Abstract group 540.76: $C_3^2:D_{30}$

• Label: 540.76
• Order: \( 2^{2} \cdot 3^{3} \cdot 5 \)
• Exponent: \( 2 \cdot 3 \cdot 5 \)
• Nilpotent: no
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{2} \)
• $\card{Z(G)}$: \( 2 \cdot 3 \)
• $\card{\Aut(G)}$: \( 2^{7} \cdot 3^{3} \cdot 5 \)
• $\card{\mathrm{Out}(G)}$: \( 2^{6} \cdot 3 \)
• Perm deg.: $16$
• Trans deg.: $90$
• Rank: $3$

Group information

Description:	$C_3^2:D_{30}$
Order:	\(540\)\(\medspace = 2^{2} \cdot 3^{3} \cdot 5 \)
Exponent:	\(30\)\(\medspace = 2 \cdot 3 \cdot 5 \)
Automorphism group:	$C_{30}:C_{12}:\GL(2,3)$, of order \(17280\)\(\medspace = 2^{7} \cdot 3^{3} \cdot 5 \)
Composition factors:	$C_2$ x 2, $C_3$ x 3, $C_5$
Derived length:	$3$

This group is nonabelian and supersolvable (hence solvable and monomial).

Group statistics

Order	1	2	3	5	6	10	15	30
Elements	1	91	26	4	206	4	104	104	540
Conjugacy classes	1	3	6	2	10	2	20	20	64
Divisions	1	3	5	1	7	1	5	5	28
Autjugacy classes	1	2	2	1	3	1	2	2	14

Dimension	1	2	3	4	6	8	24
Irr. complex chars.	4	44	8	0	8	0	0	64
Irr. rational chars.	4	8	0	2	4	8	2	28

Minimal presentations

Permutation degree:	$16$
Transitive degree:	$90$
Rank:	$3$
Inequivalent generating triples:	$3402$

Minimal degrees of faithful linear representations

	Over $\mathbb{C}$	Over $\mathbb{R}$	Over $\mathbb{Q}$
Irreducible	6	12	24
Arbitrary	5	8	10

Constructions

Presentation:	$\langle a, b, c, d \mid a^{2}=b^{3}=c^{3}=d^{30}=[a,c]=[b,c]=[c,d]=1, b^{a}=b^{2}, d^{a}=c^{2}d^{29}, d^{b}=cd \rangle$
Permutation group:	Degree $16$ $\langle(2,5)(3,7)(6,9)(11,12)(13,14)(15,16), (15,16), (1,2,5)(3,8,7)(4,6,9), (1,3,7)(2,4,5)(6,8,9), (10,11,13,14,12), (1,4,8)(2,6,7)(3,5,9)\rangle$
Direct product:	$C_2$ $\, \times\, $ $(C_3^2:D_{15})$
Semidirect product:	$C_3^2$ $\,\rtimes\,$ $D_{30}$ (4)	$\He_3$ $\,\rtimes\,$ $D_{10}$	$(C_3\times C_{30})$ $\,\rtimes\,$ $S_3$ (4)	$(C_3\times C_{15})$ $\,\rtimes\,$ $D_6$ (4)	all 10
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Non-split product:	$C_{30}$ . $(C_3:S_3)$	$C_{15}$ . $(C_6:S_3)$	$C_6$ . $(C_3:D_{15})$	$C_3$ . $(C_3:D_{30})$	more information

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

Homology

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

Subgroups

There are 850 subgroups in 110 conjugacy classes, 31 normal (13 characteristic).

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

Special subgroups

Center:	$Z \simeq$ $C_6$	$G/Z \simeq$ $C_3:D_{15}$
Commutator:	$G' \simeq$ $C_5\times \He_3$	$G/G' \simeq$ $C_2^2$
Frattini:	$\Phi \simeq$ $C_3$	$G/\Phi \simeq$ $C_3:D_{30}$
Fitting:	$\operatorname{Fit} \simeq$ $C_{10}\times \He_3$	$G/\operatorname{Fit} \simeq$ $C_2$
Radical:	$R \simeq$ $C_3^2:D_{30}$	$G/R \simeq$ $C_1$
Socle:	$\operatorname{soc} \simeq$ $C_{30}$	$G/\operatorname{soc} \simeq$ $C_3:S_3$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $C_2^2$
3-Sylow subgroup:	$P_{ 3 } \simeq$ $\He_3$
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 540: $C_3^2:D_{30}$

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

Order 180: $C_3\times D_{30}$ x 4

Order 135: $C_5\times \He_3$

Order 108: $C_3^2:D_6$

Order 90: $C_3\times D_{15}$ x 8, $C_3\times C_{30}$ x 4

Order 60: $D_{30}$ x 4, $C_3\times D_{10}$

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

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

Order 36: $C_6\times S_3$ x 4

Order 30: $D_{15}$ x 8, $C_{30}$ x 5, $C_3\times D_5$ x 2

Order 27: $\He_3$

Order 20: $D_{10}$

Order 18: $C_3\times S_3$ x 8, $C_3\times C_6$ x 4

Order 15: $C_{15}$ x 5

Order 12: $D_6$ x 4, $C_2\times C_6$

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

Order 9: $C_3^2$ x 4

Order 6: $S_3$ x 8, $C_6$ x 7

Order 5: $C_5$

Order 4: $C_2^2$

Order 3: $C_3$ x 5

Order 2: $C_2$ x 3

Order 1: $C_1$

Classes of subgroups up to automorphism

Order 540: $C_3^2:D_{30}$

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

Order 180: $C_3\times D_{30}$

Order 135: $C_5\times \He_3$

Order 108: $C_3^2:D_6$

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

Order 60: $D_{30}$, $C_3\times D_{10}$

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

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

Order 36: $C_6\times S_3$

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

Order 27: $\He_3$

Order 20: $D_{10}$

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

Order 15: $C_{15}$ x 2

Order 12: $C_2\times C_6$, $D_6$

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

Order 9: $C_3^2$

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

Order 5: $C_5$

Order 4: $C_2^2$

Order 3: $C_3$ x 2

Order 2: $C_2$ x 2

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 540: $C_3^2:D_{30}$ ($C_1$)

Order 270: $C_3^2:D_{15}$ ($C_2$) x 2, $C_{10}\times \He_3$ ($C_2$)

Order 135: $C_5\times \He_3$ ($C_2^2$)

Order 90: $C_3\times C_{30}$ ($S_3$) x 4

Order 54: $C_2\times \He_3$ ($D_5$)

Order 45: $C_3\times C_{15}$ ($D_6$) x 4

Order 30: $C_{30}$ ($C_3:S_3$)

Order 27: $\He_3$ ($D_{10}$)

Order 18: $C_3\times C_6$ ($D_{15}$) x 4

Order 15: $C_{15}$ ($C_6:S_3$)

Order 10: $C_{10}$ ($C_3^2:S_3$)

Order 9: $C_3^2$ ($D_{30}$) x 4

Order 6: $C_6$ ($C_3:D_{15}$)

Order 5: $C_5$ ($C_3^2:D_6$)

Order 3: $C_3$ ($C_3:D_{30}$)

Order 2: $C_2$ ($C_3^2:D_{15}$)

Order 1: $C_1$ ($C_3^2:D_{30}$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 540: $C_3^2:D_{30}$ ($C_1$)

Order 270: $C_{10}\times \He_3$ ($C_2$), $C_3^2:D_{15}$ ($C_2$)

Order 135: $C_5\times \He_3$ ($C_2^2$)

Order 90: $C_3\times C_{30}$ ($S_3$)

Order 54: $C_2\times \He_3$ ($D_5$)

Order 45: $C_3\times C_{15}$ ($D_6$)

Order 30: $C_{30}$ ($C_3:S_3$)

Order 27: $\He_3$ ($D_{10}$)

Order 18: $C_3\times C_6$ ($D_{15}$)

Order 15: $C_{15}$ ($C_6:S_3$)

Order 10: $C_{10}$ ($C_3^2:S_3$)

Order 9: $C_3^2$ ($D_{30}$)

Order 6: $C_6$ ($C_3:D_{15}$)

Order 5: $C_5$ ($C_3^2:D_6$)

Order 3: $C_3$ ($C_3:D_{30}$)

Order 2: $C_2$ ($C_3^2:D_{15}$)

Order 1: $C_1$ ($C_3^2:D_{30}$)

Series

Derived series

$C_3^2:D_{30}$

$\rhd$

$C_5\times \He_3$

$\rhd$

$C_3$

$\rhd$

$C_1$

Chief series

$C_3^2:D_{30}$

$\rhd$

$C_3^2:D_{15}$

$\rhd$

$C_5\times \He_3$

$\rhd$

$C_3\times C_{15}$

$\rhd$

$C_{15}$

$\rhd$

$C_3$

$\rhd$

$C_1$

Lower central series

$C_3^2:D_{30}$

$\rhd$

$C_5\times \He_3$

Upper central series

$C_1$

$\lhd$

$C_6$

Supergroups

This group is a maximal subgroup of 20 larger groups in the database.

This group is a maximal quotient of 14 larger groups in the database.

Character theory

Complex character table

See the $64 \times 64$ character table. Alternatively, you may search for characters of this group with desired properties.

Rational character table

See the $28 \times 28$ rational character table.