Abstract group 1440.2910: $C_3\times C_{12}:D_{20}$

• Label: 1440.2910
• Order: \( 2^{5} \cdot 3^{2} \cdot 5 \)
• Exponent: \( 2^{2} \cdot 3 \cdot 5 \)
• Nilpotent: no
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{3} \cdot 3 \)
• $\card{Z(G)}$: \( 2^{2} \cdot 3 \)
• $\card{\Aut(G)}$: \( 2^{12} \cdot 3 \cdot 5 \)
• $\card{\mathrm{Out}(G)}$: \( 2^{9} \)
• Perm deg.: $19$
• Trans deg.: $240$
• Rank: $3$

Group information

Description:	$C_3\times C_{12}:D_{20}$
Order:	\(1440\)\(\medspace = 2^{5} \cdot 3^{2} \cdot 5 \)
Exponent:	\(60\)\(\medspace = 2^{2} \cdot 3 \cdot 5 \)
Automorphism group:	$C_{15}:(C_2^5.C_2^6.C_2)$, of order \(61440\)\(\medspace = 2^{12} \cdot 3 \cdot 5 \)
Composition factors:	$C_2$ x 5, $C_3$ x 2, $C_5$
Derived length:	$2$

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

Group statistics

Order	1	2	3	4	5	6	10	12	15	20	30	60
Elements	1	163	8	28	4	584	12	80	32	112	96	320	1440
Conjugacy classes	1	7	5	6	2	35	6	18	10	24	30	72	216
Divisions	1	7	3	6	1	19	3	10	3	6	9	10	78
Autjugacy classes	1	4	3	2	1	10	2	4	3	2	6	4	42

Dimension	1	2	4	8	16	32
Irr. complex chars.	24	138	54	0	0	0	216
Irr. rational chars.	8	18	22	16	12	2	78

Minimal presentations

Permutation degree:	$19$
Transitive degree:	$240$
Rank:	$3$
Inequivalent generating triples:	$13104$

Minimal degrees of faithful linear representations

	Over $\mathbb{C}$	Over $\mathbb{R}$	Over $\mathbb{Q}$
Irreducible	none	none	none
Arbitrary	not computed	not computed	not computed

Constructions

Presentation:	$\langle a, b, c \mid a^{2}=b^{12}=c^{60}=1, b^{a}=b^{7}, c^{a}=c^{19}, c^{b}=c^{41} \rangle$
Permutation group:	Degree $19$ $\langle(7,8)(9,10)(12,13)(14,15)(16,17)(18,19), (1,2)(3,5)(4,6)(7,9,10,8)(16,17,19,18) \!\cdots\! \rangle$
Matrix group:	$\left\langle \left(\begin{array}{rr} 49 & 0 \\ 0 & 49 \end{array}\right), \left(\begin{array}{rr} 75 & 1 \\ 0 & 1 \end{array}\right), \left(\begin{array}{rr} 73 & 54 \\ 66 & 3 \end{array}\right), \left(\begin{array}{rr} 11 & 57 \\ 19 & 68 \end{array}\right), \left(\begin{array}{rr} 39 & 0 \\ 0 & 39 \end{array}\right), \left(\begin{array}{rr} 73 & 44 \\ 28 & 45 \end{array}\right), \left(\begin{array}{rr} 37 & 0 \\ 0 & 37 \end{array}\right), \left(\begin{array}{rr} 25 & 8 \\ 12 & 13 \end{array}\right) \right\rangle \subseteq \GL_{2}(\Z/76\Z)$
Direct product:	$C_3$ $\, \times\, $ $(C_{12}:D_{20})$
Semidirect product:	$(C_6\times D_{20})$ $\,\rtimes\,$ $S_3$	$(C_6\times D_{60})$ $\,\rtimes\,$ $C_2$	$(C_6:D_{20})$ $\,\rtimes\,$ $C_6$	$(C_6\times D_{20})$ $\,\rtimes\,$ $C_6$	all 30
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Non-split product:	$C_{30}$ . $(S_3\times D_4)$	$C_6$ . $(S_3\times D_{20})$	$(C_6\times D_{10})$ . $D_6$	$(C_2\times C_{60})$ . $D_6$	all 38

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

Homology

Abelianization:	$C_{2}^{2} \times C_{6} \simeq C_{2}^{3} \times C_{3}$
Schur multiplier:	$C_{2}^{2} \times C_{4}$
Commutator length:	$1$

Subgroups

There are 3484 subgroups in 486 conjugacy classes, 120 normal (44 characteristic).

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

Special subgroups

Center:	$Z \simeq$ $C_2\times C_6$	$G/Z \simeq$ $S_3\times D_{10}$
Commutator:	$G' \simeq$ $C_2\times C_{30}$	$G/G' \simeq$ $C_2^2\times C_6$
Frattini:	$\Phi \simeq$ $C_2^2$	$G/\Phi \simeq$ $C_{30}:D_6$
Fitting:	$\operatorname{Fit} \simeq$ $C_6\times C_{60}$	$G/\operatorname{Fit} \simeq$ $C_2^2$
Radical:	$R \simeq$ $C_3\times C_{12}:D_{20}$	$G/R \simeq$ $C_1$
Socle:	$\operatorname{soc} \simeq$ $C_6\times C_{30}$	$G/\operatorname{soc} \simeq$ $C_2^3$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $C_4:D_4$
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

No diagram available: subgroups only stored up to automorphism

Classes of subgroups up to automorphism

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Normal subgroups

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Normal subgroups up to automorphism

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Classes of subgroups up to conjugation

Order 1440: $C_3\times C_{12}:D_{20}$

Order 720: $C_6^2.D_{10}$ x 4, $C_6\times D_{60}$, $C_{20}:C_6^2$, $C_{12}:C_{60}$

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

Order 360: $D_{30}:C_6$ x 8, $C_6:C_{60}$ x 2, $C_3^2\times D_{20}$ x 2, $C_6\times D_{30}$ x 2, $D_5\times C_6^2$ x 2, $C_3\times D_{60}$ x 2, $C_6\times C_{60}$

Order 288: $C_6^2.C_2^3$

Order 240: $C_6\times D_{20}$ x 8, $C_6:D_{20}$ x 4, $C_4\times C_{60}$, $C_{12}:C_{20}$, $C_2\times D_{60}$

Order 180: $C_3\times D_{30}$ x 6, $C_3^2\times D_{10}$ x 6, $C_3:C_{60}$ x 4, $C_3\times C_{60}$ x 2, $C_6\times C_{30}$

Order 160: $C_4:D_{20}$

Order 144: $C_6^2:C_2^2$ x 4, $C_{12}:C_{12}$, $C_4:C_6^2$, $C_6\times D_{12}$

Order 120: $C_3\times D_{20}$ x 18, $C_6\times D_{10}$ x 8, $C_3:D_{20}$ x 8, $C_2\times C_{60}$ x 5, $C_2\times D_{30}$ x 2, $D_{60}$ x 2, $C_6:C_{20}$ x 2

Order 96: $C_{12}:D_4$, $C_{12}:D_4$

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

Order 80: $C_2\times D_{20}$ x 6, $C_4\times C_{20}$

Order 72: $C_6\wr C_2$ x 8, $C_2\times C_6^2$ x 2, $C_6\times D_6$ x 2, $D_4\times C_3^2$ x 2, $C_6:C_{12}$ x 2, $C_3\times D_{12}$ x 2, $C_6\times C_{12}$

Order 60: $C_3\times D_{10}$ x 26, $C_{60}$ x 10, $D_{30}$ x 6, $C_3:C_{20}$ x 4, $C_2\times C_{30}$ x 3

Order 48: $C_6\times D_4$ x 8, $C_6:D_4$ x 4, $C_2\times D_{12}$, $C_4\times C_{12}$, $C_{12}:C_4$

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

Order 40: $D_{20}$ x 12, $C_2\times D_{10}$ x 4, $C_2\times C_{20}$ x 3

Order 36: $C_6^2$ x 7, $C_6\times S_3$ x 6, $C_3:C_{12}$ x 4, $C_3\times C_{12}$ x 2

Order 32: $C_4:D_4$

Order 30: $C_3\times D_5$ x 10, $C_{30}$ x 9, $D_{15}$ x 2

Order 24: $C_3\times D_4$ x 18, $C_3:D_4$ x 8, $C_2^2\times C_6$ x 8, $C_2\times C_{12}$ x 5, $C_6:C_4$ x 2, $D_{12}$ x 2, $C_2\times D_6$ x 2

Order 20: $D_{10}$ x 12, $C_{20}$ x 6, $C_2\times C_{10}$

Order 18: $C_3\times C_6$ x 5, $C_3\times S_3$ x 2

Order 16: $C_2\times D_4$ x 6, $C_4^2$

Order 15: $C_{15}$ x 3

Order 12: $C_2\times C_6$ x 29, $C_{12}$ x 10, $D_6$ x 6, $C_3:C_4$ x 4

Order 10: $D_5$ x 4, $C_{10}$ x 3

Order 9: $C_3^2$

Order 8: $D_4$ x 12, $C_2^3$ x 4, $C_2\times C_4$ x 3

Order 6: $C_6$ x 19, $S_3$ x 2

Order 5: $C_5$

Order 4: $C_2^2$ x 13, $C_4$ x 6

Order 3: $C_3$ x 3

Order 2: $C_2$ x 7

Order 1: $C_1$

Classes of subgroups up to automorphism

Order 1440: $C_3\times C_{12}:D_{20}$

Order 720: $C_6\times D_{60}$, $C_{20}:C_6^2$, $C_6^2.D_{10}$, $C_{12}:C_{60}$

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

Order 360: $C_6:C_{60}$, $C_3^2\times D_{20}$, $D_{30}:C_6$, $C_6\times D_{30}$, $D_5\times C_6^2$, $C_6\times C_{60}$, $C_3\times D_{60}$

Order 288: $C_6^2.C_2^3$

Order 240: $C_6\times D_{20}$ x 5, $C_4\times C_{60}$, $C_{12}:C_{20}$, $C_2\times D_{60}$, $C_6:D_{20}$

Order 180: $C_3\times D_{30}$ x 2, $C_3^2\times D_{10}$ x 2, $C_6\times C_{30}$, $C_3\times C_{60}$, $C_3:C_{60}$

Order 160: $C_4:D_{20}$

Order 144: $C_{12}:C_{12}$, $C_4:C_6^2$, $C_6^2:C_2^2$, $C_6\times D_{12}$

Order 120: $C_3\times D_{20}$ x 5, $C_6\times D_{10}$ x 4, $C_2\times C_{60}$ x 4, $C_2\times D_{30}$, $D_{60}$, $C_6:C_{20}$, $C_3:D_{20}$

Order 96: $C_{12}:D_4$, $C_{12}:D_4$

Order 90: $C_3\times C_{30}$ x 2, $C_3\times D_{15}$, $C_3^2\times D_5$

Order 80: $C_2\times D_{20}$ x 3, $C_4\times C_{20}$

Order 72: $C_2\times C_6^2$, $C_6\times D_6$, $D_4\times C_3^2$, $C_6\times C_{12}$, $C_6\wr C_2$, $C_6:C_{12}$, $C_3\times D_{12}$

Order 60: $C_3\times D_{10}$ x 8, $C_{60}$ x 4, $C_2\times C_{30}$ x 3, $D_{30}$ x 2, $C_3:C_{20}$

Order 48: $C_6\times D_4$ x 5, $C_6:D_4$, $C_2\times D_{12}$, $C_4\times C_{12}$, $C_{12}:C_4$

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

Order 40: $D_{20}$ x 3, $C_2\times C_{20}$ x 2, $C_2\times D_{10}$ x 2

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

Order 32: $C_4:D_4$

Order 30: $C_{30}$ x 6, $C_3\times D_5$ x 4, $D_{15}$

Order 24: $C_3\times D_4$ x 5, $C_2\times C_{12}$ x 4, $C_2^2\times C_6$ x 4, $C_3:D_4$, $C_6:C_4$, $D_{12}$, $C_2\times D_6$

Order 20: $D_{10}$ x 4, $C_{20}$ x 2, $C_2\times C_{10}$

Order 18: $C_3\times C_6$ x 3, $C_3\times S_3$

Order 16: $C_2\times D_4$ x 3, $C_4^2$

Order 15: $C_{15}$ x 3

Order 12: $C_2\times C_6$ x 11, $C_{12}$ x 4, $D_6$ x 2, $C_3:C_4$

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

Order 9: $C_3^2$

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

Order 6: $C_6$ x 10, $S_3$

Order 5: $C_5$

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

Order 3: $C_3$ x 3

Order 2: $C_2$ x 4

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 1440: $C_3\times C_{12}:D_{20}$ ($C_1$)

Order 720: $C_6^2.D_{10}$ ($C_2$) x 4, $C_6\times D_{60}$ ($C_2$), $C_{20}:C_6^2$ ($C_2$), $C_{12}:C_{60}$ ($C_2$)

Order 480: $C_{12}:D_{20}$ ($C_3$)

Order 360: $C_6:C_{60}$ ($C_2^2$) x 2, $C_6\times D_{30}$ ($C_2^2$) x 2, $D_5\times C_6^2$ ($C_2^2$) x 2, $C_6\times C_{60}$ ($C_2^2$)

Order 240: $C_6:D_{20}$ ($C_6$) x 4, $C_{12}:C_{20}$ ($C_6$), $C_2\times D_{60}$ ($C_6$), $C_6\times D_{20}$ ($C_6$), $C_6\times D_{20}$ ($S_3$)

Order 180: $C_3:C_{60}$ ($D_4$) x 4, $C_3\times C_{60}$ ($D_4$) x 2, $C_6\times C_{30}$ ($C_2^3$)

Order 144: $C_{12}:C_{12}$ ($D_5$)

Order 120: $C_2\times D_{30}$ ($C_2\times C_6$) x 2, $C_6\times D_{10}$ ($C_2\times C_6$) x 2, $C_6\times D_{10}$ ($D_6$) x 2, $C_6:C_{20}$ ($C_2\times C_6$) x 2, $C_2\times C_{60}$ ($C_2\times C_6$), $C_2\times C_{60}$ ($D_6$)

Order 90: $C_3\times C_{30}$ ($C_2\times D_4$) x 3

Order 80: $C_2\times D_{20}$ ($C_3\times S_3$)

Order 72: $C_6:C_{12}$ ($D_{10}$) x 2, $C_6\times C_{12}$ ($D_{10}$)

Order 60: $C_3:C_{20}$ ($C_3\times D_4$) x 4, $C_{60}$ ($C_3:D_4$) x 2, $C_{60}$ ($C_3\times D_4$) x 2, $C_2\times C_{30}$ ($C_2^2\times C_6$), $C_2\times C_{30}$ ($C_2\times D_6$)

Order 48: $C_{12}:C_4$ ($C_3\times D_5$)

Order 45: $C_3\times C_{15}$ ($C_4:D_4$)

Order 40: $C_2\times D_{10}$ ($C_6\times S_3$) x 2, $C_2\times C_{20}$ ($C_6\times S_3$)

Order 36: $C_3:C_{12}$ ($D_{20}$) x 4, $C_3\times C_{12}$ ($D_{20}$) x 2, $C_6^2$ ($C_2\times D_{10}$)

Order 30: $C_{30}$ ($C_6\times D_4$) x 3, $C_{30}$ ($S_3\times D_4$) x 2, $C_{30}$ ($C_6:D_4$)

Order 24: $C_6:C_4$ ($C_3\times D_{10}$) x 2, $C_2\times C_{12}$ ($S_3\times D_5$), $C_2\times C_{12}$ ($C_3\times D_{10}$)

Order 20: $C_{20}$ ($C_6\wr C_2$) x 2, $C_2\times C_{10}$ ($C_6\times D_6$)

Order 18: $C_3\times C_6$ ($C_2\times D_{20}$) x 3

Order 15: $C_{15}$ ($C_{12}:D_4$), $C_{15}$ ($C_{12}:D_4$)

Order 12: $C_3:C_4$ ($C_3\times D_{20}$) x 4, $C_{12}$ ($C_3\times D_{20}$) x 2, $C_{12}$ ($C_3:D_{20}$) x 2, $C_2\times C_6$ ($C_6\times D_{10}$), $C_2\times C_6$ ($S_3\times D_{10}$)

Order 10: $C_{10}$ ($C_{12}:D_6$) x 2, $C_{10}$ ($C_6^2:C_2^2$)

Order 9: $C_3^2$ ($C_4:D_{20}$)

Order 8: $C_2\times C_4$ ($C_{15}:D_6$)

Order 6: $C_6$ ($C_6\times D_{20}$) x 3, $C_6$ ($S_3\times D_{20}$) x 2, $C_6$ ($C_6:D_{20}$)

Order 5: $C_5$ ($C_6^2.C_2^3$)

Order 4: $C_4$ ($D_{30}:C_6$) x 2, $C_2^2$ ($C_{30}:D_6$)

Order 3: $C_3$ ($C_{12}:D_{20}$), $C_3$ ($C_{12}:D_{20}$)

Order 2: $C_2$ ($C_{60}:D_6$) x 2, $C_2$ ($C_6^2.D_{10}$)

Order 1: $C_1$ ($C_3\times C_{12}:D_{20}$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 1440: $C_3\times C_{12}:D_{20}$ ($C_1$)

Order 720: $C_6\times D_{60}$ ($C_2$), $C_{20}:C_6^2$ ($C_2$), $C_6^2.D_{10}$ ($C_2$), $C_{12}:C_{60}$ ($C_2$)

Order 480: $C_{12}:D_{20}$ ($C_3$)

Order 360: $C_6:C_{60}$ ($C_2^2$), $C_6\times D_{30}$ ($C_2^2$), $D_5\times C_6^2$ ($C_2^2$), $C_6\times C_{60}$ ($C_2^2$)

Order 240: $C_{12}:C_{20}$ ($C_6$), $C_2\times D_{60}$ ($C_6$), $C_6\times D_{20}$ ($C_6$), $C_6\times D_{20}$ ($S_3$), $C_6:D_{20}$ ($C_6$)

Order 180: $C_6\times C_{30}$ ($C_2^3$), $C_3\times C_{60}$ ($D_4$), $C_3:C_{60}$ ($D_4$)

Order 144: $C_{12}:C_{12}$ ($D_5$)

Order 120: $C_2\times D_{30}$ ($C_2\times C_6$), $C_6\times D_{10}$ ($C_2\times C_6$), $C_6\times D_{10}$ ($D_6$), $C_2\times C_{60}$ ($C_2\times C_6$), $C_2\times C_{60}$ ($D_6$), $C_6:C_{20}$ ($C_2\times C_6$)

Order 90: $C_3\times C_{30}$ ($C_2\times D_4$) x 2

Order 80: $C_2\times D_{20}$ ($C_3\times S_3$)

Order 72: $C_6\times C_{12}$ ($D_{10}$), $C_6:C_{12}$ ($D_{10}$)

Order 60: $C_{60}$ ($C_3:D_4$), $C_{60}$ ($C_3\times D_4$), $C_2\times C_{30}$ ($C_2^2\times C_6$), $C_2\times C_{30}$ ($C_2\times D_6$), $C_3:C_{20}$ ($C_3\times D_4$)

Order 48: $C_{12}:C_4$ ($C_3\times D_5$)

Order 45: $C_3\times C_{15}$ ($C_4:D_4$)

Order 40: $C_2\times C_{20}$ ($C_6\times S_3$), $C_2\times D_{10}$ ($C_6\times S_3$)

Order 36: $C_3\times C_{12}$ ($D_{20}$), $C_3:C_{12}$ ($D_{20}$), $C_6^2$ ($C_2\times D_{10}$)

Order 30: $C_{30}$ ($C_6\times D_4$) x 2, $C_{30}$ ($C_6:D_4$), $C_{30}$ ($S_3\times D_4$)

Order 24: $C_2\times C_{12}$ ($S_3\times D_5$), $C_2\times C_{12}$ ($C_3\times D_{10}$), $C_6:C_4$ ($C_3\times D_{10}$)

Order 20: $C_2\times C_{10}$ ($C_6\times D_6$), $C_{20}$ ($C_6\wr C_2$)

Order 18: $C_3\times C_6$ ($C_2\times D_{20}$) x 2

Order 15: $C_{15}$ ($C_{12}:D_4$), $C_{15}$ ($C_{12}:D_4$)

Order 12: $C_2\times C_6$ ($C_6\times D_{10}$), $C_2\times C_6$ ($S_3\times D_{10}$), $C_{12}$ ($C_3\times D_{20}$), $C_{12}$ ($C_3:D_{20}$), $C_3:C_4$ ($C_3\times D_{20}$)

Order 10: $C_{10}$ ($C_6^2:C_2^2$), $C_{10}$ ($C_{12}:D_6$)

Order 9: $C_3^2$ ($C_4:D_{20}$)

Order 8: $C_2\times C_4$ ($C_{15}:D_6$)

Order 6: $C_6$ ($C_6\times D_{20}$) x 2, $C_6$ ($C_6:D_{20}$), $C_6$ ($S_3\times D_{20}$)

Order 5: $C_5$ ($C_6^2.C_2^3$)

Order 4: $C_2^2$ ($C_{30}:D_6$), $C_4$ ($D_{30}:C_6$)

Order 3: $C_3$ ($C_{12}:D_{20}$), $C_3$ ($C_{12}:D_{20}$)

Order 2: $C_2$ ($C_6^2.D_{10}$), $C_2$ ($C_{60}:D_6$)

Order 1: $C_1$ ($C_3\times C_{12}:D_{20}$)

Series

Derived series

$C_3\times C_{12}:D_{20}$

$\rhd$

$C_2\times C_{30}$

$\rhd$

$C_1$

Chief series

$C_3\times C_{12}:D_{20}$

$\rhd$

$C_{20}:C_6^2$

$\rhd$

$C_6\times C_{60}$

$\rhd$

$C_3\times C_{60}$

$\rhd$

$C_{60}$

$\rhd$

$C_{30}$

$\rhd$

$C_{15}$

$\rhd$

$C_5$

$\rhd$

$C_1$

Lower central series

$C_3\times C_{12}:D_{20}$

$\rhd$

$C_2\times C_{30}$

$\rhd$

$C_{15}$

Upper central series

$C_1$

$\lhd$

$C_2\times C_6$

$\lhd$

$C_2\times C_{12}$

Character theory

Complex character table

See the $216 \times 216$ 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 $78 \times 78$ rational character table.