Abstract group 480.1: $C_3:C_{160}$

• Label: 480.1
• Order: \( 2^{5} \cdot 3 \cdot 5 \)
• Exponent: \( 2^{5} \cdot 3 \cdot 5 \)
• Nilpotent: no
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{5} \cdot 5 \)
• $\card{Z(G)}$: \( 2^{4} \cdot 5 \)
• $\card{\Aut(G)}$: \( 2^{7} \cdot 3 \)
• $\card{\mathrm{Out}(G)}$: \( 2^{6} \)
• Perm deg.: $40$
• Trans deg.: $480$
• Rank: $2$

Group information

Description:	$C_3:C_{160}$
Order:	\(480\)\(\medspace = 2^{5} \cdot 3 \cdot 5 \)
Exponent:	\(480\)\(\medspace = 2^{5} \cdot 3 \cdot 5 \)
Automorphism group:	$C_4\times C_8\times D_6$, of order \(384\)\(\medspace = 2^{7} \cdot 3 \)
Composition factors:	$C_2$ x 5, $C_3$, $C_5$
Derived length:	$2$

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

Group statistics

Order	1	2	3	4	5	6	8	10	12	15	16	20	24	30	32	40	48	60	80	120	160	240
Elements	1	1	2	2	4	2	4	4	4	8	8	8	8	8	48	16	16	16	32	32	192	64	480
Conjugacy classes	1	1	1	2	4	1	4	4	2	4	8	8	4	4	16	16	8	8	32	16	64	32	240
Divisions	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	22
Autjugacy classes	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	22

Dimension	1	2	4	8	16	32	64
Irr. complex chars.	160	80	0	0	0	0	0	240
Irr. rational chars.	2	3	4	5	4	2	2	22

Minimal presentations

Permutation degree:	$40$
Transitive degree:	$480$
Rank:	$2$
Inequivalent generating pairs:	$288$

Minimal degrees of faithful linear representations

	Over $\mathbb{C}$	Over $\mathbb{R}$	Over $\mathbb{Q}$
Irreducible	2	4	128
Arbitrary	2	4	22

Constructions

Presentation:	$\langle a, b \mid a^{32}=b^{15}=1, b^{a}=b^{11} \rangle$
Permutation group:	Degree $40$ $\langle(1,2,3,7,4,8,11,17,5,9,12,18,14,20,23,27,6,10,13,19,15,21,24,28,16,22,25,29,26,30,31,32) \!\cdots\! \rangle$
Matrix group:	$\left\langle \left(\begin{array}{rr} 102 & 0 \\ 0 & 102 \end{array}\right), \left(\begin{array}{rr} 0 & 102 \\ 1 & 0 \end{array}\right), \left(\begin{array}{rr} 16 & 0 \\ 0 & 226 \end{array}\right) \right\rangle \subseteq \GL_{2}(\F_{241})$
Direct product:	$C_5$ $\, \times\, $ $(C_3:C_{32})$
Semidirect product:	$C_{15}$ $\,\rtimes\,$ $C_{32}$	$C_3$ $\,\rtimes\,$ $C_{160}$			more information
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Non-split product:	$C_{80}$ . $S_3$	$C_{60}$ . $C_8$	$C_6$ . $C_{80}$	$C_{240}$ . $C_2$	all 16

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

Homology

Abelianization:	$C_{160} \simeq C_{32} \times C_{5}$
Schur multiplier:	$C_1$
Commutator length:	$1$

Subgroups

There are 28 subgroups in 24 conjugacy classes, 22 normal, and all normal subgroups are characteristic.

Characteristic subgroups are shown in this color.

Special subgroups

Center:	$Z \simeq$ $C_{80}$	$G/Z \simeq$ $S_3$
Commutator:	$G' \simeq$ $C_3$	$G/G' \simeq$ $C_{160}$
Frattini:	$\Phi \simeq$ $C_{16}$	$G/\Phi \simeq$ $C_5\times S_3$
Fitting:	$\operatorname{Fit} \simeq$ $C_{240}$	$G/\operatorname{Fit} \simeq$ $C_2$
Radical:	$R \simeq$ $C_3:C_{160}$	$G/R \simeq$ $C_1$
Socle:	$\operatorname{soc} \simeq$ $C_{30}$	$G/\operatorname{soc} \simeq$ $C_{16}$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $C_{32}$
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

Classes of subgroups up to automorphism

Normal subgroups

Normal subgroups up to automorphism

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Classes of subgroups up to conjugation

Order 480: $C_3:C_{160}$

Order 240: $C_{240}$

Order 160: $C_{160}$

Order 120: $C_{120}$

Order 96: $C_3:C_{32}$

Order 80: $C_{80}$

Order 60: $C_{60}$

Order 48: $C_{48}$

Order 40: $C_{40}$

Order 32: $C_{32}$

Order 30: $C_{30}$

Order 24: $C_{24}$

Order 20: $C_{20}$

Order 16: $C_{16}$

Order 15: $C_{15}$

Order 12: $C_{12}$

Order 10: $C_{10}$

Order 8: $C_8$

Order 6: $C_6$

Order 5: $C_5$

Order 4: $C_4$

Order 3: $C_3$

Order 2: $C_2$

Order 1: $C_1$

Classes of subgroups up to automorphism

Order 480: $C_3:C_{160}$

Order 240: $C_{240}$

Order 160: $C_{160}$

Order 120: $C_{120}$

Order 96: $C_3:C_{32}$

Order 80: $C_{80}$

Order 60: $C_{60}$

Order 48: $C_{48}$

Order 40: $C_{40}$

Order 32: $C_{32}$

Order 30: $C_{30}$

Order 24: $C_{24}$

Order 20: $C_{20}$

Order 16: $C_{16}$

Order 15: $C_{15}$

Order 12: $C_{12}$

Order 10: $C_{10}$

Order 8: $C_8$

Order 6: $C_6$

Order 5: $C_5$

Order 4: $C_4$

Order 3: $C_3$

Order 2: $C_2$

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 480: $C_3:C_{160}$ ($C_1$)

Order 240: $C_{240}$ ($C_2$)

Order 120: $C_{120}$ ($C_4$)

Order 96: $C_3:C_{32}$ ($C_5$)

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

Order 60: $C_{60}$ ($C_8$)

Order 48: $C_{48}$ ($C_{10}$)

Order 40: $C_{40}$ ($C_3:C_4$)

Order 30: $C_{30}$ ($C_{16}$)

Order 24: $C_{24}$ ($C_{20}$)

Order 20: $C_{20}$ ($C_3:C_8$)

Order 16: $C_{16}$ ($C_5\times S_3$)

Order 15: $C_{15}$ ($C_{32}$)

Order 12: $C_{12}$ ($C_{40}$)

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

Order 8: $C_8$ ($C_3:C_{20}$)

Order 6: $C_6$ ($C_{80}$)

Order 5: $C_5$ ($C_3:C_{32}$)

Order 4: $C_4$ ($C_3:C_{40}$)

Order 3: $C_3$ ($C_{160}$)

Order 2: $C_2$ ($C_3:C_{80}$)

Order 1: $C_1$ ($C_3:C_{160}$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 480: $C_3:C_{160}$ ($C_1$)

Order 240: $C_{240}$ ($C_2$)

Order 120: $C_{120}$ ($C_4$)

Order 96: $C_3:C_{32}$ ($C_5$)

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

Order 60: $C_{60}$ ($C_8$)

Order 48: $C_{48}$ ($C_{10}$)

Order 40: $C_{40}$ ($C_3:C_4$)

Order 30: $C_{30}$ ($C_{16}$)

Order 24: $C_{24}$ ($C_{20}$)

Order 20: $C_{20}$ ($C_3:C_8$)

Order 16: $C_{16}$ ($C_5\times S_3$)

Order 15: $C_{15}$ ($C_{32}$)

Order 12: $C_{12}$ ($C_{40}$)

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

Order 8: $C_8$ ($C_3:C_{20}$)

Order 6: $C_6$ ($C_{80}$)

Order 5: $C_5$ ($C_3:C_{32}$)

Order 4: $C_4$ ($C_3:C_{40}$)

Order 3: $C_3$ ($C_{160}$)

Order 2: $C_2$ ($C_3:C_{80}$)

Order 1: $C_1$ ($C_3:C_{160}$)

Series

Derived series

$C_3:C_{160}$

$\rhd$

$C_3$

$\rhd$

$C_1$

Chief series

$C_3:C_{160}$

$\rhd$

$C_{240}$

$\rhd$

$C_{120}$

$\rhd$

$C_{60}$

$\rhd$

$C_{30}$

$\rhd$

$C_{15}$

$\rhd$

$C_5$

$\rhd$

$C_1$

Lower central series

$C_3:C_{160}$

$\rhd$

$C_3$

Upper central series

$C_1$

$\lhd$

$C_{80}$

Supergroups

This group is a maximal subgroup of 22 larger groups in the database.

This group is a maximal quotient of 8 larger groups in the database.

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 $22 \times 22$ rational character table.