Abstract group 1440.5171: $(C_6\times C_{30}).D_4$

• Label: 1440.5171
• Order: \( 2^{5} \cdot 3^{2} \cdot 5 \)
• Exponent: \( 2^{3} \cdot 3 \cdot 5 \)
• Nilpotent: no
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{3} \)
• $\card{Z(G)}$: \( 2 \)
• $\card{\Aut(G)}$: \( 2^{9} \cdot 3^{2} \cdot 5 \)
• $\card{\mathrm{Out}(G)}$: \( 2^{5} \)
• Perm deg.: $27$
• Trans deg.: $240$
• Rank: $3$

Group information

Description:	$(C_6\times C_{30}).D_4$
Order:	\(1440\)\(\medspace = 2^{5} \cdot 3^{2} \cdot 5 \)
Exponent:	\(120\)\(\medspace = 2^{3} \cdot 3 \cdot 5 \)
Automorphism group:	$(C_3\times C_{30}).C_2^6.C_2^2$, of order \(23040\)\(\medspace = 2^{9} \cdot 3^{2} \cdot 5 \)
Composition factors:	$C_2$ x 5, $C_3$ x 2, $C_5$
Derived length:	$3$

This group is nonabelian and monomial (hence solvable).

Group statistics

Order	1	2	3	4	5	6	8	10	12	15	20	30	60
Elements	1	63	8	120	4	144	360	12	168	32	240	96	192	1440
Conjugacy classes	1	3	2	5	2	6	2	6	5	6	12	18	16	84
Divisions	1	3	2	5	1	6	2	2	3	2	4	5	2	38
Autjugacy classes	1	3	2	4	1	5	1	2	2	2	3	4	1	31

Dimension	1	2	4	8	16	32
Irr. complex chars.	8	18	49	9	0	0	84
Irr. rational chars.	8	2	13	5	7	3	38

Minimal presentations

Permutation degree:	$27$
Transitive degree:	$240$
Rank:	$3$
Inequivalent generating triples:	$40320$

Minimal degrees of faithful linear representations

	Over $\mathbb{C}$	Over $\mathbb{R}$	Over $\mathbb{Q}$
Irreducible	4	8	32
Arbitrary	4	8	16

Constructions

Presentation:	${\langle a, b, c, d \mid a^{2}=c^{12}=d^{30}=1, b^{2}=c^{6}, b^{a}=bc^{6}, c^{a}= \!\cdots\! \rangle}$
Permutation group:	Degree $27$ $\langle(1,2)(3,9)(4,12)(5,7)(6,10)(8,14)(11,16)(13,15)(18,19)(20,21)(25,26), (1,3,5,11) \!\cdots\! \rangle$
Direct product:	not isomorphic to a non-trivial direct product
Semidirect product:	$C_5$ $\,\rtimes\,$ $(C_6^2.D_4)$	$C_3^2$ $\,\rtimes\,$ $(C_{20}.D_4)$	$(C_2^2.S_3^2)$ $\,\rtimes\,$ $D_5$	$(C_3^2:Q_8\times C_{10})$ $\,\rtimes\,$ $C_2$	all 8
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Non-split product:	$(D_{30}.D_6)$ . $C_2$	$(C_6.D_6)$ . $D_{10}$	$(C_6\times C_{30})$ . $D_4$	$C_6^2$ . $(C_5:D_4)$	all 20

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

Homology

Abelianization:	$C_{2}^{3} $
Schur multiplier:	$C_{2}^{2}$
Commutator length:	$1$

Subgroups

There are 1888 subgroups in 198 conjugacy classes, 35 normal (25 characteristic).

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

Special subgroups

Center:	$Z \simeq$ $C_2$	$G/Z \simeq$ $S_3^2:D_{10}$
Commutator:	$G' \simeq$ $C_3^2:C_{20}$	$G/G' \simeq$ $C_2^3$
Frattini:	$\Phi \simeq$ $C_2$	$G/\Phi \simeq$ $S_3^2:D_{10}$
Fitting:	$\operatorname{Fit} \simeq$ $C_6\times C_{30}$	$G/\operatorname{Fit} \simeq$ $D_4$
Radical:	$R \simeq$ $(C_6\times C_{30}).D_4$	$G/R \simeq$ $C_1$
Socle:	$\operatorname{soc} \simeq$ $C_3\times C_{30}$	$G/\operatorname{soc} \simeq$ $C_2\times D_4$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $Q_{16}:C_2$
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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Classes of subgroups up to conjugation

Order 1440: $(C_6\times C_{30}).D_4$

Order 720: $(C_3\times C_{15}):Q_{16}$ x 2, $C_{10}.\SOPlus(4,2)$ x 2, $D_{30}.D_6$, $C_3^2:Q_8\times C_{10}$, $(C_3\times C_{15}):\OD_{16}$

Order 360: $C_{30}.D_6$ x 3, $(C_3\times C_{15}):C_8$ x 2, $C_6:C_{60}$, $C_{30}.D_6$, $D_{30}:S_3$, $C_{30}.D_6$, $C_{30}.D_6$, $D_{30}:C_6$

Order 288: $C_6^2.D_4$

Order 240: $C_{30}:Q_8$, $D_{10}.D_6$

Order 180: $C_3^2:C_{20}$ x 2, $C_3:C_{60}$ x 2, $C_6\times C_{30}$, $C_3\times D_{30}$, $C_{15}:C_{12}$

Order 160: $C_{20}.D_4$

Order 144: $C_3^2:Q_{16}$ x 2, $C_3^2:\SD_{16}$ x 2, $C_2^2.S_3^2$, $D_6.D_6$, $C_3^2:\OD_{16}$

Order 120: $C_{15}:Q_8$ x 4, $C_6:C_{20}$ x 3, $C_6.D_{10}$, $C_2\times C_{60}$, $C_{15}:D_4$, $C_{15}:D_4$, $C_{15}:Q_8$, $C_3:D_{20}$, $D_{15}:C_4$

Order 90: $C_3\times C_{30}$ x 2, $C_3\times D_{15}$

Order 80: $C_5:Q_{16}$ x 2, $Q_8:D_5$ x 2, $Q_8\times C_{10}$, $D_{20}:C_2$, $C_5:\OD_{16}$

Order 72: $C_3^2:Q_8$ x 4, $C_3^2:C_8$ x 2, $C_6.D_6$, $C_6\wr C_2$, $C_6:C_{12}$, $D_6:S_3$, $C_6.D_6$

Order 60: $C_3:C_{20}$ x 6, $C_{60}$ x 2, $C_2\times C_{30}$ x 2, $C_{15}:C_4$, $C_5:C_{12}$, $D_{30}$, $C_3\times D_{10}$

Order 48: $D_4:S_3$, $C_6:Q_8$

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

Order 40: $C_5\times Q_8$ x 3, $C_2\times C_{20}$ x 2, $C_5:C_8$ x 2, $C_5:D_4$, $D_{20}$, $C_4\times D_5$, $C_5:Q_8$

Order 36: $C_3:C_{12}$ x 3, $C_3^2:C_4$ x 2, $C_6^2$, $C_6\times S_3$

Order 32: $Q_{16}:C_2$

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

Order 24: $C_3:Q_8$ x 5, $C_6:C_4$ x 4, $C_3:D_4$ x 2, $C_2\times C_{12}$, $C_4\times S_3$, $C_3\times D_4$

Order 20: $C_{20}$ x 4, $C_2\times C_{10}$, $D_{10}$, $C_5:C_4$

Order 18: $C_3\times C_6$ x 2, $C_3\times S_3$

Order 16: $Q_{16}$ x 2, $\SD_{16}$ x 2, $\OD_{16}$, $D_4:C_2$, $C_2\times Q_8$

Order 15: $C_{15}$ x 2

Order 12: $C_3:C_4$ x 7, $C_2\times C_6$ x 3, $C_{12}$ x 3, $D_6$

Order 10: $C_{10}$ x 2, $D_5$

Order 9: $C_3^2$

Order 8: $Q_8$ x 4, $C_2\times C_4$ x 3, $D_4$ x 2, $C_8$ x 2

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

Order 5: $C_5$

Order 4: $C_4$ x 5, $C_2^2$ x 2

Order 3: $C_3$ x 2

Order 2: $C_2$ x 3

Order 1: $C_1$

Classes of subgroups up to automorphism

Order 1440: $(C_6\times C_{30}).D_4$

Order 720: $D_{30}.D_6$, $C_3^2:Q_8\times C_{10}$, $(C_3\times C_{15}):\OD_{16}$, $(C_3\times C_{15}):Q_{16}$, $C_{10}.\SOPlus(4,2)$

Order 360: $C_{30}.D_6$ x 2, $C_6:C_{60}$, $C_{30}.D_6$, $D_{30}:S_3$, $C_{30}.D_6$, $(C_3\times C_{15}):C_8$, $C_{30}.D_6$, $D_{30}:C_6$

Order 288: $C_6^2.D_4$

Order 240: $C_{30}:Q_8$, $D_{10}.D_6$

Order 180: $C_3^2:C_{20}$ x 2, $C_6\times C_{30}$, $C_3\times D_{30}$, $C_{15}:C_{12}$, $C_3:C_{60}$

Order 160: $C_{20}.D_4$

Order 144: $C_2^2.S_3^2$, $D_6.D_6$, $C_3^2:\OD_{16}$, $C_3^2:Q_{16}$, $C_3^2:\SD_{16}$

Order 120: $C_6:C_{20}$ x 3, $C_{15}:Q_8$ x 2, $C_6.D_{10}$, $C_2\times C_{60}$, $C_{15}:D_4$, $C_{15}:D_4$, $C_{15}:Q_8$, $C_3:D_{20}$, $D_{15}:C_4$

Order 90: $C_3\times C_{30}$ x 2, $C_3\times D_{15}$

Order 80: $Q_8\times C_{10}$, $D_{20}:C_2$, $C_5:Q_{16}$, $Q_8:D_5$, $C_5:\OD_{16}$

Order 72: $C_3^2:Q_8$ x 3, $C_6.D_6$, $C_6\wr C_2$, $C_6:C_{12}$, $D_6:S_3$, $C_6.D_6$, $C_3^2:C_8$

Order 60: $C_3:C_{20}$ x 5, $C_2\times C_{30}$ x 2, $C_{60}$, $C_{15}:C_4$, $C_5:C_{12}$, $D_{30}$, $C_3\times D_{10}$

Order 48: $D_4:S_3$, $C_6:Q_8$

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

Order 40: $C_2\times C_{20}$ x 2, $C_5\times Q_8$ x 2, $C_5:D_4$, $D_{20}$, $C_4\times D_5$, $C_5:Q_8$, $C_5:C_8$

Order 36: $C_3^2:C_4$ x 2, $C_3:C_{12}$ x 2, $C_6^2$, $C_6\times S_3$

Order 32: $Q_{16}:C_2$

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

Order 24: $C_6:C_4$ x 4, $C_3:Q_8$ x 3, $C_3:D_4$ x 2, $C_2\times C_{12}$, $C_4\times S_3$, $C_3\times D_4$

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

Order 18: $C_3\times C_6$ x 2, $C_3\times S_3$

Order 16: $Q_{16}$, $\SD_{16}$, $\OD_{16}$, $D_4:C_2$, $C_2\times Q_8$

Order 15: $C_{15}$ x 2

Order 12: $C_3:C_4$ x 6, $C_2\times C_6$ x 3, $C_{12}$ x 2, $D_6$

Order 10: $C_{10}$ x 2, $D_5$

Order 9: $C_3^2$

Order 8: $Q_8$ x 3, $C_2\times C_4$ x 3, $D_4$ x 2, $C_8$

Order 6: $C_6$ x 5, $S_3$

Order 5: $C_5$

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

Order 3: $C_3$ x 2

Order 2: $C_2$ x 3

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 1440: $(C_6\times C_{30}).D_4$ ($C_1$)

Order 720: $(C_3\times C_{15}):Q_{16}$ ($C_2$) x 2, $C_{10}.\SOPlus(4,2)$ ($C_2$) x 2, $D_{30}.D_6$ ($C_2$), $C_3^2:Q_8\times C_{10}$ ($C_2$), $(C_3\times C_{15}):\OD_{16}$ ($C_2$)

Order 360: $C_{30}.D_6$ ($C_2^2$) x 2, $(C_3\times C_{15}):C_8$ ($C_2^2$) x 2, $C_{30}.D_6$ ($C_2^2$), $D_{30}:S_3$ ($C_2^2$), $C_{30}.D_6$ ($C_2^2$)

Order 180: $C_6\times C_{30}$ ($D_4$), $C_3^2:C_{20}$ ($C_2^3$), $C_3^2:C_{20}$ ($D_4$)

Order 144: $C_2^2.S_3^2$ ($D_5$)

Order 90: $C_3\times C_{30}$ ($C_2\times D_4$)

Order 72: $C_3^2:Q_8$ ($D_{10}$) x 2, $C_6.D_6$ ($D_{10}$)

Order 45: $C_3\times C_{15}$ ($Q_{16}:C_2$)

Order 36: $C_3^2:C_4$ ($C_5:D_4$), $C_3^2:C_4$ ($C_2\times D_{10}$), $C_6^2$ ($C_5:D_4$)

Order 20: $C_2\times C_{10}$ ($\SOPlus(4,2)$)

Order 18: $C_3\times C_6$ ($C_{10}:D_4$)

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

Order 9: $C_3^2$ ($C_{20}.D_4$)

Order 5: $C_5$ ($C_6^2.D_4$)

Order 4: $C_2^2$ ($S_3^2:D_5$)

Order 2: $C_2$ ($S_3^2:D_{10}$)

Order 1: $C_1$ ($(C_6\times C_{30}).D_4$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 1440: $(C_6\times C_{30}).D_4$ ($C_1$)

Order 720: $D_{30}.D_6$ ($C_2$), $C_3^2:Q_8\times C_{10}$ ($C_2$), $(C_3\times C_{15}):\OD_{16}$ ($C_2$), $(C_3\times C_{15}):Q_{16}$ ($C_2$), $C_{10}.\SOPlus(4,2)$ ($C_2$)

Order 360: $C_{30}.D_6$ ($C_2^2$), $D_{30}:S_3$ ($C_2^2$), $C_{30}.D_6$ ($C_2^2$), $(C_3\times C_{15}):C_8$ ($C_2^2$), $C_{30}.D_6$ ($C_2^2$)

Order 180: $C_6\times C_{30}$ ($D_4$), $C_3^2:C_{20}$ ($C_2^3$), $C_3^2:C_{20}$ ($D_4$)

Order 144: $C_2^2.S_3^2$ ($D_5$)

Order 90: $C_3\times C_{30}$ ($C_2\times D_4$)

Order 72: $C_6.D_6$ ($D_{10}$), $C_3^2:Q_8$ ($D_{10}$)

Order 45: $C_3\times C_{15}$ ($Q_{16}:C_2$)

Order 36: $C_3^2:C_4$ ($C_5:D_4$), $C_3^2:C_4$ ($C_2\times D_{10}$), $C_6^2$ ($C_5:D_4$)

Order 20: $C_2\times C_{10}$ ($\SOPlus(4,2)$)

Order 18: $C_3\times C_6$ ($C_{10}:D_4$)

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

Order 9: $C_3^2$ ($C_{20}.D_4$)

Order 5: $C_5$ ($C_6^2.D_4$)

Order 4: $C_2^2$ ($S_3^2:D_5$)

Order 2: $C_2$ ($S_3^2:D_{10}$)

Order 1: $C_1$ ($(C_6\times C_{30}).D_4$)

Series

Derived series

$(C_6\times C_{30}).D_4$

$\rhd$

$C_3^2:C_{20}$

$\rhd$

$C_3^2$

$\rhd$

$C_1$

Chief series

$(C_6\times C_{30}).D_4$

$\rhd$

$C_{10}.\SOPlus(4,2)$

$\rhd$

$D_{30}:S_3$

$\rhd$

$C_3^2:C_{20}$

$\rhd$

$C_3\times C_{30}$

$\rhd$

$C_3\times C_{15}$

$\rhd$

$C_5$

$\rhd$

$C_1$

Lower central series

$(C_6\times C_{30}).D_4$

$\rhd$

$C_3^2:C_{20}$

$\rhd$

$C_3\times C_{30}$

$\rhd$

$C_3\times C_{15}$

Upper central series

$C_1$

$\lhd$

$C_2$

$\lhd$

$C_2^2$

Character theory

Complex character table

See the $84 \times 84$ character table. Alternatively, you may search for characters of this group with desired properties.

Rational character table

See the $38 \times 38$ rational character table.