Abstract group 6480.by: $D_5\times S_3\wr C_3$

• Label: 6480.by
• Order: \( 2^{4} \cdot 3^{4} \cdot 5 \)
• Exponent: \( 2 \cdot 3^{2} \cdot 5 \)
• Nilpotent: no
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{2} \cdot 3 \)
• $\card{Z(G)}$: \( 1 \)
• $\card{\Aut(G)}$: \( 2^{6} \cdot 3^{4} \cdot 5 \)
• $\card{\mathrm{Out}(G)}$: \( 2^{2} \)
• Perm deg.: $14$
• Trans deg.: $45$
• Rank: $2$

Group information

Description:	$D_5\times S_3\wr C_3$
Order:	\(6480\)\(\medspace = 2^{4} \cdot 3^{4} \cdot 5 \)
Exponent:	\(90\)\(\medspace = 2 \cdot 3^{2} \cdot 5 \)
Automorphism group:	$F_5\times S_3\wr S_3$, of order \(25920\)\(\medspace = 2^{6} \cdot 3^{4} \cdot 5 \)
Composition factors:	$C_2$ x 4, $C_3$ x 4, $C_5$
Derived length:	$3$

This group is nonabelian and monomial (hence solvable).

Group statistics

Order	1	2	3	5	6	9	10	15	18	30	45
Elements	1	383	98	4	2542	144	252	392	720	1368	576	6480
Conjugacy classes	1	7	5	2	17	2	6	10	2	12	4	68
Divisions	1	7	4	1	14	1	3	4	1	5	1	42
Autjugacy classes	1	7	4	1	12	1	3	4	1	4	1	39

Dimension	1	2	3	4	6	8	12	16	24	32	48	64
Irr. complex chars.	12	12	4	0	12	6	12	6	4	0	0	0	68
Irr. rational chars.	4	4	4	2	8	4	6	2	4	1	2	1	42

Minimal presentations

Permutation degree:	$14$
Transitive degree:	$45$
Rank:	$2$
Inequivalent generating pairs:	$312$

Minimal degrees of faithful linear representations

	Over $\mathbb{C}$	Over $\mathbb{R}$	Over $\mathbb{Q}$
Irreducible	12	12	24
Arbitrary	8	8	10

Constructions

Presentation:	${\langle a, b, c, d, e \mid b^{10}=c^{6}=d^{6}=e^{3}=[b,e]=[c,e]=1, a^{6}= \!\cdots\! \rangle}$
Permutation group:	Degree $14$ $\langle(2,3,5)(4,6)(11,12)(13,14), (1,2,4,7,3,6,9,5,8)(10,11)(12,13)\rangle$
Transitive group:	45T392				more information
Direct product:	$D_5$ $\, \times\, $ $(S_3\wr C_3)$
Semidirect product:	$(D_5\times S_3^3)$ $\,\rtimes\,$ $C_3$	$S_3^3$ $\,\rtimes\,$ $(C_3\times D_5)$	$(C_5\times S_3^3)$ $\,\rtimes\,$ $C_6$	$(C_5:S_3^3)$ $\,\rtimes\,$ $C_6$	all 20
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product

Elements of the group are displayed as permutations of degree 14.

Homology

Abelianization:	$C_{2} \times C_{6} \simeq C_{2}^{2} \times C_{3}$
Schur multiplier:	$C_{2}^{2}$
Commutator length:	$1$

Subgroups

There are 23440 subgroups in 536 conjugacy classes, 24 normal, and all normal subgroups are characteristic.

Characteristic subgroups are shown in this color.

Special subgroups

Center:	$Z \simeq$ $C_1$	$G/Z \simeq$ $D_5\times S_3\wr C_3$
Commutator:	$G' \simeq$ $C_{15}:S_3^2$	$G/G' \simeq$ $C_2\times C_6$
Frattini:	$\Phi \simeq$ $C_1$	$G/\Phi \simeq$ $D_5\times S_3\wr C_3$
Fitting:	$\operatorname{Fit} \simeq$ $C_3^2\times C_{15}$	$G/\operatorname{Fit} \simeq$ $C_2^2\times A_4$
Radical:	$R \simeq$ $D_5\times S_3\wr C_3$	$G/R \simeq$ $C_1$
Socle:	$\operatorname{soc} \simeq$ $C_3^2\times C_{15}$	$G/\operatorname{soc} \simeq$ $C_2^2\times A_4$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $C_2^4$
3-Sylow subgroup:	$P_{ 3 } \simeq$ $C_3\wr C_3$
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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Classes of subgroups up to conjugation

Order 6480: $D_5\times S_3\wr C_3$

Order 3240: $C_3^3:(D_5\times A_4)$, $D_5\times C_3^3:A_4$, $S_3^3:C_{15}$

Order 2160: $D_5\times S_3^3$

Order 1620: $C_5\times C_3^3:A_4$, $C_3^3:C_6\times D_5$

Order 1296: $S_3^3:C_6$

Order 1080: $C_5:S_3^3$, $C_5:S_3^3$, $D_{15}\times S_3^2$, $C_5\times S_3^3$, $D_5\times C_3:S_3^2$, $S_3\times C_{15}:D_6$, $C_3\times D_5\times S_3^2$

Order 810: $\He_3:D_{15}$, $C_3^3:C_{30}$, $C_3\wr C_3\times D_5$

Order 720: $D_{10}\times S_3^2$

Order 648: $S_3\wr C_3$ x 2, $C_2\times C_3^3:A_4$

Order 540: $C_{15}:S_3^2$ x 2, $C_{15}:C_6^2$, $\He_3:D_{10}$, $C_{15}:S_3^2$, $C_{15}:S_3^2$, $C_{15}:S_3^2$, $C_{15}:S_3^2$, $C_{15}:S_3^2$, $C_{15}:S_3^2$, $C_3^3:D_{10}$, $C_{15}:S_3^2$, $C_{15}\times S_3^2$, $C_3^3:D_{10}$

Order 432: $C_2\times S_3^3$

Order 405: $C_3^3:C_{15}$

Order 360: $D_5\times S_3^2$ x 9, $S_3\times D_{30}$ x 2, $C_{30}:D_6$ x 2, $C_{30}:D_6$, $C_{10}\times S_3^2$, $C_{30}:D_6$

Order 324: $\He_3:D_6$, $C_3^3:A_4$

Order 270: $C_{45}:C_6$, $D_5\times \He_3$, $C_3^2:D_{15}$, $C_3^3:C_{10}$, $C_3^2:D_{15}$, $C_3^2:C_{30}$, $C_3^2\times D_{15}$, $C_3^2:C_{30}$, $D_5\times C_3^3$, $C_3^2:D_{15}$, $C_3^2:C_{30}$

Order 240: $D_6\times D_{10}$, $A_4\times D_{10}$

Order 216: $S_3^3$ x 4, $C_6:S_3^2$, $C_6:S_3^2$, $C_6\times S_3^2$

Order 180: $C_{15}:D_6$ x 11, $S_3\times D_{15}$ x 10, $C_{15}:D_6$ x 6, $C_{15}:D_6$ x 5, $C_5\times S_3^2$ x 5, $C_3\times D_{30}$ x 2, $S_3\times C_{30}$ x 2, $C_3:D_{30}$, $C_{30}:S_3$, $C_3^2\times D_{10}$

Order 162: $C_3^3:C_6$ x 2, $C_3^3:C_6$

Order 144: $D_6^2$

Order 135: $C_3^2\times C_{15}$, $C_9:C_{15}$, $C_5\times \He_3$

Order 120: $S_3\times D_{10}$ x 13, $D_5\times A_4$ x 2, $C_2\times D_{30}$, $C_{10}\times D_6$, $C_6\times D_{10}$, $C_{10}\times A_4$

Order 108: $C_3:S_3^2$ x 4, $C_3\times S_3^2$ x 4, $C_3:S_3^2$ x 2, $C_3^2:D_6$, $C_3^2:D_6$, $C_3^2\times D_6$, $C_3^2:D_6$

Order 90: $C_3\times D_{15}$ x 7, $S_3\times C_{15}$ x 7, $C_3^2\times D_5$ x 5, $C_3:D_{15}$ x 4, $C_{15}:S_3$ x 4, $C_9\times D_5$, $C_3\times C_{30}$

Order 81: $C_3\wr C_3$

Order 80: $C_2^2\times D_{10}$

Order 72: $S_3\times D_6$ x 13, $C_6\times D_6$ x 2, $C_6:D_6$

Order 60: $S_3\times D_5$ x 21, $C_3\times D_{10}$ x 8, $D_{30}$ x 7, $S_3\times C_{10}$ x 7, $C_5\times A_4$, $C_2\times C_{30}$

Order 54: $C_3^2:C_6$ x 2, $C_3^2:S_3$ x 2, $C_3^2:C_6$ x 2, $S_3\times C_3^2$ x 2, $C_3^2\times C_6$, $C_9:C_6$, $C_2\times \He_3$

Order 48: $C_2^2\times D_6$, $C_2^2\times A_4$

Order 45: $C_3\times C_{15}$ x 4, $C_{45}$

Order 40: $C_2\times D_{10}$ x 6, $C_2^2\times C_{10}$

Order 36: $S_3^2$ x 20, $C_6\times S_3$ x 15, $C_6:S_3$ x 8, $C_6^2$

Order 30: $C_3\times D_5$ x 9, $D_{15}$ x 7, $C_5\times S_3$ x 7, $C_{30}$ x 5

Order 27: $C_3^3$, $C_9:C_3$, $\He_3$

Order 24: $C_2\times D_6$ x 15, $C_2\times A_4$ x 3, $C_2^2\times C_6$

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

Order 18: $C_3\times S_3$ x 14, $C_3:S_3$ x 8, $C_3\times C_6$ x 6, $C_{18}$

Order 16: $C_2^4$

Order 15: $C_{15}$ x 4

Order 12: $D_6$ x 35, $C_2\times C_6$ x 9, $A_4$

Order 10: $D_5$ x 4, $C_{10}$ x 3

Order 9: $C_3^2$ x 4, $C_9$

Order 8: $C_2^3$ x 7

Order 6: $C_6$ x 14, $S_3$ x 14

Order 5: $C_5$

Order 4: $C_2^2$ x 13

Order 3: $C_3$ x 4

Order 2: $C_2$ x 7

Order 1: $C_1$

Classes of subgroups up to automorphism

Order 6480: $D_5\times S_3\wr C_3$

Order 3240: $C_3^3:(D_5\times A_4)$, $D_5\times C_3^3:A_4$, $S_3^3:C_{15}$

Order 2160: $D_5\times S_3^3$

Order 1620: $C_5\times C_3^3:A_4$, $C_3^3:C_6\times D_5$

Order 1296: $S_3^3:C_6$

Order 1080: $C_5:S_3^3$, $C_5:S_3^3$, $D_{15}\times S_3^2$, $C_5\times S_3^3$, $D_5\times C_3:S_3^2$, $S_3\times C_{15}:D_6$, $C_3\times D_5\times S_3^2$

Order 810: $\He_3:D_{15}$, $C_3^3:C_{30}$, $C_3\wr C_3\times D_5$

Order 720: $D_{10}\times S_3^2$

Order 648: $S_3\wr C_3$ x 2, $C_2\times C_3^3:A_4$

Order 540: $C_{15}:C_6^2$, $\He_3:D_{10}$, $C_{15}:S_3^2$, $C_{15}:S_3^2$, $C_{15}:S_3^2$, $C_{15}:S_3^2$, $C_{15}:S_3^2$, $C_{15}:S_3^2$, $C_3^3:D_{10}$, $C_{15}:S_3^2$, $C_{15}:S_3^2$, $C_{15}\times S_3^2$, $C_3^3:D_{10}$

Order 432: $C_2\times S_3^3$

Order 405: $C_3^3:C_{15}$

Order 360: $D_5\times S_3^2$ x 7, $C_{30}:D_6$, $S_3\times D_{30}$, $C_{10}\times S_3^2$, $C_{30}:D_6$, $C_{30}:D_6$

Order 324: $\He_3:D_6$, $C_3^3:A_4$

Order 270: $C_{45}:C_6$, $D_5\times \He_3$, $C_3^2:D_{15}$, $C_3^3:C_{10}$, $C_3^2:D_{15}$, $C_3^2:C_{30}$, $C_3^2\times D_{15}$, $C_3^2:C_{30}$, $D_5\times C_3^3$, $C_3^2:D_{15}$, $C_3^2:C_{30}$

Order 240: $D_6\times D_{10}$, $A_4\times D_{10}$

Order 216: $S_3^3$ x 4, $C_6:S_3^2$, $C_6:S_3^2$, $C_6\times S_3^2$

Order 180: $C_{15}:D_6$ x 7, $S_3\times D_{15}$ x 6, $C_{15}:D_6$ x 6, $C_{15}:D_6$ x 4, $C_5\times S_3^2$ x 4, $C_3:D_{30}$, $C_{30}:S_3$, $C_3\times D_{30}$, $S_3\times C_{30}$, $C_3^2\times D_{10}$

Order 162: $C_3^3:C_6$ x 2, $C_3^3:C_6$

Order 144: $D_6^2$

Order 135: $C_3^2\times C_{15}$, $C_9:C_{15}$, $C_5\times \He_3$

Order 120: $S_3\times D_{10}$ x 9, $D_5\times A_4$ x 2, $C_2\times D_{30}$, $C_{10}\times D_6$, $C_6\times D_{10}$, $C_{10}\times A_4$

Order 108: $C_3:S_3^2$ x 4, $C_3\times S_3^2$ x 3, $C_3:S_3^2$ x 2, $C_3^2:D_6$, $C_3^2:D_6$, $C_3^2\times D_6$, $C_3^2:D_6$

Order 90: $C_3\times D_{15}$ x 5, $S_3\times C_{15}$ x 5, $C_3^2\times D_5$ x 5, $C_3:D_{15}$ x 4, $C_{15}:S_3$ x 4, $C_9\times D_5$, $C_3\times C_{30}$

Order 81: $C_3\wr C_3$

Order 80: $C_2^2\times D_{10}$

Order 72: $S_3\times D_6$ x 10, $C_6:D_6$, $C_6\times D_6$

Order 60: $S_3\times D_5$ x 15, $C_3\times D_{10}$ x 6, $D_{30}$ x 5, $S_3\times C_{10}$ x 5, $C_5\times A_4$, $C_2\times C_{30}$

Order 54: $C_3^2:C_6$ x 2, $C_3^2:S_3$ x 2, $C_3^2:C_6$ x 2, $S_3\times C_3^2$ x 2, $C_3^2\times C_6$, $C_9:C_6$, $C_2\times \He_3$

Order 48: $C_2^2\times D_6$, $C_2^2\times A_4$

Order 45: $C_3\times C_{15}$ x 4, $C_{45}$

Order 40: $C_2\times D_{10}$ x 6, $C_2^2\times C_{10}$

Order 36: $S_3^2$ x 14, $C_6\times S_3$ x 9, $C_6:S_3$ x 8, $C_6^2$

Order 30: $C_3\times D_5$ x 8, $D_{15}$ x 6, $C_5\times S_3$ x 6, $C_{30}$ x 4

Order 27: $C_3^3$, $C_9:C_3$, $\He_3$

Order 24: $C_2\times D_6$ x 11, $C_2\times A_4$ x 3, $C_2^2\times C_6$

Order 20: $D_{10}$ x 9, $C_2\times C_{10}$ x 3

Order 18: $C_3\times S_3$ x 10, $C_3:S_3$ x 8, $C_3\times C_6$ x 6, $C_{18}$

Order 16: $C_2^4$

Order 15: $C_{15}$ x 4

Order 12: $D_6$ x 25, $C_2\times C_6$ x 7, $A_4$

Order 10: $D_5$ x 4, $C_{10}$ x 3

Order 9: $C_3^2$ x 4, $C_9$

Order 8: $C_2^3$ x 7

Order 6: $C_6$ x 12, $S_3$ x 12

Order 5: $C_5$

Order 4: $C_2^2$ x 12

Order 3: $C_3$ x 4

Order 2: $C_2$ x 7

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 6480: $D_5\times S_3\wr C_3$ ($C_1$)

Order 3240: $C_3^3:(D_5\times A_4)$ ($C_2$), $D_5\times C_3^3:A_4$ ($C_2$), $S_3^3:C_{15}$ ($C_2$)

Order 2160: $D_5\times S_3^3$ ($C_3$)

Order 1620: $C_5\times C_3^3:A_4$ ($C_2^2$)

Order 1080: $C_5:S_3^3$ ($C_6$), $C_5\times S_3^3$ ($C_6$), $D_5\times C_3:S_3^2$ ($C_6$)

Order 648: $S_3\wr C_3$ ($D_5$)

Order 540: $C_{15}:S_3^2$ ($C_2\times C_6$), $C_3^3:D_{10}$ ($A_4$)

Order 324: $C_3^3:A_4$ ($D_{10}$)

Order 270: $C_3^2:D_{15}$ ($C_2\times A_4$), $C_3^3:C_{10}$ ($C_2\times A_4$), $D_5\times C_3^3$ ($C_2\times A_4$)

Order 216: $S_3^3$ ($C_3\times D_5$)

Order 135: $C_3^2\times C_{15}$ ($C_2^2\times A_4$)

Order 108: $C_3:S_3^2$ ($C_3\times D_{10}$)

Order 54: $C_3^2:S_3$ ($D_5\times A_4$)

Order 27: $C_3^3$ ($A_4\times D_{10}$)

Order 10: $D_5$ ($S_3\wr C_3$)

Order 5: $C_5$ ($S_3^3:C_6$)

Order 1: $C_1$ ($D_5\times S_3\wr C_3$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 6480: $D_5\times S_3\wr C_3$ ($C_1$)

Order 3240: $C_3^3:(D_5\times A_4)$ ($C_2$), $D_5\times C_3^3:A_4$ ($C_2$), $S_3^3:C_{15}$ ($C_2$)

Order 2160: $D_5\times S_3^3$ ($C_3$)

Order 1620: $C_5\times C_3^3:A_4$ ($C_2^2$)

Order 1080: $C_5:S_3^3$ ($C_6$), $C_5\times S_3^3$ ($C_6$), $D_5\times C_3:S_3^2$ ($C_6$)

Order 648: $S_3\wr C_3$ ($D_5$)

Order 540: $C_{15}:S_3^2$ ($C_2\times C_6$), $C_3^3:D_{10}$ ($A_4$)

Order 324: $C_3^3:A_4$ ($D_{10}$)

Order 270: $C_3^2:D_{15}$ ($C_2\times A_4$), $C_3^3:C_{10}$ ($C_2\times A_4$), $D_5\times C_3^3$ ($C_2\times A_4$)

Order 216: $S_3^3$ ($C_3\times D_5$)

Order 135: $C_3^2\times C_{15}$ ($C_2^2\times A_4$)

Order 108: $C_3:S_3^2$ ($C_3\times D_{10}$)

Order 54: $C_3^2:S_3$ ($D_5\times A_4$)

Order 27: $C_3^3$ ($A_4\times D_{10}$)

Order 10: $D_5$ ($S_3\wr C_3$)

Order 5: $C_5$ ($S_3^3:C_6$)

Order 1: $C_1$ ($D_5\times S_3\wr C_3$)

Series

Derived series

$D_5\times S_3\wr C_3$

$\rhd$

$C_{15}:S_3^2$

$\rhd$

$C_3^3$

$\rhd$

$C_1$

Chief series

$D_5\times S_3\wr C_3$

$\rhd$

$D_5\times C_3^3:A_4$

$\rhd$

$C_5\times C_3^3:A_4$

$\rhd$

$C_{15}:S_3^2$

$\rhd$

$C_3^2\times C_{15}$

$\rhd$

$C_3^3$

$\rhd$

$C_1$

Lower central series

$D_5\times S_3\wr C_3$

$\rhd$

$C_{15}:S_3^2$

Upper central series

$C_1$

Supergroups

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

This group is a maximal quotient of 2 larger groups in the database.

Character theory

Complex character table

See the $68 \times 68$ character table. Alternatively, you may search for characters of this group with desired properties.

Rational character table

See the $42 \times 42$ rational character table.