Abstract group 81000.bb: $C_{15}^3.S_4$

• Label: 81000.bb
• Order: \( 2^{3} \cdot 3^{4} \cdot 5^{3} \)
• Exponent: \( 2^{2} \cdot 3^{2} \cdot 5 \)
• Nilpotent: no
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2 \)
• $\card{Z(G)}$: \( 1 \)
• $\card{\Aut(G)}$: \( 2^{6} \cdot 3^{4} \cdot 5^{3} \)
• $\card{\mathrm{Out}(G)}$: \( 2^{3} \)
• Perm deg.: $24$
• Trans deg.: $45$
• Rank: $2$

Group information

Description:	$C_{15}^3.S_4$
Order:	\(81000\)\(\medspace = 2^{3} \cdot 3^{4} \cdot 5^{3} \)
Exponent:	\(180\)\(\medspace = 2^{2} \cdot 3^{2} \cdot 5 \)
Automorphism group:	$D_{15}\wr S_3.C_4$, of order \(648000\)\(\medspace = 2^{6} \cdot 3^{4} \cdot 5^{3} \)
Composition factors:	$C_2$ x 3, $C_3$ x 4, $C_5$ x 3
Derived length:	$4$

This group is nonabelian and solvable. Whether it is monomial has not been computed.

Group statistics

Order	1	2	3	4	5	6	9	10	15	30	45
Elements	1	765	1826	20250	124	2070	3600	4860	10424	22680	14400	81000
Conjugacy classes	1	2	5	1	13	6	2	16	190	104	8	348
Divisions	1	2	4	1	5	4	1	5	32	15	1	71
Autjugacy classes	1	2	4	1	5	4	1	5	32	15	1	71

Dimension	1	2	3	4	6	8	12	16	24	32	48	64	96	192
Irr. complex chars.	2	1	2	28	28	14	182	0	91	0	0	0	0	0	348
Irr. rational chars.	2	1	2	0	2	2	5	3	6	3	12	1	25	7	71

Minimal presentations

Permutation degree:	$24$
Transitive degree:	$45$
Rank:	$2$
Inequivalent generating pairs:	$3627$

Minimal degrees of faithful linear representations

	Over $\mathbb{C}$	Over $\mathbb{R}$	Over $\mathbb{Q}$
Irreducible	4	6	24
Arbitrary	4	6	18

Constructions

Presentation:	${\langle a, b, c, d, e, f \mid c^{2}=d^{30}=e^{15}=f^{15}=[e,f]=1, a^{2}= \!\cdots\! \rangle}$
Permutation group:	Degree $24$ $\langle(1,2,3,4,6,10,14,5,8,13,15,9,7,12,11)(16,17,18,20,21,19,22,23,24), (2,3,5,9)(4,7)(6,11)(8,12,10,15)(13,14)(17,19,23,24)(18,21)(20,22)\rangle$
Transitive group:	45T966				more information
Direct product:	not isomorphic to a non-trivial direct product
Semidirect product:	$(C_{15}^3.A_4)$ $\,\rtimes\,$ $C_2$	$C_5^3$ $\,\rtimes\,$ $(C_3^3:S_4)$	$C_3^3$ $\,\rtimes\,$ $(C_5^3:S_4)$		more information
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product

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

Homology

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

Subgroups

There are 104656 subgroups in 524 conjugacy classes, 7 normal, and all normal subgroups are characteristic.

Characteristic subgroups are shown in this color.

Special subgroups

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

Hi

Subgroup diagram and profile

Classes of subgroups up to conjugation

There are too many subgroups to show. See a full page version of the diagram.

Classes of subgroups up to automorphism

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Normal subgroups

Normal subgroups up to automorphism

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Subgroup information Click on a subgroup in the diagram to see information about it.

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Classes of subgroups up to conjugation

Order 81000: $C_{15}^3.S_4$

Order 40500: $C_{15}^3.A_4$

Order 27000: $C_{15}^3.D_4$

Order 20250: $C_{15}\wr S_3$

Order 13500: $C_{15}\times C_5^2:S_3^2$, $C_{15}^3.C_2^2$, $C_{15}^3.C_4$

Order 10125: $C_{15}\wr C_3$

Order 9000: $C_5^3.S_3\wr C_2$

Order 6750: $C_{15}\times C_{15}:D_{15}$, $C_{15}\times C_{15}\times D_{15}$, $C_5\times C_{15}^2:S_3$

Order 5400: $C_{15}^2.C_3:D_4$

Order 4500: $C_5\times C_5^2:S_3^2$, $C_{15}\times D_5\times D_{15}$, $C_5^3:(C_3:S_3.C_2)$, $C_5^3:S_3^2$

Order 4050: $(C_3\times C_{15}^2):S_3$

Order 3375: $C_{15}^3$, $C_{15}^2.C_{15}$, $C_{15}^2:C_{15}$

Order 3000: $C_5^3:(C_3:D_4)$, $C_5^3:S_4$

Order 2700: $C_{15}\times S_3\times D_{15}$, $C_5^2:(C_3^3:C_2^2)$, $C_5^2:(C_3^2:(C_3:C_4))$, $C_5^2:(C_3^2:(C_3:C_4))$, $C_3\times C_5^2:S_3^2$

Order 2250: $C_5\times C_{15}\times D_{15}$ x 2, $C_{15}\times C_5:D_{15}$ x 2, $C_3\times C_{15}\times C_5\times D_5$, $C_5\times C_{15}:D_{15}$, $C_{15}\times C_5\times D_{15}$, $C_3\times C_5\wr S_3$

Order 2025: $C_{15}^2.C_3^2$

Order 1800: $D_{15}\wr C_2$

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

Order 1350: $C_{15}^2:C_6$ x 4, $C_{15}^2:C_6$ x 3, $C_{15}^2:S_3$, $C_{15}^2:C_6$, $S_3\times C_{15}^2$

Order 1125: $C_5\times C_{15}^2$ x 3, $C_5^3:C_3^2$, $C_5^2:C_{45}$

Order 1080: $S_3^2:D_{15}$

Order 1000: $C_5^3:D_4$

Order 900: $D_{15}^2$ x 2, $C_{15}\times D_{30}$, $C_{15}:D_{30}$, $C_{15}:D_{30}$, $C_{15}:D_{30}$, $D_{15}:C_{30}$, $C_{15}^2:C_4$, $C_{15}^2:C_4$

Order 810: $\He_3:C_{30}$

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

Order 675: $C_3\times C_{15}^2$ x 5, $C_{15}^2:C_3$, $C_{15}^2.C_3$

Order 648: $C_3^3:S_4$

Order 600: $D_5^2:S_3$

Order 540: $(C_3^2\times C_{15}):C_4$, $C_3^3:F_5$, $C_{15}:S_3^2$, $C_{15}:S_3^2$, $C_{15}\times S_3^2$

Order 500: $C_5:D_5^2$, $C_5\times D_5^2$, $C_5^3:C_4$

Order 450: $C_{15}\times D_{15}$ x 22, $C_{15}:C_{30}$ x 4, $C_{15}:D_{15}$ x 3, $C_{15}:C_{30}$ x 3, $C_{15}:D_{15}$ x 2, $C_{15}\times C_{30}$, $C_{15}:D_{15}$, $C_3\times C_5^2:S_3$

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

Order 375: $C_5^2\times C_{15}$ x 3, $C_5\wr C_3$

Order 360: $S_3^2:D_5$

Order 324: $C_3^3:A_4$

Order 300: $D_5\times D_{15}$ x 2, $C_5\times D_{30}$, $C_{15}\times D_{10}$, $C_{15}:D_{10}$, $C_{15}:D_{10}$, $C_3\times D_5^2$, $C_{15}:F_5$, $C_{15}:F_5$

Order 270: $C_3^2:C_{30}$ x 4, $C_3^2:D_{15}$ x 3, $C_3^2:C_{30}$, $C_3^2\times D_{15}$, $\He_3:C_{10}$

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

Order 225: $C_{15}^2$ x 31, $C_5^2:C_3^2$, $C_5^2:C_9$

Order 216: $S_3^2:S_3$

Order 200: $D_5\wr C_2$

Order 180: $S_3\times D_{15}$ x 2, $C_3\times D_{30}$, $S_3\times C_{30}$, $C_{15}:D_6$, $C_5\times S_3^2$, $C_{15}:D_6$, $C_3^2:F_5$, $(C_3\times C_{15}):C_4$

Order 162: $C_3\wr S_3$

Order 150: $D_5\times C_{15}$ x 17, $C_5\times D_{15}$ x 12, $C_5\times C_{30}$ x 3, $C_5:D_{15}$ x 2, $C_{15}:D_5$, $C_5^2:S_3$, $S_3\times C_5^2$

Order 135: $C_3^2\times C_{15}$ x 5, $C_9:C_{15}$, $C_5\times \He_3$

Order 125: $C_5^3$

Order 120: $C_{15}:D_4$

Order 108: $C_3:S_3^2$, $C_3\times S_3^2$, $C_3^3:C_4$

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

Order 90: $S_3\times C_{15}$ x 17, $C_3\times D_{15}$ x 11, $C_3\times C_{30}$ x 4, $C_3:D_{15}$ x 3, $C_{15}:S_3$, $C_3^2\times D_5$

Order 81: $C_3\wr C_3$

Order 75: $C_5\times C_{15}$ x 31, $C_5^2:C_3$

Order 72: $\SOPlus(4,2)$

Order 60: $S_3\times D_5$ x 2, $D_{30}$ x 2, $C_{15}:C_4$, $C_{15}:C_4$, $C_2\times C_{30}$, $S_3\times C_{10}$, $C_3\times D_{10}$

Order 54: $C_3^2:S_3$, $C_3^2:C_6$, $S_3\times C_3^2$

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

Order 45: $C_3\times C_{15}$ x 32, $C_{45}$

Order 40: $C_5:D_4$

Order 36: $S_3^2$ x 2, $C_3^2:C_4$, $C_6\times S_3$

Order 30: $C_{30}$ x 15, $D_{15}$ x 9, $C_5\times S_3$ x 7, $C_3\times D_5$ x 6

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

Order 25: $C_5^2$ x 5

Order 24: $C_3:D_4$, $S_4$

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

Order 18: $C_3\times S_3$ x 6, $C_3\times C_6$, $C_3:S_3$

Order 15: $C_{15}$ x 32

Order 12: $D_6$ x 2, $C_2\times C_6$, $A_4$, $C_3:C_4$

Order 10: $C_{10}$ x 5, $D_5$ x 4

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

Order 8: $D_4$

Order 6: $C_6$ x 4, $S_3$ x 4

Order 5: $C_5$ x 5

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

Order 3: $C_3$ x 4

Order 2: $C_2$ x 2

Order 1: $C_1$

Classes of subgroups up to automorphism

Order 81000: $C_{15}^3.S_4$

Order 40500: $C_{15}^3.A_4$

Order 27000: $C_{15}^3.D_4$

Order 20250: $C_{15}\wr S_3$

Order 13500: $C_{15}\times C_5^2:S_3^2$, $C_{15}^3.C_2^2$, $C_{15}^3.C_4$

Order 10125: $C_{15}\wr C_3$

Order 9000: $C_5^3.S_3\wr C_2$

Order 6750: $C_{15}\times C_{15}:D_{15}$, $C_{15}\times C_{15}\times D_{15}$, $C_5\times C_{15}^2:S_3$

Order 5400: $C_{15}^2.C_3:D_4$

Order 4500: $C_5\times C_5^2:S_3^2$, $C_{15}\times D_5\times D_{15}$, $C_5^3:(C_3:S_3.C_2)$, $C_5^3:S_3^2$

Order 4050: $(C_3\times C_{15}^2):S_3$

Order 3375: $C_{15}^3$, $C_{15}^2.C_{15}$, $C_{15}^2:C_{15}$

Order 3000: $C_5^3:(C_3:D_4)$, $C_5^3:S_4$

Order 2700: $C_{15}\times S_3\times D_{15}$, $C_5^2:(C_3^3:C_2^2)$, $C_5^2:(C_3^2:(C_3:C_4))$, $C_5^2:(C_3^2:(C_3:C_4))$, $C_3\times C_5^2:S_3^2$

Order 2250: $C_5\times C_{15}\times D_{15}$ x 2, $C_{15}\times C_5:D_{15}$ x 2, $C_3\times C_{15}\times C_5\times D_5$, $C_5\times C_{15}:D_{15}$, $C_{15}\times C_5\times D_{15}$, $C_3\times C_5\wr S_3$

Order 2025: $C_{15}^2.C_3^2$

Order 1800: $D_{15}\wr C_2$

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

Order 1350: $C_{15}^2:C_6$ x 4, $C_{15}^2:C_6$ x 3, $C_{15}^2:S_3$, $C_{15}^2:C_6$, $S_3\times C_{15}^2$

Order 1125: $C_5\times C_{15}^2$ x 3, $C_5^3:C_3^2$, $C_5^2:C_{45}$

Order 1080: $S_3^2:D_{15}$

Order 1000: $C_5^3:D_4$

Order 900: $D_{15}^2$ x 2, $C_{15}\times D_{30}$, $C_{15}:D_{30}$, $C_{15}:D_{30}$, $C_{15}:D_{30}$, $D_{15}:C_{30}$, $C_{15}^2:C_4$, $C_{15}^2:C_4$

Order 810: $\He_3:C_{30}$

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

Order 675: $C_3\times C_{15}^2$ x 5, $C_{15}^2:C_3$, $C_{15}^2.C_3$

Order 648: $C_3^3:S_4$

Order 600: $D_5^2:S_3$

Order 540: $(C_3^2\times C_{15}):C_4$, $C_3^3:F_5$, $C_{15}:S_3^2$, $C_{15}:S_3^2$, $C_{15}\times S_3^2$

Order 500: $C_5:D_5^2$, $C_5\times D_5^2$, $C_5^3:C_4$

Order 450: $C_{15}\times D_{15}$ x 22, $C_{15}:C_{30}$ x 4, $C_{15}:D_{15}$ x 3, $C_{15}:C_{30}$ x 3, $C_{15}:D_{15}$ x 2, $C_{15}\times C_{30}$, $C_{15}:D_{15}$, $C_3\times C_5^2:S_3$

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

Order 375: $C_5^2\times C_{15}$ x 3, $C_5\wr C_3$

Order 360: $S_3^2:D_5$

Order 324: $C_3^3:A_4$

Order 300: $D_5\times D_{15}$ x 2, $C_5\times D_{30}$, $C_{15}\times D_{10}$, $C_{15}:D_{10}$, $C_{15}:D_{10}$, $C_3\times D_5^2$, $C_{15}:F_5$, $C_{15}:F_5$

Order 270: $C_3^2:C_{30}$ x 4, $C_3^2:D_{15}$ x 3, $C_3^2:C_{30}$, $C_3^2\times D_{15}$, $\He_3:C_{10}$

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

Order 225: $C_{15}^2$ x 31, $C_5^2:C_3^2$, $C_5^2:C_9$

Order 216: $S_3^2:S_3$

Order 200: $D_5\wr C_2$

Order 180: $S_3\times D_{15}$ x 2, $C_3\times D_{30}$, $S_3\times C_{30}$, $C_{15}:D_6$, $C_5\times S_3^2$, $C_{15}:D_6$, $C_3^2:F_5$, $(C_3\times C_{15}):C_4$

Order 162: $C_3\wr S_3$

Order 150: $D_5\times C_{15}$ x 17, $C_5\times D_{15}$ x 12, $C_5\times C_{30}$ x 3, $C_5:D_{15}$ x 2, $C_{15}:D_5$, $C_5^2:S_3$, $S_3\times C_5^2$

Order 135: $C_3^2\times C_{15}$ x 5, $C_9:C_{15}$, $C_5\times \He_3$

Order 125: $C_5^3$

Order 120: $C_{15}:D_4$

Order 108: $C_3:S_3^2$, $C_3\times S_3^2$, $C_3^3:C_4$

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

Order 90: $S_3\times C_{15}$ x 17, $C_3\times D_{15}$ x 11, $C_3\times C_{30}$ x 4, $C_3:D_{15}$ x 3, $C_{15}:S_3$, $C_3^2\times D_5$

Order 81: $C_3\wr C_3$

Order 75: $C_5\times C_{15}$ x 31, $C_5^2:C_3$

Order 72: $\SOPlus(4,2)$

Order 60: $S_3\times D_5$ x 2, $D_{30}$ x 2, $C_{15}:C_4$, $C_{15}:C_4$, $C_2\times C_{30}$, $S_3\times C_{10}$, $C_3\times D_{10}$

Order 54: $C_3^2:S_3$, $C_3^2:C_6$, $S_3\times C_3^2$

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

Order 45: $C_3\times C_{15}$ x 32, $C_{45}$

Order 40: $C_5:D_4$

Order 36: $S_3^2$ x 2, $C_3^2:C_4$, $C_6\times S_3$

Order 30: $C_{30}$ x 15, $D_{15}$ x 9, $C_5\times S_3$ x 7, $C_3\times D_5$ x 6

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

Order 25: $C_5^2$ x 5

Order 24: $C_3:D_4$, $S_4$

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

Order 18: $C_3\times S_3$ x 6, $C_3\times C_6$, $C_3:S_3$

Order 15: $C_{15}$ x 32

Order 12: $D_6$ x 2, $C_2\times C_6$, $A_4$, $C_3:C_4$

Order 10: $C_{10}$ x 5, $D_5$ x 4

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

Order 8: $D_4$

Order 6: $C_6$ x 4, $S_3$ x 4

Order 5: $C_5$ x 5

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

Order 3: $C_3$ x 4

Order 2: $C_2$ x 2

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 81000: $C_{15}^3.S_4$ ($C_1$)

Order 40500: $C_{15}^3.A_4$ ($C_2$)

Order 13500: $C_{15}^3.C_2^2$ ($S_3$)

Order 3375: $C_{15}^3$ ($S_4$)

Order 125: $C_5^3$ ($C_3^3:S_4$)

Order 27: $C_3^3$ ($C_5^3:S_4$)

Order 1: $C_1$ ($C_{15}^3.S_4$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 81000: $C_{15}^3.S_4$ ($C_1$)

Order 40500: $C_{15}^3.A_4$ ($C_2$)

Order 13500: $C_{15}^3.C_2^2$ ($S_3$)

Order 3375: $C_{15}^3$ ($S_4$)

Order 125: $C_5^3$ ($C_3^3:S_4$)

Order 27: $C_3^3$ ($C_5^3:S_4$)

Order 1: $C_1$ ($C_{15}^3.S_4$)

Series

Derived series

$C_{15}^3.S_4$

$\rhd$

$C_{15}^3.A_4$

$\rhd$

$C_{15}^3.C_2^2$

$\rhd$

$C_{15}^3$

$\rhd$

$C_1$

Chief series

$C_{15}^3.S_4$

$\rhd$

$C_{15}^3.A_4$

$\rhd$

$C_{15}^3.C_2^2$

$\rhd$

$C_{15}^3$

$\rhd$

$C_5^3$

$\rhd$

$C_1$

Lower central series

$C_{15}^3.S_4$

$\rhd$

$C_{15}^3.A_4$

Upper central series

$C_1$

Supergroups

This group is a maximal subgroup of 5 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 $348 \times 348$ 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 $71 \times 71$ rational character table.