Abstract group 960.319: $C_{60}.D_8$

• Label: 960.319
• Order: \( 2^{6} \cdot 3 \cdot 5 \)
• Exponent: \( 2^{3} \cdot 3 \cdot 5 \)
• Nilpotent: no
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{3} \cdot 3 \)
• $\card{Z(G)}$: \( 2^{2} \cdot 3 \)
• $\card{\Aut(G)}$: \( 2^{11} \cdot 5 \)
• $\card{\mathrm{Out}(G)}$: \( 2^{7} \)
• Perm deg.: $24$
• Trans deg.: $960$
• Rank: $2$

Group information

Description:	$C_{60}.D_8$
Order:	\(960\)\(\medspace = 2^{6} \cdot 3 \cdot 5 \)
Exponent:	\(120\)\(\medspace = 2^{3} \cdot 3 \cdot 5 \)
Automorphism group:	$C_5:(C_2^4.C_2^6.C_2)$, of order \(10240\)\(\medspace = 2^{11} \cdot 5 \)
Composition factors:	$C_2$ x 6, $C_3$, $C_5$
Derived length:	$2$

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

Group statistics

Order	1	2	3	4	5	6	8	10	12	15	20	24	30	60
Elements	1	3	2	28	4	6	160	12	56	8	112	320	24	224	960
Conjugacy classes	1	3	2	7	2	6	8	6	14	4	20	16	12	40	141
Divisions	1	3	1	7	1	3	2	3	7	1	7	2	3	7	48
Autjugacy classes	1	3	1	6	1	3	2	3	6	1	6	2	3	6	44

Dimension	1	2	4	8	16	32
Irr. complex chars.	24	78	39	0	0	0	141
Irr. rational chars.	4	8	13	15	7	1	48

Minimal presentations

Permutation degree:	$24$
Transitive degree:	$960$
Rank:	$2$
Inequivalent generating pairs:	$24$

Minimal degrees of faithful linear representations

	Over $\mathbb{C}$	Over $\mathbb{R}$	Over $\mathbb{Q}$
Irreducible	none	none	none
Arbitrary	6	8	14

Constructions

Presentation:	$\langle a, b, c \mid b^{8}=c^{60}=1, a^{2}=b^{4}, b^{a}=b^{3}c^{15}, c^{a}=c^{31}, c^{b}=c^{19} \rangle$
Permutation group:	Degree $24$ $\langle(1,2,4,7,6,3,5,8)(9,10,11,14)(12,13,16,15)(18,19)(20,21), (1,3,6,2)(4,7,5,8) \!\cdots\! \rangle$
Matrix group:	$\left\langle \left(\begin{array}{rr} 64 & 35 \\ 35 & 29 \end{array}\right), \left(\begin{array}{rr} 71 & 0 \\ 0 & 71 \end{array}\right), \left(\begin{array}{rr} 16 & 0 \\ 0 & 16 \end{array}\right), \left(\begin{array}{rr} 94 & 90 \\ 45 & 4 \end{array}\right), \left(\begin{array}{rr} 76 & 0 \\ 0 & 76 \end{array}\right), \left(\begin{array}{rr} 1 & 21 \\ 0 & 1 \end{array}\right), \left(\begin{array}{rr} 62 & 77 \\ 50 & 78 \end{array}\right), \left(\begin{array}{rr} 94 & 100 \\ 10 & 74 \end{array}\right) \right\rangle \subseteq \GL_{2}(\Z/105\Z)$
Direct product:	$C_3$ $\, \times\, $ $(C_{20}.D_8)$
Semidirect product:	$C_{15}$ $\,\rtimes\,$ $(C_4.D_8)$	$C_5$ $\,\rtimes\,$ $(C_{12}.D_8)$			more information
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Non-split product:	$C_{60}$ . $D_8$	$C_{60}$ . $Q_{16}$	$C_{60}$ . $\SD_{16}$ (2)	$(C_{12}:Q_8)$ . $D_5$	all 52

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

Homology

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

Subgroups

There are 348 subgroups in 128 conjugacy classes, 70 normal (62 characteristic).

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

Special subgroups

Center:	$Z \simeq$ $C_2\times C_6$	$G/Z \simeq$ $C_{10}.D_4$
Commutator:	$G' \simeq$ $C_2\times C_{20}$	$G/G' \simeq$ $C_2\times C_{12}$
Frattini:	$\Phi \simeq$ $C_4^2$	$G/\Phi \simeq$ $C_3\times D_{10}$
Fitting:	$\operatorname{Fit} \simeq$ $C_{60}:Q_8$	$G/\operatorname{Fit} \simeq$ $C_2$
Radical:	$R \simeq$ $C_{60}.D_8$	$G/R \simeq$ $C_1$
Socle:	$\operatorname{soc} \simeq$ $C_2\times C_{30}$	$G/\operatorname{soc} \simeq$ $C_2^2:C_4$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $C_4.D_8$
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 Click on a subgroup in the diagram to see information about it.

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Classes of subgroups up to conjugation

Order 960: $C_{60}.D_8$

Order 480: $C_{20}:C_{24}$ x 2, $C_{60}:Q_8$

Order 320: $C_{20}.D_8$

Order 240: $C_4:C_{60}$ x 3, $C_{10}:C_{24}$ x 2, $C_4\times C_{60}$, $Q_8\times C_{30}$

Order 192: $C_{12}.D_8$

Order 160: $C_{20}:C_8$ x 2, $C_{20}:Q_8$

Order 120: $C_2\times C_{60}$ x 5, $Q_8\times C_{15}$ x 2, $C_5:C_{24}$ x 2

Order 96: $C_4:C_{24}$ x 2, $C_{12}:Q_8$

Order 80: $C_4:C_{20}$ x 3, $C_{10}:C_8$ x 2, $Q_8\times C_{10}$, $C_4\times C_{20}$

Order 64: $C_4.D_8$

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

Order 48: $C_4:C_{12}$ x 3, $C_2\times C_{24}$ x 2, $C_6\times Q_8$, $C_4\times C_{12}$

Order 40: $C_2\times C_{20}$ x 5, $C_5\times Q_8$ x 2, $C_5:C_8$ x 2

Order 32: $C_4:C_8$ x 2, $C_4:Q_8$

Order 30: $C_{30}$ x 3

Order 24: $C_2\times C_{12}$ x 5, $C_{24}$ x 2, $C_3\times Q_8$ x 2

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

Order 16: $C_4:C_4$ x 3, $C_2\times C_8$ x 2, $C_4^2$, $C_2\times Q_8$

Order 15: $C_{15}$

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

Order 10: $C_{10}$ x 3

Order 8: $C_2\times C_4$ x 5, $Q_8$ x 2, $C_8$ x 2

Order 6: $C_6$ x 3

Order 5: $C_5$

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

Order 3: $C_3$

Order 2: $C_2$ x 3

Order 1: $C_1$

Classes of subgroups up to automorphism

Order 960: $C_{60}.D_8$

Order 480: $C_{20}:C_{24}$ x 2, $C_{60}:Q_8$

Order 320: $C_{20}.D_8$

Order 240: $C_4:C_{60}$ x 2, $C_{10}:C_{24}$ x 2, $C_4\times C_{60}$, $Q_8\times C_{30}$

Order 192: $C_{12}.D_8$

Order 160: $C_{20}:C_8$ x 2, $C_{20}:Q_8$

Order 120: $C_2\times C_{60}$ x 4, $Q_8\times C_{15}$ x 2, $C_5:C_{24}$ x 2

Order 96: $C_4:C_{24}$ x 2, $C_{12}:Q_8$

Order 80: $C_{10}:C_8$ x 2, $C_4:C_{20}$ x 2, $Q_8\times C_{10}$, $C_4\times C_{20}$

Order 64: $C_4.D_8$

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

Order 48: $C_2\times C_{24}$ x 2, $C_4:C_{12}$ x 2, $C_6\times Q_8$, $C_4\times C_{12}$

Order 40: $C_2\times C_{20}$ x 4, $C_5\times Q_8$ x 2, $C_5:C_8$ x 2

Order 32: $C_4:C_8$ x 2, $C_4:Q_8$

Order 30: $C_{30}$ x 3

Order 24: $C_2\times C_{12}$ x 4, $C_{24}$ x 2, $C_3\times Q_8$ x 2

Order 20: $C_{20}$ x 6, $C_2\times C_{10}$

Order 16: $C_2\times C_8$ x 2, $C_4:C_4$ x 2, $C_4^2$, $C_2\times Q_8$

Order 15: $C_{15}$

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

Order 10: $C_{10}$ x 3

Order 8: $C_2\times C_4$ x 4, $Q_8$ x 2, $C_8$ x 2

Order 6: $C_6$ x 3

Order 5: $C_5$

Order 4: $C_4$ x 6, $C_2^2$

Order 3: $C_3$

Order 2: $C_2$ x 3

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 960: $C_{60}.D_8$ ($C_1$)

Order 480: $C_{20}:C_{24}$ ($C_2$) x 2, $C_{60}:Q_8$ ($C_2$)

Order 320: $C_{20}.D_8$ ($C_3$)

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

Order 160: $C_{20}:C_8$ ($C_6$) x 2, $C_{20}:Q_8$ ($C_6$)

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

Order 96: $C_{12}:Q_8$ ($D_5$)

Order 80: $C_4:C_{20}$ ($C_{12}$) x 2, $C_4\times C_{20}$ ($C_2\times C_6$)

Order 60: $C_{60}$ ($\SD_{16}$) x 2, $C_{60}$ ($Q_{16}$), $C_{60}$ ($D_8$), $C_2\times C_{30}$ ($C_2^2:C_4$)

Order 48: $C_4:C_{12}$ ($C_5:C_4$) x 2, $C_4\times C_{12}$ ($D_{10}$)

Order 40: $C_2\times C_{20}$ ($C_3\times D_4$) x 2, $C_2\times C_{20}$ ($C_2\times C_{12}$)

Order 32: $C_4:Q_8$ ($C_3\times D_5$)

Order 30: $C_{30}$ ($D_4:C_4$), $C_{30}$ ($C_4.D_4$), $C_{30}$ ($Q_8:C_4$)

Order 24: $C_2\times C_{12}$ ($C_5:D_4$) x 2, $C_2\times C_{12}$ ($C_{10}:C_4$)

Order 20: $C_{20}$ ($C_3\times \SD_{16}$) x 2, $C_2\times C_{10}$ ($C_2^2:C_{12}$), $C_{20}$ ($C_3\times Q_{16}$), $C_{20}$ ($C_3\times D_8$)

Order 16: $C_4:C_4$ ($C_5:C_{12}$) x 2, $C_4^2$ ($C_3\times D_{10}$)

Order 15: $C_{15}$ ($C_4.D_8$)

Order 12: $C_2\times C_6$ ($C_{10}.D_4$), $C_{12}$ ($C_5:Q_{16}$), $C_{12}$ ($Q_8:D_5$), $C_{12}$ ($C_5:\SD_{16}$), $C_{12}$ ($C_5:D_8$)

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

Order 8: $C_2\times C_4$ ($C_{15}:D_4$) x 2, $C_2\times C_4$ ($C_{10}:C_{12}$)

Order 6: $C_6$ ($C_{20}.D_4$), $C_6$ ($C_{10}.Q_{16}$), $C_6$ ($C_{10}.D_8$)

Order 5: $C_5$ ($C_{12}.D_8$)

Order 4: $C_2^2$ ($C_{30}.D_4$), $C_4$ ($C_{15}:Q_{16}$), $C_4$ ($C_{15}:\SD_{16}$), $C_4$ ($C_{15}:\SD_{16}$), $C_4$ ($C_{15}:D_8$)

Order 3: $C_3$ ($C_{20}.D_8$)

Order 2: $C_2$ ($C_{60}.D_4$), $C_2$ ($C_{60}.D_4$), $C_2$ ($C_{30}.D_8$)

Order 1: $C_1$ ($C_{60}.D_8$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 960: $C_{60}.D_8$ ($C_1$)

Order 480: $C_{20}:C_{24}$ ($C_2$) x 2, $C_{60}:Q_8$ ($C_2$)

Order 320: $C_{20}.D_8$ ($C_3$)

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

Order 160: $C_{20}:C_8$ ($C_6$) x 2, $C_{20}:Q_8$ ($C_6$)

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

Order 96: $C_{12}:Q_8$ ($D_5$)

Order 80: $C_4:C_{20}$ ($C_{12}$), $C_4\times C_{20}$ ($C_2\times C_6$)

Order 60: $C_{60}$ ($\SD_{16}$) x 2, $C_{60}$ ($Q_{16}$), $C_{60}$ ($D_8$), $C_2\times C_{30}$ ($C_2^2:C_4$)

Order 48: $C_4:C_{12}$ ($C_5:C_4$), $C_4\times C_{12}$ ($D_{10}$)

Order 40: $C_2\times C_{20}$ ($C_3\times D_4$) x 2, $C_2\times C_{20}$ ($C_2\times C_{12}$)

Order 32: $C_4:Q_8$ ($C_3\times D_5$)

Order 30: $C_{30}$ ($D_4:C_4$), $C_{30}$ ($C_4.D_4$), $C_{30}$ ($Q_8:C_4$)

Order 24: $C_2\times C_{12}$ ($C_5:D_4$) x 2, $C_2\times C_{12}$ ($C_{10}:C_4$)

Order 20: $C_{20}$ ($C_3\times \SD_{16}$) x 2, $C_2\times C_{10}$ ($C_2^2:C_{12}$), $C_{20}$ ($C_3\times Q_{16}$), $C_{20}$ ($C_3\times D_8$)

Order 16: $C_4:C_4$ ($C_5:C_{12}$), $C_4^2$ ($C_3\times D_{10}$)

Order 15: $C_{15}$ ($C_4.D_8$)

Order 12: $C_2\times C_6$ ($C_{10}.D_4$), $C_{12}$ ($C_5:Q_{16}$), $C_{12}$ ($Q_8:D_5$), $C_{12}$ ($C_5:\SD_{16}$), $C_{12}$ ($C_5:D_8$)

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

Order 8: $C_2\times C_4$ ($C_{15}:D_4$) x 2, $C_2\times C_4$ ($C_{10}:C_{12}$)

Order 6: $C_6$ ($C_{20}.D_4$), $C_6$ ($C_{10}.Q_{16}$), $C_6$ ($C_{10}.D_8$)

Order 5: $C_5$ ($C_{12}.D_8$)

Order 4: $C_2^2$ ($C_{30}.D_4$), $C_4$ ($C_{15}:Q_{16}$), $C_4$ ($C_{15}:\SD_{16}$), $C_4$ ($C_{15}:\SD_{16}$), $C_4$ ($C_{15}:D_8$)

Order 3: $C_3$ ($C_{20}.D_8$)

Order 2: $C_2$ ($C_{60}.D_4$), $C_2$ ($C_{60}.D_4$), $C_2$ ($C_{30}.D_8$)

Order 1: $C_1$ ($C_{60}.D_8$)

Series

Derived series

$C_{60}.D_8$

$\rhd$

$C_2\times C_{20}$

$\rhd$

$C_1$

Chief series

$C_{60}.D_8$

$\rhd$

$C_{60}:Q_8$

$\rhd$

$C_4\times C_{60}$

$\rhd$

$C_2\times C_{60}$

$\rhd$

$C_2\times C_{30}$

$\rhd$

$C_{30}$

$\rhd$

$C_{15}$

$\rhd$

$C_5$

$\rhd$

$C_1$

Lower central series

$C_{60}.D_8$

$\rhd$

$C_2\times C_{20}$

$\rhd$

$C_2\times C_{10}$

$\rhd$

$C_5$

Upper central series

$C_1$

$\lhd$

$C_2\times C_6$

$\lhd$

$C_4\times C_{12}$

$\lhd$

$C_{12}:Q_8$

Character theory

Complex character table

See the $141 \times 141$ 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 $48 \times 48$ rational character table.