Abstract group 480.394: $C_{30}:Q_{16}$

• Label: 480.394
• Order: \( 2^{5} \cdot 3 \cdot 5 \)
• Exponent: \( 2^{3} \cdot 3 \cdot 5 \)
• Nilpotent: no
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{3} \)
• $\card{Z(G)}$: \( 2^{2} \)
• $\card{\Aut(G)}$: \( 2^{10} \cdot 3 \cdot 5 \)
• $\card{\mathrm{Out}(G)}$: \( 2^{7} \)
• Perm deg.: $26$
• Trans deg.: $480$
• Rank: $3$

Group information

Description:	$C_{30}:Q_{16}$
Order:	\(480\)\(\medspace = 2^{5} \cdot 3 \cdot 5 \)
Exponent:	\(120\)\(\medspace = 2^{3} \cdot 3 \cdot 5 \)
Automorphism group:	$C_{15}:(C_2^3.C_2^6.C_2)$, of order \(15360\)\(\medspace = 2^{10} \cdot 3 \cdot 5 \)
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	8	10	12	15	20	30	60
Elements	1	3	2	68	4	6	120	12	88	8	112	24	32	480
Conjugacy classes	1	3	1	6	2	3	4	6	6	2	12	6	8	60
Divisions	1	3	1	6	1	3	2	3	4	1	4	3	2	34
Autjugacy classes	1	2	1	4	1	2	1	2	3	1	3	2	2	25

Dimension	1	2	4	8	16
Irr. complex chars.	8	30	22	0	0	60
Irr. rational chars.	8	6	10	8	2	34

Minimal presentations

Permutation degree:	$26$
Transitive degree:	$480$
Rank:	$3$
Inequivalent generating triples:	$2016$

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:	$\langle a, b, c \mid b^{8}=c^{30}=1, a^{2}=b^{4}, b^{a}=b^{7}, c^{a}=c^{11}, c^{b}=c^{29} \rangle$
Permutation group:	Degree $26$ $\langle(1,2,5,8)(3,9,11,16)(4,12,14,7)(6,13,15,10)(21,22)(23,24)(25,26), (1,3,5,11) \!\cdots\! \rangle$
Matrix group:	$\left\langle \left(\begin{array}{rr} 1 & 14 \\ 0 & 1 \end{array}\right), \left(\begin{array}{rr} 31 & 20 \\ 10 & 11 \end{array}\right), \left(\begin{array}{rr} 49 & 18 \\ 40 & 21 \end{array}\right), \left(\begin{array}{rr} 29 & 0 \\ 0 & 29 \end{array}\right), \left(\begin{array}{rr} 59 & 10 \\ 55 & 39 \end{array}\right), \left(\begin{array}{rr} 36 & 35 \\ 35 & 1 \end{array}\right), \left(\begin{array}{rr} 69 & 0 \\ 0 & 69 \end{array}\right) \right\rangle \subseteq \GL_{2}(\Z/70\Z)$
Direct product:	$C_2$ $\, \times\, $ $(C_{15}:Q_{16})$
Semidirect product:	$C_{30}$ $\,\rtimes\,$ $Q_{16}$ (2)	$C_6$ $\,\rtimes\,$ $(C_5:Q_{16})$ (2)	$C_5$ $\,\rtimes\,$ $(C_6:Q_{16})$	$C_{15}$ $\,\rtimes\,$ $(C_2\times Q_{16})$	all 6
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Non-split product:	$C_{60}$ . $D_4$	$C_{60}$ . $C_2^3$	$(C_6:Q_8)$ . $D_5$	$(C_5:Q_8)$ . $D_6$ (2)	all 30

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

Homology

Abelianization:	$C_{2}^{3} $
Schur multiplier:	$C_{2}^{2}$
Commutator length:	$2$

Subgroups

There are 444 subgroups in 120 conjugacy classes, 52 normal (30 characteristic).

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

Special subgroups

Center:	$Z \simeq$ $C_2^2$	$G/Z \simeq$ $C_{15}:D_4$
Commutator:	$G' \simeq$ $C_{60}$	$G/G' \simeq$ $C_2^3$
Frattini:	$\Phi \simeq$ $C_4$	$G/\Phi \simeq$ $S_3\times D_{10}$
Fitting:	$\operatorname{Fit} \simeq$ $C_2\times C_{60}$	$G/\operatorname{Fit} \simeq$ $C_2^2$
Radical:	$R \simeq$ $C_{30}:Q_{16}$	$G/R \simeq$ $C_1$
Socle:	$\operatorname{soc} \simeq$ $C_2\times C_{30}$	$G/\operatorname{soc} \simeq$ $D_4$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $C_2\times Q_{16}$
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 480: $C_{30}:Q_{16}$

Order 240: $C_{15}:Q_{16}$ x 4, $C_{30}:C_8$, $C_{30}:Q_8$, $C_{30}:Q_8$

Order 160: $C_{10}:Q_{16}$

Order 120: $C_{15}:Q_8$ x 3, $C_{15}:Q_8$ x 3, $C_{15}:C_8$ x 2, $C_2\times C_{60}$, $C_6:C_{20}$, $C_{10}:C_{12}$

Order 96: $C_6:Q_{16}$

Order 80: $C_5:Q_{16}$ x 4, $C_{10}:C_8$, $Q_8\times C_{10}$, $C_{10}:Q_8$

Order 60: $C_{60}$ x 2, $C_5:C_{12}$ x 2, $C_3:C_{20}$ x 2, $C_2\times C_{30}$

Order 48: $C_3:Q_{16}$ x 4, $C_6:C_8$, $C_6\times Q_8$, $C_6:Q_8$

Order 40: $C_5:Q_8$ x 3, $C_5\times Q_8$ x 3, $C_2\times C_{20}$ x 2, $C_5:C_8$ x 2, $C_{10}:C_4$

Order 32: $C_2\times Q_{16}$

Order 30: $C_{30}$ x 3

Order 24: $C_3:Q_8$ x 3, $C_3\times Q_8$ x 3, $C_2\times C_{12}$ x 2, $C_3:C_8$ x 2, $C_6:C_4$

Order 20: $C_{20}$ x 4, $C_5:C_4$ x 2, $C_2\times C_{10}$

Order 16: $Q_{16}$ x 4, $C_2\times Q_8$ x 2, $C_2\times C_8$

Order 15: $C_{15}$

Order 12: $C_{12}$ x 4, $C_3:C_4$ x 2, $C_2\times C_6$

Order 10: $C_{10}$ x 3

Order 8: $Q_8$ x 6, $C_2\times C_4$ x 3, $C_8$ 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: $C_1$

Classes of subgroups up to automorphism

Order 480: $C_{30}:Q_{16}$

Order 240: $C_{30}:C_8$, $C_{15}:Q_{16}$, $C_{30}:Q_8$, $C_{30}:Q_8$

Order 160: $C_{10}:Q_{16}$

Order 120: $C_{15}:Q_8$ x 2, $C_{15}:Q_8$ x 2, $C_2\times C_{60}$, $C_{15}:C_8$, $C_6:C_{20}$, $C_{10}:C_{12}$

Order 96: $C_6:Q_{16}$

Order 80: $C_{10}:C_8$, $Q_8\times C_{10}$, $C_{10}:Q_8$, $C_5:Q_{16}$

Order 60: $C_{60}$ x 2, $C_5:C_{12}$, $C_2\times C_{30}$, $C_3:C_{20}$

Order 48: $C_6:C_8$, $C_6\times Q_8$, $C_6:Q_8$, $C_3:Q_{16}$

Order 40: $C_2\times C_{20}$ x 2, $C_5:Q_8$ x 2, $C_5\times Q_8$ x 2, $C_{10}:C_4$, $C_5:C_8$

Order 32: $C_2\times Q_{16}$

Order 30: $C_{30}$ x 2

Order 24: $C_2\times C_{12}$ x 2, $C_3:Q_8$ x 2, $C_3\times Q_8$ x 2, $C_6:C_4$, $C_3:C_8$

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

Order 16: $C_2\times Q_8$ x 2, $Q_{16}$, $C_2\times C_8$

Order 15: $C_{15}$

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

Order 10: $C_{10}$ x 2

Order 8: $Q_8$ x 4, $C_2\times C_4$ x 3, $C_8$

Order 6: $C_6$ x 2

Order 5: $C_5$

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

Order 3: $C_3$

Order 2: $C_2$ x 2

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 480: $C_{30}:Q_{16}$ ($C_1$)

Order 240: $C_{15}:Q_{16}$ ($C_2$) x 4, $C_{30}:C_8$ ($C_2$), $C_{30}:Q_8$ ($C_2$), $C_{30}:Q_8$ ($C_2$)

Order 120: $C_{15}:C_8$ ($C_2^2$) x 2, $C_{15}:Q_8$ ($C_2^2$) x 2, $C_{15}:Q_8$ ($C_2^2$) x 2, $C_2\times C_{60}$ ($C_2^2$)

Order 80: $C_{10}:Q_8$ ($S_3$)

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

Order 48: $C_6:Q_8$ ($D_5$)

Order 40: $C_5:Q_8$ ($D_6$) x 2, $C_2\times C_{20}$ ($D_6$)

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

Order 24: $C_3:Q_8$ ($D_{10}$) x 2, $C_2\times C_{12}$ ($D_{10}$)

Order 20: $C_2\times C_{10}$ ($C_3:D_4$), $C_{20}$ ($C_3:D_4$), $C_{20}$ ($C_2\times D_6$)

Order 15: $C_{15}$ ($C_2\times Q_{16}$)

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

Order 10: $C_{10}$ ($C_3:Q_{16}$) x 2, $C_{10}$ ($C_6:D_4$)

Order 8: $C_2\times C_4$ ($S_3\times D_5$)

Order 6: $C_6$ ($C_5:Q_{16}$) x 2, $C_6$ ($C_{10}:D_4$)

Order 5: $C_5$ ($C_6:Q_{16}$)

Order 4: $C_2^2$ ($C_{15}:D_4$), $C_4$ ($S_3\times D_{10}$), $C_4$ ($C_{15}:D_4$)

Order 3: $C_3$ ($C_{10}:Q_{16}$)

Order 2: $C_2$ ($C_{15}:Q_{16}$) x 2, $C_2$ ($C_{30}:D_4$)

Order 1: $C_1$ ($C_{30}:Q_{16}$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 480: $C_{30}:Q_{16}$ ($C_1$)

Order 240: $C_{30}:C_8$ ($C_2$), $C_{15}:Q_{16}$ ($C_2$), $C_{30}:Q_8$ ($C_2$), $C_{30}:Q_8$ ($C_2$)

Order 120: $C_2\times C_{60}$ ($C_2^2$), $C_{15}:C_8$ ($C_2^2$), $C_{15}:Q_8$ ($C_2^2$), $C_{15}:Q_8$ ($C_2^2$)

Order 80: $C_{10}:Q_8$ ($S_3$)

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

Order 48: $C_6:Q_8$ ($D_5$)

Order 40: $C_2\times C_{20}$ ($D_6$), $C_5:Q_8$ ($D_6$)

Order 30: $C_{30}$ ($Q_{16}$), $C_{30}$ ($C_2\times D_4$)

Order 24: $C_2\times C_{12}$ ($D_{10}$), $C_3:Q_8$ ($D_{10}$)

Order 20: $C_2\times C_{10}$ ($C_3:D_4$), $C_{20}$ ($C_3:D_4$), $C_{20}$ ($C_2\times D_6$)

Order 15: $C_{15}$ ($C_2\times Q_{16}$)

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

Order 10: $C_{10}$ ($C_6:D_4$), $C_{10}$ ($C_3:Q_{16}$)

Order 8: $C_2\times C_4$ ($S_3\times D_5$)

Order 6: $C_6$ ($C_{10}:D_4$), $C_6$ ($C_5:Q_{16}$)

Order 5: $C_5$ ($C_6:Q_{16}$)

Order 4: $C_2^2$ ($C_{15}:D_4$), $C_4$ ($S_3\times D_{10}$), $C_4$ ($C_{15}:D_4$)

Order 3: $C_3$ ($C_{10}:Q_{16}$)

Order 2: $C_2$ ($C_{15}:Q_{16}$), $C_2$ ($C_{30}:D_4$)

Order 1: $C_1$ ($C_{30}:Q_{16}$)

Series

Derived series

$C_{30}:Q_{16}$

$\rhd$

$C_{60}$

$\rhd$

$C_1$

Chief series

$C_{30}:Q_{16}$

$\rhd$

$C_{15}:Q_{16}$

$\rhd$

$C_{15}:Q_8$

$\rhd$

$C_{60}$

$\rhd$

$C_{30}$

$\rhd$

$C_{15}$

$\rhd$

$C_5$

$\rhd$

$C_1$

Lower central series

$C_{30}:Q_{16}$

$\rhd$

$C_{60}$

$\rhd$

$C_{30}$

$\rhd$

$C_{15}$

Upper central series

$C_1$

$\lhd$

$C_2^2$

$\lhd$

$C_2\times C_4$

Supergroups

This group is a maximal subgroup of 94 larger groups in the database.

This group is a maximal quotient of 24 larger groups in the database.

Character theory

Complex character table

See the $60 \times 60$ character table. Alternatively, you may search for characters of this group with desired properties.

Rational character table

See the $34 \times 34$ rational character table.