Abstract group 480.767: $C_{60}:Q_8$

• Label: 480.767
• Order: \( 2^{5} \cdot 3 \cdot 5 \)
• Exponent: \( 2^{2} \cdot 3 \cdot 5 \)
• Nilpotent: no
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{3} \cdot 5 \)
• $\card{Z(G)}$: \( 2^{2} \cdot 5 \)
• $\card{\Aut(G)}$: \( 2^{10} \cdot 3 \)
• $\card{\mathrm{Out}(G)}$: \( 2^{7} \)
• Perm deg.: $20$
• Trans deg.: $480$
• Rank: $3$

Group information

Description:	$C_{60}:Q_8$
Order:	\(480\)\(\medspace = 2^{5} \cdot 3 \cdot 5 \)
Exponent:	\(60\)\(\medspace = 2^{2} \cdot 3 \cdot 5 \)
Automorphism group:	$C_3:(C_2^5.C_2^5)$, of order \(3072\)\(\medspace = 2^{10} \cdot 3 \)
Composition factors:	$C_2$ x 5, $C_3$, $C_5$
Derived length:	$2$

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

Group statistics

Order	1	2	3	4	5	6	10	12	15	20	30	60
Elements	1	3	2	60	4	6	12	24	8	240	24	96	480
Conjugacy classes	1	3	1	10	4	3	12	6	4	40	12	24	120
Divisions	1	3	1	10	1	3	3	4	1	10	3	4	44
Autjugacy classes	1	3	1	5	1	3	3	2	1	5	3	2	30

Dimension	1	2	4	8	16
Irr. complex chars.	40	70	10	0	0	120
Irr. rational chars.	8	10	12	10	4	44

Minimal presentations

Permutation degree:	$20$
Transitive degree:	$480$
Rank:	$3$
Inequivalent generating triples:	$10416$

Minimal degrees of faithful linear representations

	Over $\mathbb{C}$	Over $\mathbb{R}$	Over $\mathbb{Q}$
Irreducible	none	none	none
Arbitrary	4	6	12

Constructions

Presentation:	$\langle a, b, c \mid b^{12}=c^{20}=[a,c]=1, a^{2}=b^{6}, b^{a}=b^{11}, c^{b}=c^{11} \rangle$
Permutation group:	Degree $20$ $\langle(1,2,4,6)(3,7,8,5)(9,10)(11,12)(14,15), (9,11,10,12), (1,3,4,8)(2,5,6,7) \!\cdots\! \rangle$
Matrix group:	$\left\langle \left(\begin{array}{rr} 25 & 0 \\ 0 & 25 \end{array}\right), \left(\begin{array}{rr} 23 & 0 \\ 0 & 23 \end{array}\right), \left(\begin{array}{rr} 10 & 0 \\ 0 & 10 \end{array}\right), \left(\begin{array}{rr} 22 & 4 \\ 30 & 11 \end{array}\right), \left(\begin{array}{rr} 23 & 11 \\ 11 & 1 \end{array}\right), \left(\begin{array}{rr} 4 & 15 \\ 3 & 28 \end{array}\right), \left(\begin{array}{rr} 4 & 27 \\ 12 & 7 \end{array}\right) \right\rangle \subseteq \GL_{2}(\Z/33\Z)$
Direct product:	$C_5$ $\, \times\, $ $(C_{12}:Q_8)$
Semidirect product:	$C_{60}$ $\,\rtimes\,$ $Q_8$ (2)	$(C_3:C_{20})$ $\,\rtimes\,$ $Q_8$ (2)	$C_{20}$ $\,\rtimes\,$ $(C_3:Q_8)$ (2)	$C_{15}$ $\,\rtimes\,$ $(C_4:Q_8)$	all 8
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Non-split product:	$C_{10}$ . $(S_3\times Q_8)$	$C_{10}$ . $(S_3\times D_4)$	$(C_4:C_{20})$ . $S_3$	$(C_{30}:Q_8)$ . $C_2$ (2)	all 34

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

Homology

Abelianization:	$C_{2}^{2} \times C_{10} \simeq C_{2}^{3} \times C_{5}$
Schur multiplier:	$C_{2}^{2}$
Commutator length:	$1$

Subgroups

There are 276 subgroups in 136 conjugacy classes, 74 normal (38 characteristic).

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

Special subgroups

Center:	$Z \simeq$ $C_2\times C_{10}$	$G/Z \simeq$ $C_2\times D_6$
Commutator:	$G' \simeq$ $C_2\times C_6$	$G/G' \simeq$ $C_2^2\times C_{10}$
Frattini:	$\Phi \simeq$ $C_2^2$	$G/\Phi \simeq$ $C_{10}\times D_6$
Fitting:	$\operatorname{Fit} \simeq$ $C_4:C_{60}$	$G/\operatorname{Fit} \simeq$ $C_2$
Radical:	$R \simeq$ $C_{60}:Q_8$	$G/R \simeq$ $C_1$
Socle:	$\operatorname{soc} \simeq$ $C_2\times C_{30}$	$G/\operatorname{soc} \simeq$ $C_2^3$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $C_4:Q_8$
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

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Classes of subgroups up to automorphism

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Normal subgroups

Normal subgroups up to automorphism

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Subgroup information

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Classes of subgroups up to conjugation

Order 480: $C_{60}:Q_8$

Order 240: $C_{30}.D_4$ x 2, $C_{30}:Q_8$ x 2, $C_4:C_{60}$, $C_{12}:C_{20}$, $C_{12}:C_{20}$

Order 160: $C_{20}:Q_8$

Order 120: $C_6:C_{20}$ x 4, $C_{15}:Q_8$ x 4, $C_2\times C_{60}$ x 3

Order 96: $C_{12}:Q_8$

Order 80: $C_4:C_{20}$ x 4, $Q_8\times C_{10}$ x 2, $C_4\times C_{20}$

Order 60: $C_3:C_{20}$ x 6, $C_{60}$ x 4, $C_2\times C_{30}$

Order 48: $C_6:Q_8$ x 2, $C_6.D_4$ x 2, $C_4:C_{12}$, $C_{12}:C_4$, $C_{12}:C_4$

Order 40: $C_2\times C_{20}$ x 7, $C_5\times Q_8$ x 4

Order 32: $C_4:Q_8$

Order 30: $C_{30}$ x 3

Order 24: $C_6:C_4$ x 4, $C_3:Q_8$ x 4, $C_2\times C_{12}$ x 3

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

Order 16: $C_4:C_4$ x 4, $C_2\times Q_8$ x 2, $C_4^2$

Order 15: $C_{15}$

Order 12: $C_3:C_4$ x 6, $C_{12}$ x 4, $C_2\times C_6$

Order 10: $C_{10}$ x 3

Order 8: $C_2\times C_4$ x 7, $Q_8$ x 4

Order 6: $C_6$ x 3

Order 5: $C_5$

Order 4: $C_4$ x 10, $C_2^2$

Order 3: $C_3$

Order 2: $C_2$ x 3

Order 1: $C_1$

Classes of subgroups up to automorphism

Order 480: $C_{60}:Q_8$

Order 240: $C_4:C_{60}$, $C_{12}:C_{20}$, $C_{30}.D_4$, $C_{12}:C_{20}$, $C_{30}:Q_8$

Order 160: $C_{20}:Q_8$

Order 120: $C_6:C_{20}$ x 3, $C_2\times C_{60}$ x 2, $C_{15}:Q_8$

Order 96: $C_{12}:Q_8$

Order 80: $C_4:C_{20}$ x 3, $Q_8\times C_{10}$, $C_4\times C_{20}$

Order 60: $C_3:C_{20}$ x 3, $C_{60}$ x 2, $C_2\times C_{30}$

Order 48: $C_6:Q_8$, $C_4:C_{12}$, $C_{12}:C_4$, $C_6.D_4$, $C_{12}:C_4$

Order 40: $C_2\times C_{20}$ x 5, $C_5\times Q_8$

Order 32: $C_4:Q_8$

Order 30: $C_{30}$ x 3

Order 24: $C_6:C_4$ x 3, $C_2\times C_{12}$ x 2, $C_3:Q_8$

Order 20: $C_{20}$ x 5, $C_2\times C_{10}$

Order 16: $C_4:C_4$ x 3, $C_4^2$, $C_2\times Q_8$

Order 15: $C_{15}$

Order 12: $C_3:C_4$ x 3, $C_{12}$ x 2, $C_2\times C_6$

Order 10: $C_{10}$ x 3

Order 8: $C_2\times C_4$ x 5, $Q_8$

Order 6: $C_6$ x 3

Order 5: $C_5$

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

Order 3: $C_3$

Order 2: $C_2$ x 3

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 480: $C_{60}:Q_8$ ($C_1$)

Order 240: $C_{30}.D_4$ ($C_2$) x 2, $C_{30}:Q_8$ ($C_2$) x 2, $C_4:C_{60}$ ($C_2$), $C_{12}:C_{20}$ ($C_2$), $C_{12}:C_{20}$ ($C_2$)

Order 120: $C_6:C_{20}$ ($C_2^2$) x 4, $C_2\times C_{60}$ ($C_2^2$) x 3

Order 96: $C_{12}:Q_8$ ($C_5$)

Order 80: $C_4:C_{20}$ ($S_3$)

Order 60: $C_{60}$ ($Q_8$) x 2, $C_3:C_{20}$ ($Q_8$) x 2, $C_3:C_{20}$ ($D_4$) x 2, $C_2\times C_{30}$ ($C_2^3$)

Order 48: $C_6:Q_8$ ($C_{10}$) x 2, $C_6.D_4$ ($C_{10}$) x 2, $C_4:C_{12}$ ($C_{10}$), $C_{12}:C_4$ ($C_{10}$), $C_{12}:C_4$ ($C_{10}$)

Order 40: $C_2\times C_{20}$ ($D_6$) x 3

Order 30: $C_{30}$ ($C_2\times Q_8$) x 2, $C_{30}$ ($C_2\times D_4$)

Order 24: $C_6:C_4$ ($C_2\times C_{10}$) x 4, $C_2\times C_{12}$ ($C_2\times C_{10}$) x 3

Order 20: $C_{20}$ ($C_3:Q_8$) x 2, $C_2\times C_{10}$ ($C_2\times D_6$)

Order 16: $C_4:C_4$ ($C_5\times S_3$)

Order 15: $C_{15}$ ($C_4:Q_8$)

Order 12: $C_{12}$ ($C_5\times Q_8$) x 2, $C_3:C_4$ ($C_5\times Q_8$) x 2, $C_3:C_4$ ($C_5\times D_4$) x 2, $C_2\times C_6$ ($C_2^2\times C_{10}$)

Order 10: $C_{10}$ ($S_3\times Q_8$), $C_{10}$ ($S_3\times D_4$), $C_{10}$ ($C_6:Q_8$)

Order 8: $C_2\times C_4$ ($S_3\times C_{10}$) x 3

Order 6: $C_6$ ($Q_8\times C_{10}$) x 2, $C_6$ ($D_4\times C_{10}$)

Order 5: $C_5$ ($C_{12}:Q_8$)

Order 4: $C_4$ ($C_{15}:Q_8$) x 2, $C_2^2$ ($C_{10}\times D_6$)

Order 3: $C_3$ ($C_{20}:Q_8$)

Order 2: $C_2$ ($C_{20}.D_6$), $C_2$ ($C_{20}:D_6$), $C_2$ ($C_{30}:Q_8$)

Order 1: $C_1$ ($C_{60}:Q_8$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 480: $C_{60}:Q_8$ ($C_1$)

Order 240: $C_4:C_{60}$ ($C_2$), $C_{12}:C_{20}$ ($C_2$), $C_{30}.D_4$ ($C_2$), $C_{12}:C_{20}$ ($C_2$), $C_{30}:Q_8$ ($C_2$)

Order 120: $C_6:C_{20}$ ($C_2^2$) x 3, $C_2\times C_{60}$ ($C_2^2$) x 2

Order 96: $C_{12}:Q_8$ ($C_5$)

Order 80: $C_4:C_{20}$ ($S_3$)

Order 60: $C_{60}$ ($Q_8$), $C_2\times C_{30}$ ($C_2^3$), $C_3:C_{20}$ ($Q_8$), $C_3:C_{20}$ ($D_4$)

Order 48: $C_6:Q_8$ ($C_{10}$), $C_4:C_{12}$ ($C_{10}$), $C_{12}:C_4$ ($C_{10}$), $C_6.D_4$ ($C_{10}$), $C_{12}:C_4$ ($C_{10}$)

Order 40: $C_2\times C_{20}$ ($D_6$) x 2

Order 30: $C_{30}$ ($C_2\times Q_8$) x 2, $C_{30}$ ($C_2\times D_4$)

Order 24: $C_6:C_4$ ($C_2\times C_{10}$) x 3, $C_2\times C_{12}$ ($C_2\times C_{10}$) x 2

Order 20: $C_2\times C_{10}$ ($C_2\times D_6$), $C_{20}$ ($C_3:Q_8$)

Order 16: $C_4:C_4$ ($C_5\times S_3$)

Order 15: $C_{15}$ ($C_4:Q_8$)

Order 12: $C_2\times C_6$ ($C_2^2\times C_{10}$), $C_{12}$ ($C_5\times Q_8$), $C_3:C_4$ ($C_5\times Q_8$), $C_3:C_4$ ($C_5\times D_4$)

Order 10: $C_{10}$ ($S_3\times Q_8$), $C_{10}$ ($S_3\times D_4$), $C_{10}$ ($C_6:Q_8$)

Order 8: $C_2\times C_4$ ($S_3\times C_{10}$) x 2

Order 6: $C_6$ ($Q_8\times C_{10}$) x 2, $C_6$ ($D_4\times C_{10}$)

Order 5: $C_5$ ($C_{12}:Q_8$)

Order 4: $C_2^2$ ($C_{10}\times D_6$), $C_4$ ($C_{15}:Q_8$)

Order 3: $C_3$ ($C_{20}:Q_8$)

Order 2: $C_2$ ($C_{20}.D_6$), $C_2$ ($C_{20}:D_6$), $C_2$ ($C_{30}:Q_8$)

Order 1: $C_1$ ($C_{60}:Q_8$)

Series

Derived series

$C_{60}:Q_8$

$\rhd$

$C_2\times C_6$

$\rhd$

$C_1$

Chief series

$C_{60}:Q_8$

$\rhd$

$C_{12}:C_{20}$

$\rhd$

$C_2\times C_{60}$

$\rhd$

$C_{60}$

$\rhd$

$C_{20}$

$\rhd$

$C_{10}$

$\rhd$

$C_5$

$\rhd$

$C_1$

Lower central series

$C_{60}:Q_8$

$\rhd$

$C_2\times C_6$

$\rhd$

$C_3$

Upper central series

$C_1$

$\lhd$

$C_2\times C_{10}$

$\lhd$

$C_4:C_{20}$

Supergroups

This group is a maximal subgroup of 133 larger groups in the database.

This group is a maximal quotient of 23 larger groups in the database.

Character theory

Complex character table

See the $120 \times 120$ character table. Alternatively, you may search for characters of this group with desired properties.

Rational character table

See the $44 \times 44$ rational character table.