C_{30}$ ($C_2^2\times C_4$)

Order 24: $C_2\times C_{12}$ ($C_2\times C_{10}$) x 4, $C_2^2\times C_6$ ($C_2\times C_{10}$), $C_3\times D_4$ ($C_{20}$)

Order 20: $C_2\times C_{10}$ ($C_2\times C_{12}$), $C_2\times C_{10}$ ($C_2^2\times C_6$), $C_{20}$ ($C_2\times C_{12}$), $C_{20}$ ($C_3\times D_4$)

Order 16: $C_4:C_4$ ($C_{30}$), $C_2^2:C_4$ ($C_{30}$), $C_4^2$ ($C_{30}$), $C_2\times D_4$ ($C_{30}$), $C_2^2\times C_4$ ($C_{30}$)

Order 15: $C_{15}$ ($C_4\times D_4$)

Order 12: $C_2\times C_6$ ($C_2\times C_{20}$), $C_2\times C_6$ ($C_2^2\times C_{10}$), $C_{12}$ ($C_2\times C_{20}$), $C_{12}$ ($C_5\times D_4$)

Order 10: $C_{10}$ ($D_4:C_6$), $C_{10}$ ($C_6\times D_4$), $C_{10}$ ($C_2^2\times C_{12}$)

Order 8: $C_2\times C_4$ ($C_2\times C_{30}$) x 4, $C_2^3$ ($C_2\times C_{30}$), $D_4$ ($C_{60}$)

Order 6: $C_6$ ($D_4:C_{10}$), $C_6$ ($D_4\times C_{10}$), $C_6$ ($C_2^2\times C_{20}$)

Order 5: $C_5$ ($D_4\times C_{12}$)

Order 4: $C_2^2$ ($C_2^2\times C_{30}$), $C_2^2$ ($C_2\times C_{60}$), $C_4$ ($D_4\times C_{15}$), $C_4$ ($C_2\times C_{60}$)

Order 3: $C_3$ ($D_4\times C_{20}$)

Order 2: $C_2$ ($D_4:C_{30}$), $C_2$ ($D_4\times C_{30}$), $C_2$ ($C_2^2\times C_{60}$)

Order 1: $C_1$ ($D_4\times C_{60}$)

Series

Derived series

$D_4\times C_{60}$

$\rhd$

$C_2$

$\rhd$

$C_1$

Chief series

$D_4\times C_{60}$

$\rhd$

$C_2^2\times C_{60}$

$\rhd$

$C_2^2\times C_{30}$

$\rhd$

$C_2\times C_{30}$

$\rhd$

$C_{30}$

$\rhd$

$C_{15}$

$\rhd$

$C_5$

$\rhd$

$C_1$

Lower central series

$D_4\times C_{60}$

$\rhd$

$C_2$

$\rhd$

$C_1$

Upper central series

$C_1$

$\lhd$

$C_2\times C_{60}$

$\lhd$

$D_4\times C_{60}$

Supergroups

This group is a maximal subgroup of 178 larger groups in the database.

This group is a maximal quotient of 27 larger groups in the database.

Character theory

Complex character table

See the $300 \times 300$ 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 $60 \times 60$ rational character table.