Abstract group 720.256: $C_5\times C_3^2:C_4^2$

• Label: 720.256
• Order: \( 2^{4} \cdot 3^{2} \cdot 5 \)
• Exponent: \( 2^{2} \cdot 3 \cdot 5 \)
• Nilpotent: no
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{4} \cdot 5 \)
• $\card{Z(G)}$: \( 2^{2} \cdot 5 \)
• $\card{\Aut(G)}$: \( 2^{9} \cdot 3^{2} \)
• $\card{\mathrm{Out}(G)}$: \( 2^{7} \)
• Perm deg.: $19$
• Trans deg.: $240$
• Rank: $2$

Group information

Description:	$C_5\times C_3^2:C_4^2$
Order:	\(720\)\(\medspace = 2^{4} \cdot 3^{2} \cdot 5 \)
Exponent:	\(60\)\(\medspace = 2^{2} \cdot 3 \cdot 5 \)
Automorphism group:	$C_3^2.C_2^6.C_2^3$, of order \(4608\)\(\medspace = 2^{9} \cdot 3^{2} \)
Composition factors:	$C_2$ x 4, $C_3$ x 2, $C_5$
Derived length:	$2$

This group is nonabelian, supersolvable (hence solvable and monomial), metabelian, and an A-group.

Group statistics

Order	1	2	3	4	5	6	10	12	15	20	30	60
Elements	1	3	8	60	4	24	12	48	32	240	96	192	720
Conjugacy classes	1	3	3	12	4	9	12	8	12	48	36	32	180
Divisions	1	3	3	6	1	9	3	4	3	6	9	4	52
Autjugacy classes	1	2	2	2	1	5	2	1	2	2	5	1	26

Dimension	1	2	4	8	16
Irr. complex chars.	80	80	20	0	0	180
Irr. rational chars.	4	14	12	14	8	52

Minimal presentations

Permutation degree:	$19$
Transitive degree:	$240$
Rank:	$2$
Inequivalent generating pairs:	$18$

Minimal degrees of faithful linear representations

	Over $\mathbb{C}$	Over $\mathbb{R}$	Over $\mathbb{Q}$
Irreducible	none	none	none
Arbitrary	4	6	10

Constructions

Presentation:	$\langle a, b, c \mid a^{4}=b^{12}=c^{15}=[a,c]=1, b^{a}=b^{5}, c^{b}=c^{11} \rangle$
Permutation group:	Degree $19$ $\langle(4,5)(6,7)(14,15)(16,17,18,19), (2,3)(4,6,5,7), (8,9,10,11,12), (4,5)(6,7), (16,18)(17,19), (1,2,3), (13,14,15)\rangle$
Matrix group:	$\left\langle \left(\begin{array}{rr} 23 & 0 \\ 0 & 23 \end{array}\right), \left(\begin{array}{rr} 37 & 20 \\ 36 & 29 \end{array}\right), \left(\begin{array}{rr} 9 & 0 \\ 0 & 9 \end{array}\right), \left(\begin{array}{rr} 15 & 5 \\ 31 & 28 \end{array}\right), \left(\begin{array}{rr} 21 & 33 \\ 22 & 43 \end{array}\right), \left(\begin{array}{rr} 21 & 0 \\ 0 & 21 \end{array}\right), \left(\begin{array}{rr} 23 & 33 \\ 11 & 12 \end{array}\right) \right\rangle \subseteq \GL_{2}(\Z/44\Z)$
Direct product:	$C_5$ $\, \times\, $ $(C_3:C_4)$ ${}^2$
Semidirect product:	$(C_3:C_{60})$ $\,\rtimes\,$ $C_4$ (4)	$(C_3:C_{12})$ $\,\rtimes\,$ $C_{20}$ (4)	$C_{15}$ $\,\rtimes\,$ $(C_{12}:C_4)$ (2)	$C_3$ $\,\rtimes\,$ $(C_{12}:C_{20})$ (2)	all 8
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Non-split product:	$(C_6:C_{20})$ . $S_3$ (2)	$C_{30}$ . $(C_4\times S_3)$ (4)	$C_6$ . $(S_3\times C_{20})$ (4)	$(C_{30}.D_6)$ . $C_2$	all 22

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

Homology

Abelianization:	$C_{4} \times C_{20} \simeq C_{4}^{2} \times C_{5}$
Schur multiplier:	$C_{4}$
Commutator length:	$1$

Subgroups

There are 320 subgroups in 136 conjugacy classes, 72 normal (16 characteristic).

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

Special subgroups

Center:	$Z \simeq$ $C_2\times C_{10}$	$G/Z \simeq$ $S_3^2$
Commutator:	$G' \simeq$ $C_3^2$	$G/G' \simeq$ $C_4\times C_{20}$
Frattini:	$\Phi \simeq$ $C_2^2$	$G/\Phi \simeq$ $C_5\times S_3^2$
Fitting:	$\operatorname{Fit} \simeq$ $C_6\times C_{30}$	$G/\operatorname{Fit} \simeq$ $C_2^2$
Radical:	$R \simeq$ $C_5\times C_3^2:C_4^2$	$G/R \simeq$ $C_1$
Socle:	$\operatorname{soc} \simeq$ $C_6\times C_{30}$	$G/\operatorname{soc} \simeq$ $C_2^2$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $C_4^2$
3-Sylow subgroup:	$P_{ 3 } \simeq$ $C_3^2$
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 720: $C_5\times C_3^2:C_4^2$

Order 360: $C_6:C_{60}$ x 2, $C_{30}.D_6$

Order 240: $C_{12}:C_{20}$ x 2

Order 180: $C_3:C_{60}$ x 4, $C_3^2:C_{20}$ x 2, $C_6\times C_{30}$

Order 144: $C_3^2:C_4^2$

Order 120: $C_6:C_{20}$ x 5, $C_2\times C_{60}$ x 2

Order 90: $C_3\times C_{30}$ x 3

Order 80: $C_4\times C_{20}$

Order 72: $C_6:C_{12}$ x 2, $C_6.D_6$

Order 60: $C_3:C_{20}$ x 10, $C_{60}$ x 4, $C_2\times C_{30}$ x 3

Order 48: $C_{12}:C_4$ x 2

Order 45: $C_3\times C_{15}$

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

Order 36: $C_3:C_{12}$ x 4, $C_3^2:C_4$ x 2, $C_6^2$

Order 30: $C_{30}$ x 9

Order 24: $C_6:C_4$ x 5, $C_2\times C_{12}$ x 2

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

Order 18: $C_3\times C_6$ x 3

Order 16: $C_4^2$

Order 15: $C_{15}$ x 3

Order 12: $C_3:C_4$ x 10, $C_{12}$ x 4, $C_2\times C_6$ x 3

Order 10: $C_{10}$ x 3

Order 9: $C_3^2$

Order 8: $C_2\times C_4$ x 3

Order 6: $C_6$ x 9

Order 5: $C_5$

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

Order 3: $C_3$ x 3

Order 2: $C_2$ x 3

Order 1: $C_1$

Classes of subgroups up to automorphism

Order 720: $C_5\times C_3^2:C_4^2$

Order 360: $C_6:C_{60}$, $C_{30}.D_6$

Order 240: $C_{12}:C_{20}$

Order 180: $C_6\times C_{30}$, $C_3^2:C_{20}$, $C_3:C_{60}$

Order 144: $C_3^2:C_4^2$

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

Order 90: $C_3\times C_{30}$ x 2

Order 80: $C_4\times C_{20}$

Order 72: $C_6.D_6$, $C_6:C_{12}$

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

Order 48: $C_{12}:C_4$

Order 45: $C_3\times C_{15}$

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

Order 36: $C_3^2:C_4$, $C_3:C_{12}$, $C_6^2$

Order 30: $C_{30}$ x 5

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

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

Order 18: $C_3\times C_6$ x 2

Order 16: $C_4^2$

Order 15: $C_{15}$ x 2

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

Order 10: $C_{10}$ x 2

Order 9: $C_3^2$

Order 8: $C_2\times C_4$ x 2

Order 6: $C_6$ x 5

Order 5: $C_5$

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

Order 3: $C_3$ x 2

Order 2: $C_2$ x 2

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 720: $C_5\times C_3^2:C_4^2$ ($C_1$)

Order 360: $C_6:C_{60}$ ($C_2$) x 2, $C_{30}.D_6$ ($C_2$)

Order 180: $C_3:C_{60}$ ($C_4$) x 4, $C_3^2:C_{20}$ ($C_4$) x 2, $C_6\times C_{30}$ ($C_2^2$)

Order 144: $C_3^2:C_4^2$ ($C_5$)

Order 120: $C_6:C_{20}$ ($S_3$) x 2

Order 90: $C_3\times C_{30}$ ($C_2\times C_4$) x 3

Order 72: $C_6:C_{12}$ ($C_{10}$) x 2, $C_6.D_6$ ($C_{10}$)

Order 60: $C_3:C_{20}$ ($C_3:C_4$) x 4, $C_2\times C_{30}$ ($D_6$) x 2

Order 45: $C_3\times C_{15}$ ($C_4^2$)

Order 36: $C_3:C_{12}$ ($C_{20}$) x 4, $C_3^2:C_4$ ($C_{20}$) x 2, $C_6^2$ ($C_2\times C_{10}$)

Order 30: $C_{30}$ ($C_4\times S_3$) x 4, $C_{30}$ ($C_6:C_4$) x 2

Order 24: $C_6:C_4$ ($C_5\times S_3$) x 2

Order 20: $C_2\times C_{10}$ ($S_3^2$)

Order 18: $C_3\times C_6$ ($C_2\times C_{20}$) x 3

Order 15: $C_{15}$ ($C_{12}:C_4$) x 2

Order 12: $C_3:C_4$ ($C_3:C_{20}$) x 4, $C_2\times C_6$ ($S_3\times C_{10}$) x 2

Order 10: $C_{10}$ ($C_6.D_6$) x 2, $C_{10}$ ($C_6.D_6$)

Order 9: $C_3^2$ ($C_4\times C_{20}$)

Order 6: $C_6$ ($S_3\times C_{20}$) x 4, $C_6$ ($C_6:C_{20}$) x 2

Order 5: $C_5$ ($C_3^2:C_4^2$)

Order 4: $C_2^2$ ($C_5\times S_3^2$)

Order 3: $C_3$ ($C_{12}:C_{20}$) x 2

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

Order 1: $C_1$ ($C_5\times C_3^2:C_4^2$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 720: $C_5\times C_3^2:C_4^2$ ($C_1$)

Order 360: $C_6:C_{60}$ ($C_2$), $C_{30}.D_6$ ($C_2$)

Order 180: $C_6\times C_{30}$ ($C_2^2$), $C_3^2:C_{20}$ ($C_4$), $C_3:C_{60}$ ($C_4$)

Order 144: $C_3^2:C_4^2$ ($C_5$)

Order 120: $C_6:C_{20}$ ($S_3$)

Order 90: $C_3\times C_{30}$ ($C_2\times C_4$) x 2

Order 72: $C_6.D_6$ ($C_{10}$), $C_6:C_{12}$ ($C_{10}$)

Order 60: $C_2\times C_{30}$ ($D_6$), $C_3:C_{20}$ ($C_3:C_4$)

Order 45: $C_3\times C_{15}$ ($C_4^2$)

Order 36: $C_3^2:C_4$ ($C_{20}$), $C_3:C_{12}$ ($C_{20}$), $C_6^2$ ($C_2\times C_{10}$)

Order 30: $C_{30}$ ($C_4\times S_3$) x 2, $C_{30}$ ($C_6:C_4$)

Order 24: $C_6:C_4$ ($C_5\times S_3$)

Order 20: $C_2\times C_{10}$ ($S_3^2$)

Order 18: $C_3\times C_6$ ($C_2\times C_{20}$) x 2

Order 15: $C_{15}$ ($C_{12}:C_4$)

Order 12: $C_2\times C_6$ ($S_3\times C_{10}$), $C_3:C_4$ ($C_3:C_{20}$)

Order 10: $C_{10}$ ($C_6.D_6$), $C_{10}$ ($C_6.D_6$)

Order 9: $C_3^2$ ($C_4\times C_{20}$)

Order 6: $C_6$ ($S_3\times C_{20}$) x 2, $C_6$ ($C_6:C_{20}$)

Order 5: $C_5$ ($C_3^2:C_4^2$)

Order 4: $C_2^2$ ($C_5\times S_3^2$)

Order 3: $C_3$ ($C_{12}:C_{20}$)

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

Order 1: $C_1$ ($C_5\times C_3^2:C_4^2$)

Series

Derived series

$C_5\times C_3^2:C_4^2$

$\rhd$

$C_3^2$

$\rhd$

$C_1$

Chief series

$C_5\times C_3^2:C_4^2$

$\rhd$

$C_6:C_{60}$

$\rhd$

$C_6\times C_{30}$

$\rhd$

$C_3\times C_{30}$

$\rhd$

$C_3\times C_{15}$

$\rhd$

$C_{15}$

$\rhd$

$C_5$

$\rhd$

$C_1$

Lower central series

$C_5\times C_3^2:C_4^2$

$\rhd$

$C_3^2$

Upper central series

$C_1$

$\lhd$

$C_2\times C_{10}$

Supergroups

This group is a maximal subgroup of 74 larger groups in the database.

This group is a maximal quotient of 6 larger groups in the database.

Character theory

Complex character table

See the $180 \times 180$ 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 $52 \times 52$ rational character table.