Abstract group 18000.o: $C_5^3:(S_3\times S_4)$

• Label: 18000.o
• Order: \( 2^{4} \cdot 3^{2} \cdot 5^{3} \)
• Exponent: \( 2^{2} \cdot 3 \cdot 5 \)
• Nilpotent: no
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{2} \)
• $\card{Z(G)}$: \( 1 \)
• $\card{\Aut(G)}$: \( 2^{6} \cdot 3^{2} \cdot 5^{3} \)
• $\card{\mathrm{Out}(G)}$: \( 2^{2} \)
• Perm deg.: $18$
• Trans deg.: $45$
• Rank: $2$

Group information

Description:	$C_5^3:(S_3\times S_4)$
Order:	\(18000\)\(\medspace = 2^{4} \cdot 3^{2} \cdot 5^{3} \)
Exponent:	\(60\)\(\medspace = 2^{2} \cdot 3 \cdot 5 \)
Automorphism group:	$S_3\times D_5^3.D_6$, of order \(72000\)\(\medspace = 2^{6} \cdot 3^{2} \cdot 5^{3} \)
Composition factors:	$C_2$ x 4, $C_3$ x 2, $C_5$ x 3
Derived length:	$4$

This group is nonabelian and monomial (hence solvable).

Group statistics

Order	1	2	3	4	5	6	10	12	15	20	30	60
Elements	1	735	602	2400	124	3450	4140	300	2648	600	1800	1200	18000
Conjugacy classes	1	5	3	2	9	3	17	1	15	2	4	4	66
Divisions	1	5	3	2	5	3	9	1	7	1	2	1	40
Autjugacy classes	1	5	3	2	5	3	9	1	7	1	2	1	40

Dimension	1	2	3	4	6	8	12	16	24	32	48	64	96
Irr. complex chars.	4	4	4	1	10	4	18	8	10	0	3	0	0	66
Irr. rational chars.	4	4	4	1	2	0	4	2	9	2	6	1	1	40

Minimal presentations

Permutation degree:	$18$
Transitive degree:	$45$
Rank:	$2$
Inequivalent generating pairs:	$558$

Minimal degrees of faithful linear representations

	Over $\mathbb{C}$	Over $\mathbb{R}$	Over $\mathbb{Q}$
Irreducible	12	12	24
Arbitrary	8	8	14

Constructions

Presentation:	${\langle a, b, c, d, e, f \mid b^{6}=c^{6}=d^{10}=e^{5}=f^{5}=[c,e]=[e,f]= \!\cdots\! \rangle}$
Permutation group:	Degree $18$ $\langle(1,2,4,6,7,10)(3,5,8)(9,11,12,13,14,15)(16,17), (1,3)(2,4,7,10)(5,8)(6,9)(11,12,14,15)(17,18)\rangle$
Transitive group:	45T611				more information
Direct product:	not isomorphic to a non-trivial direct product
Semidirect product:	$(C_5^3:S_4)$ $\,\rtimes\,$ $S_3$	$C_5^3$ $\,\rtimes\,$ $(S_3\times S_4)$	$C_3$ $\,\rtimes\,$ $(D_5\wr S_3)$	$(C_5^3:A_4)$ $\,\rtimes\,$ $D_6$	all 12
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product

Elements of the group are displayed as permutations of degree 18.

Homology

Abelianization:	$C_{2}^{2} $
Schur multiplier:	$C_{2}^{2}$
Commutator length:	$1$

Subgroups

There are 53800 subgroups in 408 conjugacy classes, 15 normal, and all normal subgroups are characteristic.

Characteristic subgroups are shown in this color.

Special subgroups

Center:	$Z \simeq$ $C_1$	$G/Z \simeq$ $C_5^3:(S_3\times S_4)$
Commutator:	$G' \simeq$ $C_3\times C_5^3:A_4$	$G/G' \simeq$ $C_2^2$
Frattini:	$\Phi \simeq$ $C_1$	$G/\Phi \simeq$ $C_5^3:(S_3\times S_4)$
Fitting:	$\operatorname{Fit} \simeq$ $C_5^2\times C_{15}$	$G/\operatorname{Fit} \simeq$ $C_2\times S_4$
Radical:	$R \simeq$ $C_5^3:(S_3\times S_4)$	$G/R \simeq$ $C_1$
Socle:	$\operatorname{soc} \simeq$ $C_5^2\times C_{15}$	$G/\operatorname{soc} \simeq$ $C_2\times S_4$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $C_2\times D_4$
3-Sylow subgroup:	$P_{ 3 } \simeq$ $C_3^2$
5-Sylow subgroup:	$P_{ 5 } \simeq$ $C_5^3$

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 18000: $C_5^3:(S_3\times S_4)$

Order 9000: $(C_5^2\times C_{15}):S_4$, $C_5^3:(S_3\times A_4)$, $C_3\times C_5^3:S_4$

Order 6000: $C_5^2.C_{20}:D_6$, $D_5\wr S_3$

Order 4500: $C_5^3:S_3^2$, $C_3\times C_5^3:A_4$

Order 3000: $C_5^3:(C_2^2\times S_3)$ x 2, $C_5^3:(C_3:D_4)$ x 2, $C_5^3:S_4$ x 2, $C_5^3:(C_4\times S_3)$, $C_5\times C_5^2:D_{12}$, $C_3\times C_5^3:D_4$, $D_5\wr C_3$, $C_5^3:S_4$

Order 2250: $(C_5^2\times C_{15}):S_3$, $C_3\times C_5^2:D_{15}$, $C_5^2:(C_3\times D_{15})$

Order 2000: $D_5^3:C_2$

Order 1500: $C_{15}:D_5^2$ x 3, $C_5^2:D_{30}$ x 2, $C_{15}:D_5^2$ x 2, $C_5^3:A_4$ x 2, $(C_5\times C_{15}):F_5$, $C_5^2:C_{60}$, $C_5^3:D_6$

Order 1200: $D_5^2:D_6$

Order 1125: $C_5^3:C_3^2$

Order 1000: $D_5^3$ x 2, $C_5^3:D_4$ x 2, $C_5^2:D_{20}$, $D_5^2:C_{10}$, $D_5\times C_5:F_5$

Order 900: $C_5^2:S_3^2$

Order 750: $C_5^2\times D_{15}$ x 2, $C_5^2:C_{30}$ x 2, $C_5\wr S_3$ x 2, $C_5^2:D_{15}$, $C_5^3:C_6$, $C_5^2:D_{15}$

Order 600: $S_3\times D_5^2$ x 3, $D_5^2:S_3$ x 2, $C_5^2:D_{12}$, $D_5^2:C_6$, $F_5\times D_{15}$, $S_3\times C_5:F_5$

Order 500: $C_5\times D_5^2$ x 3, $C_5:D_5^2$ x 2, $C_5:D_5^2$ x 2, $C_5^3:C_4$, $C_5^2:C_{20}$

Order 450: $(C_5\times C_{15}):C_6$, $(C_5\times C_{15}):S_3$, $C_3\times C_5^2:S_3$

Order 400: $D_5^2:C_2^2$

Order 375: $C_5\wr C_3$ x 2, $C_5^2\times C_{15}$

Order 300: $D_5\times D_{15}$ x 7, $C_{15}:D_{10}$ x 4, $C_{15}:D_{10}$ x 3, $C_3\times D_5^2$ x 3, $C_{15}:D_{10}$ x 2, $C_{15}:F_5$, $C_{15}:F_5$, $C_{15}:F_5$, $C_{15}\times F_5$, $C_5^2:D_6$

Order 250: $C_5^2:C_{10}$ x 2, $D_5\times C_5^2$ x 2, $C_5^3:C_2$

Order 240: $C_{12}:D_{10}$

Order 225: $C_5^2:C_3^2$

Order 200: $D_5\wr C_2$ x 4, $D_5\times D_{10}$ x 3, $C_{10}:F_5$, $D_5\times F_5$

Order 180: $C_{15}:D_6$

Order 150: $D_5\times C_{15}$ x 7, $C_5\times D_{15}$ x 7, $C_5:D_{15}$ x 5, $C_5^2:S_3$ x 3, $C_{15}:D_5$ x 2, $S_3\times C_5^2$ x 2, $C_5^2:C_6$

Order 144: $S_3\times S_4$

Order 125: $C_5^3$

Order 120: $S_3\times D_{10}$ x 3, $C_{15}:D_4$ x 2, $S_3\times F_5$, $C_4\times D_{15}$, $C_5\times D_{12}$, $C_3\times D_{20}$

Order 100: $D_5^2$ x 13, $C_5\times D_{10}$ x 4, $C_5:D_{10}$ x 2, $C_5:F_5$ x 2, $C_5\times F_5$, $C_5:F_5$

Order 90: $C_3\times D_{15}$ x 2, $C_{15}:S_3$

Order 80: $D_4\times D_5$

Order 75: $C_5\times C_{15}$ x 5, $C_5^2:C_3$ x 2

Order 72: $S_3\times A_4$, $C_3:S_4$, $C_3\times S_4$

Order 60: $S_3\times D_5$ x 12, $S_3\times C_{10}$ x 3, $C_3\times D_{10}$ x 3, $D_{30}$ x 2, $C_{15}:C_4$, $C_3\times F_5$, $C_{60}$, $C_{15}:C_4$

Order 50: $C_5\times D_5$ x 14, $C_5:D_5$ x 7, $C_5\times C_{10}$ x 2

Order 48: $C_2\times S_4$, $S_3\times D_4$

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

Order 40: $C_2\times D_{10}$ x 3, $C_5:D_4$ x 2, $D_{20}$, $C_4\times D_5$, $C_2\times F_5$, $C_5\times D_4$

Order 36: $C_3\times A_4$, $S_3^2$

Order 30: $C_5\times S_3$ x 9, $D_{15}$ x 8, $C_3\times D_5$ x 8, $C_{30}$ x 2

Order 25: $C_5^2$ x 5

Order 24: $S_4$ x 3, $C_3:D_4$ x 2, $C_2\times D_6$ x 2, $D_{12}$, $C_4\times S_3$, $C_2\times A_4$, $C_3\times D_4$

Order 20: $D_{10}$ x 16, $C_2\times C_{10}$ x 3, $F_5$ x 2, $C_{20}$, $C_5:C_4$

Order 18: $C_3\times S_3$ x 2, $C_3:S_3$

Order 16: $C_2\times D_4$

Order 15: $C_{15}$ x 7

Order 12: $D_6$ x 6, $C_2\times C_6$ x 2, $A_4$ x 2, $C_{12}$, $C_3:C_4$

Order 10: $D_5$ x 14, $C_{10}$ x 9

Order 9: $C_3^2$

Order 8: $D_4$ x 4, $C_2^3$ x 2, $C_2\times C_4$

Order 6: $S_3$ x 6, $C_6$ x 3

Order 5: $C_5$ x 5

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

Order 3: $C_3$ x 3

Order 2: $C_2$ x 5

Order 1: $C_1$

Classes of subgroups up to automorphism

Order 18000: $C_5^3:(S_3\times S_4)$

Order 9000: $(C_5^2\times C_{15}):S_4$, $C_5^3:(S_3\times A_4)$, $C_3\times C_5^3:S_4$

Order 6000: $C_5^2.C_{20}:D_6$, $D_5\wr S_3$

Order 4500: $C_5^3:S_3^2$, $C_3\times C_5^3:A_4$

Order 3000: $C_5^3:(C_2^2\times S_3)$ x 2, $C_5^3:(C_3:D_4)$ x 2, $C_5^3:S_4$ x 2, $C_5^3:(C_4\times S_3)$, $C_5\times C_5^2:D_{12}$, $C_3\times C_5^3:D_4$, $D_5\wr C_3$, $C_5^3:S_4$

Order 2250: $(C_5^2\times C_{15}):S_3$, $C_3\times C_5^2:D_{15}$, $C_5^2:(C_3\times D_{15})$

Order 2000: $D_5^3:C_2$

Order 1500: $C_{15}:D_5^2$ x 3, $C_5^2:D_{30}$ x 2, $C_{15}:D_5^2$ x 2, $C_5^3:A_4$ x 2, $(C_5\times C_{15}):F_5$, $C_5^2:C_{60}$, $C_5^3:D_6$

Order 1200: $D_5^2:D_6$

Order 1125: $C_5^3:C_3^2$

Order 1000: $D_5^3$ x 2, $C_5^3:D_4$ x 2, $C_5^2:D_{20}$, $D_5^2:C_{10}$, $D_5\times C_5:F_5$

Order 900: $C_5^2:S_3^2$

Order 750: $C_5^2\times D_{15}$ x 2, $C_5^2:C_{30}$ x 2, $C_5\wr S_3$ x 2, $C_5^2:D_{15}$, $C_5^3:C_6$, $C_5^2:D_{15}$

Order 600: $S_3\times D_5^2$ x 3, $D_5^2:S_3$ x 2, $C_5^2:D_{12}$, $D_5^2:C_6$, $F_5\times D_{15}$, $S_3\times C_5:F_5$

Order 500: $C_5\times D_5^2$ x 3, $C_5:D_5^2$ x 2, $C_5:D_5^2$ x 2, $C_5^3:C_4$, $C_5^2:C_{20}$

Order 450: $(C_5\times C_{15}):C_6$, $(C_5\times C_{15}):S_3$, $C_3\times C_5^2:S_3$

Order 400: $D_5^2:C_2^2$

Order 375: $C_5\wr C_3$ x 2, $C_5^2\times C_{15}$

Order 300: $D_5\times D_{15}$ x 7, $C_{15}:D_{10}$ x 4, $C_{15}:D_{10}$ x 3, $C_3\times D_5^2$ x 3, $C_{15}:D_{10}$ x 2, $C_{15}:F_5$, $C_{15}:F_5$, $C_{15}:F_5$, $C_{15}\times F_5$, $C_5^2:D_6$

Order 250: $C_5^2:C_{10}$ x 2, $D_5\times C_5^2$ x 2, $C_5^3:C_2$

Order 240: $C_{12}:D_{10}$

Order 225: $C_5^2:C_3^2$

Order 200: $D_5\wr C_2$ x 4, $D_5\times D_{10}$ x 3, $C_{10}:F_5$, $D_5\times F_5$

Order 180: $C_{15}:D_6$

Order 150: $D_5\times C_{15}$ x 7, $C_5\times D_{15}$ x 7, $C_5:D_{15}$ x 5, $C_5^2:S_3$ x 3, $C_{15}:D_5$ x 2, $S_3\times C_5^2$ x 2, $C_5^2:C_6$

Order 144: $S_3\times S_4$

Order 125: $C_5^3$

Order 120: $S_3\times D_{10}$ x 3, $C_{15}:D_4$ x 2, $S_3\times F_5$, $C_4\times D_{15}$, $C_5\times D_{12}$, $C_3\times D_{20}$

Order 100: $D_5^2$ x 13, $C_5\times D_{10}$ x 4, $C_5:D_{10}$ x 2, $C_5:F_5$ x 2, $C_5\times F_5$, $C_5:F_5$

Order 90: $C_3\times D_{15}$ x 2, $C_{15}:S_3$

Order 80: $D_4\times D_5$

Order 75: $C_5\times C_{15}$ x 5, $C_5^2:C_3$ x 2

Order 72: $S_3\times A_4$, $C_3:S_4$, $C_3\times S_4$

Order 60: $S_3\times D_5$ x 12, $S_3\times C_{10}$ x 3, $C_3\times D_{10}$ x 3, $D_{30}$ x 2, $C_{15}:C_4$, $C_3\times F_5$, $C_{60}$, $C_{15}:C_4$

Order 50: $C_5\times D_5$ x 14, $C_5:D_5$ x 7, $C_5\times C_{10}$ x 2

Order 48: $C_2\times S_4$, $S_3\times D_4$

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

Order 40: $C_2\times D_{10}$ x 3, $C_5:D_4$ x 2, $D_{20}$, $C_4\times D_5$, $C_2\times F_5$, $C_5\times D_4$

Order 36: $C_3\times A_4$, $S_3^2$

Order 30: $C_5\times S_3$ x 9, $D_{15}$ x 8, $C_3\times D_5$ x 8, $C_{30}$ x 2

Order 25: $C_5^2$ x 5

Order 24: $S_4$ x 3, $C_3:D_4$ x 2, $C_2\times D_6$ x 2, $D_{12}$, $C_4\times S_3$, $C_2\times A_4$, $C_3\times D_4$

Order 20: $D_{10}$ x 16, $C_2\times C_{10}$ x 3, $F_5$ x 2, $C_{20}$, $C_5:C_4$

Order 18: $C_3\times S_3$ x 2, $C_3:S_3$

Order 16: $C_2\times D_4$

Order 15: $C_{15}$ x 7

Order 12: $D_6$ x 6, $C_2\times C_6$ x 2, $A_4$ x 2, $C_{12}$, $C_3:C_4$

Order 10: $D_5$ x 14, $C_{10}$ x 9

Order 9: $C_3^2$

Order 8: $D_4$ x 4, $C_2^3$ x 2, $C_2\times C_4$

Order 6: $S_3$ x 6, $C_6$ x 3

Order 5: $C_5$ x 5

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

Order 3: $C_3$ x 3

Order 2: $C_2$ x 5

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 18000: $C_5^3:(S_3\times S_4)$ ($C_1$)

Order 9000: $(C_5^2\times C_{15}):S_4$ ($C_2$), $C_5^3:(S_3\times A_4)$ ($C_2$), $C_3\times C_5^3:S_4$ ($C_2$)

Order 4500: $C_3\times C_5^3:A_4$ ($C_2^2$)

Order 3000: $C_5^3:(C_2^2\times S_3)$ ($S_3$), $C_5^3:S_4$ ($S_3$)

Order 1500: $C_{15}:D_5^2$ ($D_6$), $C_5^3:A_4$ ($D_6$)

Order 750: $C_5^2:D_{15}$ ($S_4$)

Order 500: $C_5:D_5^2$ ($S_3^2$)

Order 375: $C_5^2\times C_{15}$ ($C_2\times S_4$)

Order 125: $C_5^3$ ($S_3\times S_4$)

Order 3: $C_3$ ($D_5\wr S_3$)

Order 1: $C_1$ ($C_5^3:(S_3\times S_4)$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 18000: $C_5^3:(S_3\times S_4)$ ($C_1$)

Order 9000: $(C_5^2\times C_{15}):S_4$ ($C_2$), $C_5^3:(S_3\times A_4)$ ($C_2$), $C_3\times C_5^3:S_4$ ($C_2$)

Order 4500: $C_3\times C_5^3:A_4$ ($C_2^2$)

Order 3000: $C_5^3:(C_2^2\times S_3)$ ($S_3$), $C_5^3:S_4$ ($S_3$)

Order 1500: $C_{15}:D_5^2$ ($D_6$), $C_5^3:A_4$ ($D_6$)

Order 750: $C_5^2:D_{15}$ ($S_4$)

Order 500: $C_5:D_5^2$ ($S_3^2$)

Order 375: $C_5^2\times C_{15}$ ($C_2\times S_4$)

Order 125: $C_5^3$ ($S_3\times S_4$)

Order 3: $C_3$ ($D_5\wr S_3$)

Order 1: $C_1$ ($C_5^3:(S_3\times S_4)$)

Series

Derived series

$C_5^3:(S_3\times S_4)$

$\rhd$

$C_3\times C_5^3:A_4$

$\rhd$

$C_5:D_5^2$

$\rhd$

$C_5^3$

$\rhd$

$C_1$

Chief series

$C_5^3:(S_3\times S_4)$

$\rhd$

$C_5^3:(S_3\times A_4)$

$\rhd$

$C_3\times C_5^3:A_4$

$\rhd$

$C_5^3:A_4$

$\rhd$

$C_5:D_5^2$

$\rhd$

$C_5^3$

$\rhd$

$C_1$

Lower central series

$C_5^3:(S_3\times S_4)$

$\rhd$

$C_3\times C_5^3:A_4$

Upper central series

$C_1$

Supergroups

This group is a maximal subgroup of 3 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 $66 \times 66$ character table. Alternatively, you may search for characters of this group with desired properties.

Rational character table

See the $40 \times 40$ rational character table.