Abstract group 360.148: $C_2\times C_3^2:C_{20}$

• Label: 360.148
• Order: \( 2^{3} \cdot 3^{2} \cdot 5 \)
• Exponent: \( 2^{2} \cdot 3 \cdot 5 \)
• Nilpotent: no
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{3} \cdot 5 \)
• $\card{Z(G)}$: \( 2 \cdot 5 \)
• $\card{\Aut(G)}$: \( 2^{7} \cdot 3^{2} \)
• $\card{\mathrm{Out}(G)}$: \( 2^{5} \)
• Perm deg.: $13$
• Trans deg.: $60$
• Rank: $2$

Group information

Description:	$C_2\times C_3^2:C_{20}$
Order:	\(360\)\(\medspace = 2^{3} \cdot 3^{2} \cdot 5 \)
Exponent:	\(60\)\(\medspace = 2^{2} \cdot 3 \cdot 5 \)
Automorphism group:	$C_4\times F_9:C_2^2$, of order \(1152\)\(\medspace = 2^{7} \cdot 3^{2} \)
Composition factors:	$C_2$ x 3, $C_3$ x 2, $C_5$
Derived length:	$2$

This group is nonabelian, monomial (hence solvable), metabelian, and an A-group.

Group statistics

Order	1	2	3	4	5	6	10	15	20	30
Elements	1	19	8	36	4	8	76	32	144	32	360
Conjugacy classes	1	3	2	4	4	2	12	8	16	8	60
Divisions	1	3	2	2	1	2	3	2	2	2	20
Autjugacy classes	1	3	1	1	1	1	3	1	1	1	14

Dimension	1	2	4	8	16
Irr. complex chars.	40	0	20	0	0	60
Irr. rational chars.	4	2	8	2	4	20

Minimal presentations

Permutation degree:	$13$
Transitive degree:	$60$
Rank:	$2$
Inequivalent generating pairs:	$36$

Minimal degrees of faithful linear representations

	Over $\mathbb{C}$	Over $\mathbb{R}$	Over $\mathbb{Q}$
Irreducible	4	8	16
Arbitrary	4	6	8

Constructions

Presentation:	$\langle a, b, c \mid a^{4}=b^{6}=c^{15}=[b,c]=1, b^{a}=b^{5}c^{5}, c^{a}=b^{2}c \rangle$
Permutation group:	Degree $13$ $\langle(1,2)(3,5,6,4), (12,13), (7,8,9,10,11), (3,6)(4,5), (1,3,6)(2,4,5), (2,5,4)\rangle$
Matrix group:	$\left\langle \left(\begin{array}{ll}\alpha^{36} & \alpha^{29} \\ \alpha^{23} & \alpha^{66} \\ \end{array}\right), \left(\begin{array}{ll}\alpha^{64} & 0 \\ \alpha^{51} & \alpha^{4} \\ \end{array}\right), \left(\begin{array}{ll}\alpha^{48} & 0 \\ 0 & \alpha^{48} \\ \end{array}\right), \left(\begin{array}{ll}\alpha^{2} & \alpha^{35} \\ \alpha^{29} & \alpha^{22} \\ \end{array}\right), \left(\begin{array}{ll}\alpha^{40} & 0 \\ 0 & \alpha^{40} \\ \end{array}\right), \left(\begin{array}{ll}1 & 0 \\ \alpha^{57} & \alpha^{40} \\ \end{array}\right) \right\rangle \subseteq \GL_{2}(\F_{81}) = \GL_{2}(\F_{3}[\alpha]/(\alpha^{4} + 2 \alpha^{3} + 2))$
Direct product:	$C_2$ $\, \times\, $ $C_5$ $\, \times\, $ $(C_3^2:C_4)$
Semidirect product:	$(C_{15}:S_3)$ $\,\rtimes\,$ $C_4$	$(C_3:S_3)$ $\,\rtimes\,$ $C_{20}$	$(C_3\times C_{30})$ $\,\rtimes\,$ $C_4$	$(C_3\times C_6)$ $\,\rtimes\,$ $C_{20}$	all 6
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Non-split product:	$(C_{30}:S_3)$ . $C_2$	$(C_6:S_3)$ . $C_{10}$	$(C_{15}:S_3)$ . $C_2^2$	$(C_3:S_3)$ . $(C_2\times C_{10})$	more information

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

Homology

Abelianization:	$C_{2} \times C_{20} \simeq C_{2} \times C_{4} \times C_{5}$
Schur multiplier:	$C_{6}$
Commutator length:	$1$

Subgroups

There are 216 subgroups in 52 conjugacy classes, 20 normal (16 characteristic).

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

Special subgroups

Center:	$Z \simeq$ $C_{10}$	$G/Z \simeq$ $C_3^2:C_4$
Commutator:	$G' \simeq$ $C_3^2$	$G/G' \simeq$ $C_2\times C_{20}$
Frattini:	$\Phi \simeq$ $C_1$	$G/\Phi \simeq$ $C_2\times C_3^2:C_{20}$
Fitting:	$\operatorname{Fit} \simeq$ $C_3\times C_{30}$	$G/\operatorname{Fit} \simeq$ $C_4$
Radical:	$R \simeq$ $C_2\times C_3^2:C_{20}$	$G/R \simeq$ $C_1$
Socle:	$\operatorname{soc} \simeq$ $C_3\times C_{30}$	$G/\operatorname{soc} \simeq$ $C_4$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $C_2\times C_4$
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

Classes of subgroups up to automorphism

Normal subgroups

Normal subgroups up to automorphism

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Classes of subgroups up to conjugation

Order 360: $C_2\times C_3^2:C_{20}$

Order 180: $C_3^2:C_{20}$ x 2, $C_{30}:S_3$

Order 90: $C_{15}:S_3$ x 2, $C_3\times C_{30}$

Order 72: $C_2\times C_3^2:C_4$

Order 60: $S_3\times C_{10}$ x 2

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

Order 40: $C_2\times C_{20}$

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

Order 30: $C_5\times S_3$ x 4, $C_{30}$ x 2

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

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

Order 15: $C_{15}$ x 2

Order 12: $D_6$ x 2

Order 10: $C_{10}$ x 3

Order 9: $C_3^2$

Order 8: $C_2\times C_4$

Order 6: $S_3$ x 4, $C_6$ x 2

Order 5: $C_5$

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

Order 3: $C_3$ x 2

Order 2: $C_2$ x 3

Order 1: $C_1$

Classes of subgroups up to automorphism

Order 360: $C_2\times C_3^2:C_{20}$

Order 180: $C_{30}:S_3$, $C_3^2:C_{20}$

Order 90: $C_{15}:S_3$ x 2, $C_3\times C_{30}$

Order 72: $C_2\times C_3^2:C_4$

Order 60: $S_3\times C_{10}$

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

Order 40: $C_2\times C_{20}$

Order 36: $C_3^2:C_4$, $C_6:S_3$

Order 30: $C_5\times S_3$ x 2, $C_{30}$

Order 20: $C_2\times C_{10}$, $C_{20}$

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

Order 15: $C_{15}$

Order 12: $D_6$

Order 10: $C_{10}$ x 3

Order 9: $C_3^2$

Order 8: $C_2\times C_4$

Order 6: $S_3$ x 2, $C_6$

Order 5: $C_5$

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

Order 3: $C_3$

Order 2: $C_2$ x 3

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 360: $C_2\times C_3^2:C_{20}$ ($C_1$)

Order 180: $C_3^2:C_{20}$ ($C_2$) x 2, $C_{30}:S_3$ ($C_2$)

Order 90: $C_{15}:S_3$ ($C_2^2$), $C_{15}:S_3$ ($C_4$), $C_3\times C_{30}$ ($C_4$)

Order 72: $C_2\times C_3^2:C_4$ ($C_5$)

Order 45: $C_3\times C_{15}$ ($C_2\times C_4$)

Order 36: $C_3^2:C_4$ ($C_{10}$) x 2, $C_6:S_3$ ($C_{10}$)

Order 18: $C_3\times C_6$ ($C_{20}$), $C_3:S_3$ ($C_2\times C_{10}$), $C_3:S_3$ ($C_{20}$)

Order 10: $C_{10}$ ($C_3^2:C_4$)

Order 9: $C_3^2$ ($C_2\times C_{20}$)

Order 5: $C_5$ ($C_2\times C_3^2:C_4$)

Order 2: $C_2$ ($C_3^2:C_{20}$)

Order 1: $C_1$ ($C_2\times C_3^2:C_{20}$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 360: $C_2\times C_3^2:C_{20}$ ($C_1$)

Order 180: $C_{30}:S_3$ ($C_2$), $C_3^2:C_{20}$ ($C_2$)

Order 90: $C_{15}:S_3$ ($C_2^2$), $C_{15}:S_3$ ($C_4$), $C_3\times C_{30}$ ($C_4$)

Order 72: $C_2\times C_3^2:C_4$ ($C_5$)

Order 45: $C_3\times C_{15}$ ($C_2\times C_4$)

Order 36: $C_3^2:C_4$ ($C_{10}$), $C_6:S_3$ ($C_{10}$)

Order 18: $C_3\times C_6$ ($C_{20}$), $C_3:S_3$ ($C_2\times C_{10}$), $C_3:S_3$ ($C_{20}$)

Order 10: $C_{10}$ ($C_3^2:C_4$)

Order 9: $C_3^2$ ($C_2\times C_{20}$)

Order 5: $C_5$ ($C_2\times C_3^2:C_4$)

Order 2: $C_2$ ($C_3^2:C_{20}$)

Order 1: $C_1$ ($C_2\times C_3^2:C_{20}$)

Series

Derived series

$C_2\times C_3^2:C_{20}$

$\rhd$

$C_3^2$

$\rhd$

$C_1$

Chief series

$C_2\times C_3^2:C_{20}$

$\rhd$

$C_3^2:C_{20}$

$\rhd$

$C_{15}:S_3$

$\rhd$

$C_3\times C_{15}$

$\rhd$

$C_5$

$\rhd$

$C_1$

Lower central series

$C_2\times C_3^2:C_{20}$

$\rhd$

$C_3^2$

Upper central series

$C_1$

$\lhd$

$C_{10}$

Supergroups

This group is a maximal subgroup of 33 larger groups in the database.

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

Character theory

Complex character table

See the $60 \times 60$ character table. Alternatively, you may search for characters of this group with desired properties.

Rational character table

See the $20 \times 20$ rational character table.