Abstract group 480.51: $C_{12}.D_{20}$

• Label: 480.51
• Order: \( 2^{5} \cdot 3 \cdot 5 \)
• Exponent: \( 2^{3} \cdot 3 \cdot 5 \)
• Nilpotent: no
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{3} \)
• $\card{Z(G)}$: \( 2^{2} \)
• $\card{\Aut(G)}$: \( 2^{9} \cdot 3 \cdot 5 \)
• $\card{\mathrm{Out}(G)}$: \( 2^{6} \)
• Perm deg.: $20$
• Trans deg.: $480$
• Rank: $2$

Group information

Description:	$C_{12}.D_{20}$
Order:	\(480\)\(\medspace = 2^{5} \cdot 3 \cdot 5 \)
Exponent:	\(120\)\(\medspace = 2^{3} \cdot 3 \cdot 5 \)
Automorphism group:	$C_{15}:(C_2^4\times C_4\times D_4)$, of order \(7680\)\(\medspace = 2^{9} \cdot 3 \cdot 5 \)
Composition factors:	$C_2$ x 5, $C_3$, $C_5$
Derived length:	$2$

This group is nonabelian, supersolvable (hence solvable and monomial), hyperelementary for $p = 2$, and metabelian.

Group statistics

Order	1	2	3	4	5	6	8	10	12	15	20	24	30	60
Elements	1	3	2	148	4	6	40	12	8	8	112	80	24	32	480
Conjugacy classes	1	3	1	6	2	3	4	6	4	2	12	8	6	8	66
Divisions	1	3	1	5	1	3	1	3	2	1	3	1	3	2	30
Autjugacy classes	1	3	1	4	1	3	1	3	2	1	3	1	3	2	29

Dimension	1	2	4	8	16
Irr. complex chars.	8	38	20	0	0	66
Irr. rational chars.	4	6	7	11	2	30

Minimal presentations

Permutation degree:	$20$
Transitive degree:	$480$
Rank:	$2$
Inequivalent generating pairs:	$6$

Minimal degrees of faithful linear representations

	Over $\mathbb{C}$	Over $\mathbb{R}$	Over $\mathbb{Q}$
Irreducible	none	none	none
Arbitrary	4	6	12

Constructions

Presentation:	$\langle a, b, c \mid b^{20}=c^{12}=[a,c]=1, a^{2}=c^{3}, b^{a}=b^{19}c^{9}, c^{b}=c^{11} \rangle$
Permutation group:	Degree $20$ $\langle(1,2,3,5,4,6,7,8)(13,14)(15,16)(17,18)(19,20), (2,5)(3,7)(6,8)(10,11)(17,19,20,18) \!\cdots\! \rangle$
Matrix group:	$\left\langle \left(\begin{array}{rr} 1 & 20 \\ 20 & 11 \end{array}\right), \left(\begin{array}{rr} 23 & 0 \\ 25 & 13 \end{array}\right), \left(\begin{array}{rr} 3 & 8 \\ 20 & 7 \end{array}\right), \left(\begin{array}{rr} 16 & 15 \\ 15 & 1 \end{array}\right), \left(\begin{array}{rr} 11 & 0 \\ 0 & 11 \end{array}\right), \left(\begin{array}{rr} 1 & 6 \\ 0 & 1 \end{array}\right), \left(\begin{array}{rr} 19 & 0 \\ 0 & 19 \end{array}\right) \right\rangle \subseteq \GL_{2}(\Z/30\Z)$
Direct product:	not isomorphic to a non-trivial direct product
Semidirect product:	$(C_{15}:Q_8)$ $\,\rtimes\,$ $C_4$ (2)	$C_{15}$ $\,\rtimes\,$ $(Q_8:C_4)$	$C_5$ $\,\rtimes\,$ $(C_6.Q_{16})$	$C_3$ $\,\rtimes\,$ $(C_4.D_{20})$	more information
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Non-split product:	$C_{60}$ . $D_4$	$C_{30}$ . $Q_{16}$	$C_{12}$ . $D_{20}$	$C_{30}$ . $\SD_{16}$	all 33

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

Homology

Abelianization:	$C_{2} \times C_{4} $
Schur multiplier:	$C_{2}$
Commutator length:	$1$

Subgroups

There are 412 subgroups in 84 conjugacy classes, 40 normal (38 characteristic).

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

Special subgroups

Center:	$Z \simeq$ $C_2^2$	$G/Z \simeq$ $C_5:D_{12}$
Commutator:	$G' \simeq$ $C_{60}$	$G/G' \simeq$ $C_2\times C_4$
Frattini:	$\Phi \simeq$ $C_2\times C_4$	$G/\Phi \simeq$ $S_3\times D_5$
Fitting:	$\operatorname{Fit} \simeq$ $C_2\times C_{60}$	$G/\operatorname{Fit} \simeq$ $C_2^2$
Radical:	$R \simeq$ $C_{12}.D_{20}$	$G/R \simeq$ $C_1$
Socle:	$\operatorname{soc} \simeq$ $C_2\times C_{30}$	$G/\operatorname{soc} \simeq$ $D_4$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $Q_8:C_4$
3-Sylow subgroup:	$P_{ 3 } \simeq$ $C_3$
5-Sylow subgroup:	$P_{ 5 } \simeq$ $C_5$

Hi

Subgroup diagram and profile

Classes of subgroups up to conjugation

Classes of subgroups up to automorphism

Normal subgroups

Normal subgroups up to automorphism

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Subgroup information

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Classes of subgroups up to conjugation

Order 480: $C_{12}.D_{20}$

Order 240: $C_{12}:C_{20}$, $C_{10}:C_{24}$, $C_{30}:Q_8$

Order 160: $C_4.D_{20}$

Order 120: $C_{15}:Q_8$ x 3, $C_2\times C_{60}$, $C_{30}:C_4$, $C_6:C_{20}$, $C_5:C_{24}$

Order 96: $C_6.Q_{16}$

Order 80: $C_{10}:C_8$, $C_{10}:Q_8$, $C_4:C_{20}$

Order 60: $C_{60}$ x 2, $C_{15}:C_4$ x 2, $C_2\times C_{30}$, $C_3:C_{20}$

Order 48: $C_6:Q_8$, $C_2\times C_{24}$, $C_{12}:C_4$

Order 40: $C_5:Q_8$ x 3, $C_2\times C_{20}$ x 2, $C_{10}:C_4$, $C_5:C_8$

Order 32: $Q_8:C_4$

Order 30: $C_{30}$ x 3

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

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

Order 16: $C_2\times C_8$, $C_4:C_4$, $C_2\times Q_8$

Order 15: $C_{15}$

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

Order 10: $C_{10}$ x 3

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

Order 6: $C_6$ x 3

Order 5: $C_5$

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

Order 3: $C_3$

Order 2: $C_2$ x 3

Order 1: $C_1$

Classes of subgroups up to automorphism

Order 480: $C_{12}.D_{20}$

Order 240: $C_{12}:C_{20}$, $C_{10}:C_{24}$, $C_{30}:Q_8$

Order 160: $C_4.D_{20}$

Order 120: $C_{15}:Q_8$ x 2, $C_2\times C_{60}$, $C_{30}:C_4$, $C_6:C_{20}$, $C_5:C_{24}$

Order 96: $C_6.Q_{16}$

Order 80: $C_{10}:C_8$, $C_{10}:Q_8$, $C_4:C_{20}$

Order 60: $C_{60}$ x 2, $C_{15}:C_4$, $C_2\times C_{30}$, $C_3:C_{20}$

Order 48: $C_6:Q_8$, $C_2\times C_{24}$, $C_{12}:C_4$

Order 40: $C_2\times C_{20}$ x 2, $C_5:Q_8$ x 2, $C_{10}:C_4$, $C_5:C_8$

Order 32: $Q_8:C_4$

Order 30: $C_{30}$ x 3

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

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

Order 16: $C_2\times C_8$, $C_4:C_4$, $C_2\times Q_8$

Order 15: $C_{15}$

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

Order 10: $C_{10}$ x 3

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

Order 6: $C_6$ x 3

Order 5: $C_5$

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

Order 3: $C_3$

Order 2: $C_2$ x 3

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 480: $C_{12}.D_{20}$ ($C_1$)

Order 240: $C_{12}:C_{20}$ ($C_2$), $C_{10}:C_{24}$ ($C_2$), $C_{30}:Q_8$ ($C_2$)

Order 120: $C_{15}:Q_8$ ($C_4$) x 2, $C_2\times C_{60}$ ($C_2^2$)

Order 80: $C_{10}:C_8$ ($S_3$)

Order 60: $C_{60}$ ($D_4$), $C_{60}$ ($C_2\times C_4$), $C_2\times C_{30}$ ($D_4$)

Order 48: $C_{12}:C_4$ ($D_5$)

Order 40: $C_2\times C_{20}$ ($D_6$)

Order 30: $C_{30}$ ($Q_{16}$), $C_{30}$ ($\SD_{16}$), $C_{30}$ ($C_2^2:C_4$)

Order 24: $C_2\times C_{12}$ ($D_{10}$)

Order 20: $C_2\times C_{10}$ ($D_{12}$), $C_{20}$ ($C_3:D_4$), $C_{20}$ ($C_4\times S_3$)

Order 15: $C_{15}$ ($Q_8:C_4$)

Order 12: $C_2\times C_6$ ($C_5:D_4$), $C_{12}$ ($D_{20}$), $C_{12}$ ($C_4\times D_5$)

Order 10: $C_{10}$ ($C_3:Q_{16}$), $C_{10}$ ($C_{24}:C_2$), $C_{10}$ ($D_6:C_4$)

Order 8: $C_2\times C_4$ ($S_3\times D_5$)

Order 6: $C_6$ ($C_5:Q_{16}$), $C_6$ ($C_5:\SD_{16}$), $C_6$ ($D_{10}:C_4$)

Order 5: $C_5$ ($C_6.Q_{16}$)

Order 4: $C_2^2$ ($C_5:D_{12}$), $C_4$ ($C_3:D_{20}$), $C_4$ ($D_{15}:C_4$)

Order 3: $C_3$ ($C_4.D_{20}$)

Order 2: $C_2$ ($D_{30}:C_4$), $C_2$ ($C_{15}:Q_{16}$), $C_2$ ($C_{15}:\SD_{16}$)

Order 1: $C_1$ ($C_{12}.D_{20}$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 480: $C_{12}.D_{20}$ ($C_1$)

Order 240: $C_{12}:C_{20}$ ($C_2$), $C_{10}:C_{24}$ ($C_2$), $C_{30}:Q_8$ ($C_2$)

Order 120: $C_2\times C_{60}$ ($C_2^2$), $C_{15}:Q_8$ ($C_4$)

Order 80: $C_{10}:C_8$ ($S_3$)

Order 60: $C_{60}$ ($D_4$), $C_{60}$ ($C_2\times C_4$), $C_2\times C_{30}$ ($D_4$)

Order 48: $C_{12}:C_4$ ($D_5$)

Order 40: $C_2\times C_{20}$ ($D_6$)

Order 30: $C_{30}$ ($Q_{16}$), $C_{30}$ ($\SD_{16}$), $C_{30}$ ($C_2^2:C_4$)

Order 24: $C_2\times C_{12}$ ($D_{10}$)

Order 20: $C_2\times C_{10}$ ($D_{12}$), $C_{20}$ ($C_3:D_4$), $C_{20}$ ($C_4\times S_3$)

Order 15: $C_{15}$ ($Q_8:C_4$)

Order 12: $C_2\times C_6$ ($C_5:D_4$), $C_{12}$ ($D_{20}$), $C_{12}$ ($C_4\times D_5$)

Order 10: $C_{10}$ ($C_3:Q_{16}$), $C_{10}$ ($C_{24}:C_2$), $C_{10}$ ($D_6:C_4$)

Order 8: $C_2\times C_4$ ($S_3\times D_5$)

Order 6: $C_6$ ($C_5:Q_{16}$), $C_6$ ($C_5:\SD_{16}$), $C_6$ ($D_{10}:C_4$)

Order 5: $C_5$ ($C_6.Q_{16}$)

Order 4: $C_2^2$ ($C_5:D_{12}$), $C_4$ ($C_3:D_{20}$), $C_4$ ($D_{15}:C_4$)

Order 3: $C_3$ ($C_4.D_{20}$)

Order 2: $C_2$ ($D_{30}:C_4$), $C_2$ ($C_{15}:Q_{16}$), $C_2$ ($C_{15}:\SD_{16}$)

Order 1: $C_1$ ($C_{12}.D_{20}$)

Series

Derived series

$C_{12}.D_{20}$

$\rhd$

$C_{60}$

$\rhd$

$C_1$

Chief series

$C_{12}.D_{20}$

$\rhd$

$C_{10}:C_{24}$

$\rhd$

$C_2\times C_{60}$

$\rhd$

$C_{60}$

$\rhd$

$C_{12}$

$\rhd$

$C_6$

$\rhd$

$C_3$

$\rhd$

$C_1$

Lower central series

$C_{12}.D_{20}$

$\rhd$

$C_{60}$

$\rhd$

$C_{30}$

$\rhd$

$C_{15}$

Upper central series

$C_1$

$\lhd$

$C_2^2$

$\lhd$

$C_2\times C_4$

Supergroups

This group is a maximal subgroup of 131 larger groups in the database.

This group is a maximal quotient of 11 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 $30 \times 30$ rational character table.