Abstract group 25200.d: $C_{60}.D_{210}$

• Label: 25200.d
• Order: \( 2^{4} \cdot 3^{2} \cdot 5^{2} \cdot 7 \)
• Exponent: \( 2^{2} \cdot 3 \cdot 5 \cdot 7 \)
• Nilpotent: no
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{3} \cdot 3 \cdot 5 \)
• $\card{Z(G)}$: \( 2^{2} \cdot 3 \cdot 5 \)
• $\card{\Aut(G)}$: \( 2^{11} \cdot 3^{2} \cdot 5 \cdot 7 \)
• $\card{\mathrm{Out}(G)}$: \( 2^{9} \cdot 3 \)
• Perm deg.: $31$
• Trans deg.: $840$
• Rank: $3$

Group information

Description:	$C_{60}.D_{210}$
Order:	\(25200\)\(\medspace = 2^{4} \cdot 3^{2} \cdot 5^{2} \cdot 7 \)
Exponent:	\(420\)\(\medspace = 2^{2} \cdot 3 \cdot 5 \cdot 7 \)
Automorphism group:	$C_{210}.C_6.C_2^6.C_2^3$, of order \(645120\)\(\medspace = 2^{11} \cdot 3^{2} \cdot 5 \cdot 7 \)
Composition factors:	$C_2$ x 4, $C_3$ x 2, $C_5$ x 2, $C_7$
Derived length:	$2$

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

Group statistics

Order	1	2	3	4	5	6	7	10	12	14	15	20	21	28	30	35	42	60	70	84	105	140	210	420
Elements	1	423	8	424	24	864	6	1752	872	18	192	1776	48	24	3936	144	144	4128	432	192	1152	576	3456	4608	25200
Conjugacy classes	1	4	5	5	14	17	3	46	22	9	100	60	24	12	308	72	72	408	216	96	576	288	1728	2304	6390
Divisions	1	4	3	4	4	9	1	12	8	2	14	10	3	2	40	4	7	30	10	6	14	8	38	28	262
Autjugacy classes	1	3	3	3	3	7	1	7	7	2	9	7	3	2	19	3	6	19	6	6	9	6	18	18	168

Minimal presentations

Permutation degree:	$31$
Transitive degree:	$840$
Rank:	$3$
Inequivalent generating triples:	not computed

Minimal degrees of linear representations for this group have not been computed

Constructions

Presentation:	$\langle a, b, c \mid a^{2}=b^{30}=c^{420}=[b,c]=1, b^{a}=bc^{406}, c^{a}=c^{209} \rangle$
Permutation group:	Degree $31$ $\langle(1,2,3)(9,10,13,11)(12,14,16,15)(17,18,20,19,21)(25,26,27,29,31,30,28), (2,3) \!\cdots\! \rangle$
Matrix group:	$\left\langle \left(\begin{array}{rr} 2 & 0 \\ 0 & 211 \end{array}\right), \left(\begin{array}{rr} 0 & 1 \\ 1 & 0 \end{array}\right), \left(\begin{array}{rr} 1 & 0 \\ 0 & 386 \end{array}\right) \right\rangle \subseteq \GL_{2}(\F_{421})$
Direct product:	not computed
Semidirect product:	not isomorphic to a non-trivial semidirect product
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Possibly split product:	$D_{420}$ . $C_{30}$	$C_{420}$ . $D_{30}$ (2)	$C_{60}$ . $D_{210}$ (2)	$C_{30}^2$ . $D_{14}$	all 177

Elements of the group are displayed as matrices in $\GL_{2}(\F_{421})$.

Homology

Abelianization:	$C_{2}^{2} \times C_{30} \simeq C_{2}^{3} \times C_{3} \times C_{5}$
Schur multiplier:	$C_{2}^{2}$
Commutator length:	not computed

Subgroups

There are 12288 subgroups in 896 conjugacy classes, 236 normal (204 characteristic).

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

Special subgroups

Center:	$Z \simeq$ $C_{60}$	$G/Z \simeq$ $D_{210}$
Commutator:	$G' \simeq$ $C_{210}$	$G/G' \simeq$ $C_2^2\times C_{30}$
Frattini:	$\Phi \simeq$ $C_2$	$G/\Phi \simeq$ $C_{35}.(C_{30}\times S_3).C_2$
Fitting:	$\operatorname{Fit} \simeq$ $C_{30}\times C_{420}$	$G/\operatorname{Fit} \simeq$ $C_2$
Radical:	$R \simeq$ $C_{60}.D_{210}$	$G/R \simeq$ $C_1$
Socle:	$\operatorname{soc} \simeq$ $C_{15}\times C_{210}$	$G/\operatorname{soc} \simeq$ $C_2^3$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $D_4:C_2$
3-Sylow subgroup:	$P_{ 3 } \simeq$ $C_3^2$
5-Sylow subgroup:	$P_{ 5 } \simeq$ $C_5^2$
7-Sylow subgroup:	$P_{ 7 } \simeq$ $C_7$

Hi

Subgroup diagram and profile

Classes of subgroups up to conjugation

No diagram available: inclusions not computed

Classes of subgroups up to automorphism

No diagram available: inclusions not computed

Normal subgroups

No diagram available: inclusions not computed

Normal subgroups up to automorphism

No diagram available: inclusions not computed

Classes of subgroups up to conjugation

Order 25200: $C_{60}.D_{210}$

Order 12600: $C_{60}\times D_{105}$ x 2, $C_{30}.D_{210}$ x 2, $C_{420}.C_{30}$, $C_{15}\times D_{420}$, $C_{30}\times C_{420}$

Order 8400: $D_{140}:C_{30}$, $D_{420}:C_{10}$

Order 6300: $C_{15}\times C_{420}$ x 2, $C_{105}:C_{60}$ x 2, $C_{15}\times D_{210}$ x 2, $C_{30}\times C_{210}$

Order 5040: $D_{420}:C_6$, $D_{84}:C_{30}$

Order 4200: $C_{10}\times C_{420}$ x 3, $C_{20}\times D_{105}$ x 2, $C_{60}\times D_{35}$ x 2, $D_{70}:C_{30}$ x 2, $D_{210}:C_{10}$ x 2, $C_{140}.C_{30}$, $C_{15}\times D_{140}$, $C_{420}.C_{10}$, $C_5\times D_{420}$

Order 3600: $D_{60}:C_{30}$

Order 3150: $C_{15}\times D_{105}$ x 2, $C_{15}\times C_{210}$ x 2

Order 2800: $D_{140}:C_{10}$

Order 2520: $C_6\times C_{420}$ x 4, $D_{21}\times C_{60}$ x 2, $C_{12}\times D_{105}$ x 2, $D_{42}:C_{30}$ x 2, $D_{210}:C_6$ x 2, $C_{420}.C_6$, $C_3\times D_{420}$, $C_{15}\times D_{84}$, $C_{420}.C_6$

Order 2100: $C_5\times C_{420}$ x 6, $C_{10}\times C_{210}$ x 3, $C_{15}\times D_{70}$ x 2, $C_{35}:C_{60}$ x 2, $C_{105}:C_{20}$ x 2, $C_5\times D_{210}$ x 2

Order 1800: $C_{30}\wr C_2$ x 2, $D_{15}\times C_{60}$ x 2, $C_{30}\times C_{60}$, $C_{15}\times D_{60}$, $C_{15}^2:Q_8$

Order 1680: $D_{420}:C_2$, $D_{84}:C_{10}$, $D_{140}:C_6$, $D_{28}:C_{30}$

Order 1575: $C_{15}\times C_{105}$

Order 1400: $C_{20}\times D_{35}$ x 2, $D_{70}:C_{10}$ x 2, $C_5\times D_{140}$, $C_{140}.C_{10}$, $C_{10}\times C_{140}$

Order 1260: $C_3\times C_{420}$ x 8, $C_6\times C_{210}$ x 4, $C_{105}:C_{12}$ x 2, $C_{21}:C_{60}$ x 2, $C_3\times D_{210}$ x 2, $C_{15}\times D_{42}$ x 2

Order 1200: $D_{60}:C_{10}$, $D_{20}:C_{30}$

Order 1050: $C_5\times C_{210}$ x 7, $C_5\times D_{105}$ x 2, $C_{15}\times D_{35}$ x 2

Order 1008: $D_{84}:C_6$

Order 900: $C_{15}\times C_{60}$ x 2, $C_{15}:C_{60}$ x 2, $C_{15}\times D_{30}$ x 2, $C_{30}^2$

Order 840: $C_2\times C_{420}$ x 14, $D_7\times C_{60}$ x 2, $C_{105}:D_4$ x 2, $C_4\times D_{105}$ x 2, $C_{105}:D_4$ x 2, $C_{20}\times D_{21}$ x 2, $C_{105}:D_4$ x 2, $C_{12}\times D_{35}$ x 2, $D_{14}:C_{30}$ x 2, $C_{15}\times D_{28}$, $C_{105}:Q_8$, $D_{420}$, $C_{105}:Q_8$, $C_5\times D_{84}$, $C_{105}:Q_8$, $C_3\times D_{140}$, $C_{105}:Q_8$

Order 720: $D_{60}:C_6$, $D_{12}:C_{30}$

Order 700: $C_5\times D_{70}$ x 2, $C_5\times C_{140}$ x 2, $C_{35}:C_{20}$ x 2, $C_{10}\times C_{70}$

Order 630: $C_3\times C_{210}$ x 10, $C_3\times D_{105}$ x 2, $C_{15}\times D_{21}$ x 2

Order 600: $C_{10}\times C_{60}$ x 3, $D_{30}:C_{10}$ x 2, $C_{20}\times D_{15}$ x 2, $D_{10}:C_{30}$ x 2, $D_5\times C_{60}$ x 2, $C_5\times D_{60}$, $C_{60}.C_{10}$, $C_{15}\times D_{20}$, $C_{20}.C_{30}$

Order 560: $D_{140}:C_2$, $D_{28}:C_{10}$

Order 525: $C_5\times C_{105}$ x 3

Order 504: $D_{42}:C_6$ x 2, $C_{12}\times D_{21}$ x 2, $C_6\times C_{84}$, $C_3\times D_{84}$, $C_{84}.C_6$

Order 450: $C_{15}\times C_{30}$ x 2, $C_{15}\times D_{15}$ x 2

Order 420: $C_{420}$ x 28, $C_2\times C_{210}$ x 14, $C_{105}:C_4$ x 2, $C_{105}:C_4$ x 2, $C_7:C_{60}$ x 2, $D_{210}$ x 2, $C_5\times D_{42}$ x 2, $C_3\times D_{70}$ x 2, $C_{15}\times D_{14}$ x 2, $C_{105}:C_4$ x 2

Order 400: $D_{20}:C_{10}$

Order 360: $C_6\times C_{60}$ x 4, $D_6:C_{30}$ x 2, $S_3\times C_{60}$ x 2, $D_{30}:C_6$ x 2, $C_{12}\times D_{15}$ x 2, $C_{15}\times D_{12}$, $C_{12}.C_{30}$, $C_3\times D_{60}$, $C_{60}.C_6$

Order 350: $C_5\times D_{35}$ x 2, $C_5\times C_{70}$ x 2

Order 336: $D_{84}:C_2$, $D_{28}:C_6$

Order 315: $C_3\times C_{105}$ x 4

Order 300: $C_5\times C_{60}$ x 6, $C_{10}\times C_{30}$ x 3, $C_5\times D_{30}$ x 2, $C_{15}\times D_{10}$ x 2, $C_{15}:C_{20}$ x 2, $C_5:C_{60}$ x 2

Order 280: $C_2\times C_{140}$ x 4, $C_{35}:D_4$ x 2, $C_4\times D_{35}$ x 2, $C_{35}:D_4$ x 2, $D_7\times C_{20}$ x 2, $D_{140}$, $C_{35}:Q_8$, $C_5\times D_{28}$, $C_{35}:Q_8$

Order 252: $C_3\times D_{42}$ x 2, $C_3\times C_{84}$ x 2, $C_{21}:C_{12}$ x 2, $C_6\times C_{42}$

Order 240: $D_4:C_{30}$, $D_{60}:C_2$, $D_{12}:C_{10}$, $D_{20}:C_6$

Order 225: $C_{15}^2$

Order 210: $C_{210}$ x 38, $C_5\times D_{21}$ x 2, $C_3\times D_{35}$ x 2, $D_7\times C_{15}$ x 2, $D_{105}$ x 2

Order 200: $C_{10}\wr C_2$ x 2, $D_5\times C_{20}$ x 2, $C_{10}\times C_{20}$, $C_5\times D_{20}$, $C_5^2:Q_8$

Order 180: $C_3\times C_{60}$ x 8, $C_6\times C_{30}$ x 4, $C_3\times D_{30}$ x 2, $S_3\times C_{30}$ x 2, $C_{15}:C_{12}$ x 2, $C_3:C_{60}$ x 2

Order 175: $C_5\times C_{35}$

Order 168: $C_2\times C_{84}$ x 3, $C_{21}:D_4$ x 2, $C_4\times D_{21}$ x 2, $C_{21}:D_4$ x 2, $C_{12}\times D_7$ x 2, $D_{84}$, $C_{21}:Q_8$, $C_3\times D_{28}$, $C_{21}:Q_8$

Order 150: $C_5\times C_{30}$ x 7, $D_5\times C_{15}$ x 2, $C_5\times D_{15}$ x 2

Order 144: $D_{12}:C_6$

Order 140: $C_{140}$ x 8, $C_2\times C_{70}$ x 4, $C_5\times D_{14}$ x 2, $C_{35}:C_4$ x 2, $C_7:C_{20}$ x 2, $D_{70}$ x 2

Order 126: $C_3\times C_{42}$ x 2, $C_3\times D_{21}$ x 2

Order 120: $C_2\times C_{60}$ x 16, $D_4\times C_{15}$ x 3, $C_{15}:D_4$ x 2, $C_4\times D_{15}$ x 2, $C_{15}:D_4$ x 2, $S_3\times C_{20}$ x 2, $C_{15}:D_4$ x 2, $D_5\times C_{12}$ x 2, $Q_8\times C_{15}$, $D_{60}$, $C_{15}:Q_8$, $C_5\times D_{12}$, $C_{15}:Q_8$, $C_3\times D_{20}$, $C_{15}:Q_8$

Order 112: $D_{28}:C_2$

Order 105: $C_{105}$ x 14

Order 100: $C_5\times C_{20}$ x 2, $C_5:C_{20}$ x 2, $C_5\times D_{10}$ x 2, $C_{10}^2$

Order 90: $C_3\times C_{30}$ x 10, $C_3\times D_{15}$ x 2, $S_3\times C_{15}$ x 2

Order 84: $C_{84}$ x 6, $C_2\times C_{42}$ x 3, $C_{21}:C_4$ x 2, $C_7:C_{12}$ x 2, $D_{42}$ x 2, $C_3\times D_{14}$ x 2

Order 80: $D_4:C_{10}$, $D_{20}:C_2$

Order 75: $C_5\times C_{15}$ x 3

Order 72: $C_6\wr C_2$ x 2, $S_3\times C_{12}$ x 2, $C_6\times C_{12}$, $C_3\times D_{12}$, $C_3^2:Q_8$

Order 70: $C_{70}$ x 10, $D_{35}$ x 2, $C_5\times D_7$ x 2

Order 63: $C_3\times C_{21}$

Order 60: $C_{60}$ x 30, $C_2\times C_{30}$ x 16, $C_{15}:C_4$ x 2, $C_5:C_{12}$ x 2, $D_{30}$ x 2, $S_3\times C_{10}$ x 2, $C_3\times D_{10}$ x 2, $C_3:C_{20}$ x 2

Order 56: $C_7:D_4$ x 2, $C_4\times D_7$ x 2, $C_2\times C_{28}$, $D_{28}$, $C_7:Q_8$

Order 50: $C_5\times C_{10}$ x 2, $C_5\times D_5$ x 2

Order 48: $D_4:C_6$, $D_{12}:C_2$

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

Order 42: $C_{42}$ x 7, $D_{21}$ x 2, $C_3\times D_7$ x 2

Order 40: $C_2\times C_{20}$ x 6, $C_5\times D_4$ x 3, $C_5:D_4$ x 2, $C_4\times D_5$ x 2, $D_{20}$, $C_5:Q_8$, $C_5\times Q_8$

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

Order 35: $C_{35}$ x 4

Order 30: $C_{30}$ x 40, $D_{15}$ x 2, $C_3\times D_5$ x 2, $C_5\times S_3$ x 2

Order 28: $D_{14}$ x 2, $C_{28}$ x 2, $C_7:C_4$ x 2, $C_2\times C_{14}$

Order 25: $C_5^2$

Order 24: $C_2\times C_{12}$ x 5, $C_3\times D_4$ x 3, $C_3:D_4$ x 2, $C_4\times S_3$ x 2, $D_{12}$, $C_3:Q_8$, $C_3\times Q_8$

Order 21: $C_{21}$ x 3

Order 20: $C_{20}$ x 10, $C_2\times C_{10}$ x 6, $D_{10}$ x 2, $C_5:C_4$ x 2

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

Order 16: $D_4:C_2$

Order 15: $C_{15}$ x 14

Order 14: $C_{14}$ x 2, $D_7$ x 2

Order 12: $C_{12}$ x 8, $C_2\times C_6$ x 5, $D_6$ x 2, $C_3:C_4$ x 2

Order 10: $C_{10}$ x 12, $D_5$ x 2

Order 9: $C_3^2$

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

Order 7: $C_7$

Order 6: $C_6$ x 9, $S_3$ x 2

Order 5: $C_5$ x 4

Order 4: $C_4$ x 4, $C_2^2$ x 3

Order 3: $C_3$ x 3

Order 2: $C_2$ x 4

Order 1: $C_1$

Classes of subgroups up to automorphism

Order 25200: $C_{60}.D_{210}$

Order 12600: $C_{60}\times D_{105}$

Order 3600: $D_{60}:C_{30}$

Order 2800: $D_{140}:C_{10}$

Order 1575: $C_{15}\times C_{105}$

Order 1008: $D_{84}:C_6$

Order 400: $D_{20}:C_{10}$

Order 225: $C_{15}^2$

Order 175: $C_5\times C_{35}$

Order 144: $D_{12}:C_6$

Order 112: $D_{28}:C_2$

Order 63: $C_3\times C_{21}$

Order 25: $C_5^2$

Order 16: $D_4:C_2$

Order 9: $C_3^2$

Order 7: $C_7$

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 25200: $C_{60}.D_{210}$ ($C_1$)

Order 12600: $C_{60}\times D_{105}$ ($C_2$) x 2, $C_{30}.D_{210}$ ($C_2$) x 2, $C_{420}.C_{30}$ ($C_2$), $C_{15}\times D_{420}$ ($C_2$), $C_{30}\times C_{420}$ ($C_2$)

Order 8400: $D_{420}:C_{10}$ ($C_3$)

Order 6300: $C_{15}\times C_{420}$ ($C_2^2$) x 2, $C_{105}:C_{60}$ ($C_2^2$) x 2, $C_{15}\times D_{210}$ ($C_2^2$) x 2, $C_{30}\times C_{210}$ ($C_2^2$)

Order 5040: $D_{420}:C_6$ ($C_5$)

Order 4200: $C_{20}\times D_{105}$ ($C_6$) x 2, $D_{210}:C_{10}$ ($C_6$) x 2, $C_{420}.C_{10}$ ($C_6$), $C_5\times D_{420}$ ($C_6$), $C_{10}\times C_{420}$ ($C_6$), $C_{10}\times C_{420}$ ($S_3$)

Order 3150: $C_{15}\times C_{210}$ ($C_2^3$)

Order 2520: $C_{12}\times D_{105}$ ($C_{10}$) x 2, $D_{210}:C_6$ ($C_{10}$) x 2, $C_{420}.C_6$ ($C_{10}$), $C_3\times D_{420}$ ($C_{10}$), $C_6\times C_{420}$ ($C_{10}$), $C_6\times C_{420}$ ($D_5$)

Order 2100: $C_5\times C_{420}$ ($C_2\times C_6$) x 2, $C_5\times C_{420}$ ($D_6$) x 2, $C_{105}:C_{20}$ ($C_2\times C_6$) x 2, $C_5\times D_{210}$ ($C_2\times C_6$) x 2, $C_{10}\times C_{210}$ ($C_2\times C_6$), $C_{10}\times C_{210}$ ($D_6$)

Order 1800: $C_{30}\times C_{60}$ ($D_7$)

Order 1680: $D_{420}:C_2$ ($C_{15}$)

Order 1575: $C_{15}\times C_{105}$ ($D_4:C_2$)

Order 1400: $C_{10}\times C_{140}$ ($C_3\times S_3$)

Order 1260: $C_3\times C_{420}$ ($C_2\times C_{10}$) x 2, $C_3\times C_{420}$ ($D_{10}$) x 2, $C_{105}:C_{12}$ ($C_2\times C_{10}$) x 2, $C_3\times D_{210}$ ($C_2\times C_{10}$) x 2, $C_6\times C_{210}$ ($C_2\times C_{10}$), $C_6\times C_{210}$ ($D_{10}$)

Order 1050: $C_5\times C_{210}$ ($C_2^2\times C_6$), $C_5\times C_{210}$ ($C_2\times D_6$)

Order 900: $C_{15}\times C_{60}$ ($D_{14}$) x 2, $C_{30}^2$ ($D_{14}$)

Order 840: $C_{105}:D_4$ ($C_{30}$) x 2, $C_4\times D_{105}$ ($C_{30}$) x 2, $C_2\times C_{420}$ ($C_{30}$), $C_2\times C_{420}$ ($D_{15}$), $C_2\times C_{420}$ ($C_3\times D_5$), $C_2\times C_{420}$ ($C_5\times S_3$), $D_{420}$ ($C_{30}$), $C_{105}:Q_8$ ($C_{30}$)

Order 700: $C_5\times C_{140}$ ($C_6\times S_3$) x 2, $C_{10}\times C_{70}$ ($C_6\times S_3$)

Order 630: $C_3\times C_{210}$ ($C_2^2\times C_{10}$), $C_3\times C_{210}$ ($C_2\times D_{10}$)

Order 600: $C_{10}\times C_{60}$ ($D_{21}$), $C_{10}\times C_{60}$ ($C_3\times D_7$)

Order 525: $C_5\times C_{105}$ ($D_4:C_6$), $C_5\times C_{105}$ ($D_{12}:C_2$)

Order 504: $C_6\times C_{84}$ ($C_5\times D_5$)

Order 450: $C_{15}\times C_{30}$ ($C_2\times D_{14}$)

Order 420: $D_{210}$ ($C_2\times C_{30}$) x 2, $C_{420}$ ($C_2\times C_{30}$) x 2, $C_{420}$ ($D_{30}$) x 2, $C_{420}$ ($S_3\times C_{10}$) x 2, $C_{420}$ ($C_3\times D_{10}$) x 2, $C_{105}:C_4$ ($C_2\times C_{30}$) x 2, $C_2\times C_{210}$ ($C_2\times C_{30}$), $C_2\times C_{210}$ ($D_{30}$), $C_2\times C_{210}$ ($S_3\times C_{10}$), $C_2\times C_{210}$ ($C_3\times D_{10}$)

Order 360: $C_6\times C_{60}$ ($D_{35}$), $C_6\times C_{60}$ ($C_5\times D_7$)

Order 350: $C_5\times C_{70}$ ($C_6\times D_6$)

Order 315: $C_3\times C_{105}$ ($D_4:C_{10}$), $C_3\times C_{105}$ ($D_{20}:C_2$)

Order 300: $C_5\times C_{60}$ ($D_{42}$) x 2, $C_5\times C_{60}$ ($C_3\times D_{14}$) x 2, $C_{10}\times C_{30}$ ($D_{42}$), $C_{10}\times C_{30}$ ($C_3\times D_{14}$)

Order 280: $C_2\times C_{140}$ ($C_3\times D_{15}$), $C_2\times C_{140}$ ($S_3\times C_{15}$)

Order 252: $C_3\times C_{84}$ ($C_5\times D_{10}$) x 2, $C_6\times C_{42}$ ($C_5\times D_{10}$)

Order 225: $C_{15}^2$ ($D_{28}:C_2$)

Order 210: $C_{210}$ ($C_2^2\times C_{30}$), $C_{210}$ ($C_2\times D_{30}$), $C_{210}$ ($C_{10}\times D_6$), $C_{210}$ ($C_6\times D_{10}$)

Order 200: $C_{10}\times C_{20}$ ($C_3\times D_{21}$)

Order 180: $C_3\times C_{60}$ ($C_5\times D_{14}$) x 2, $C_3\times C_{60}$ ($D_{70}$) x 2, $C_6\times C_{30}$ ($C_5\times D_{14}$), $C_6\times C_{30}$ ($D_{70}$)

Order 175: $C_5\times C_{35}$ ($D_{12}:C_6$)

Order 168: $C_2\times C_{84}$ ($D_5\times C_{15}$), $C_2\times C_{84}$ ($C_5\times D_{15}$)

Order 150: $C_5\times C_{30}$ ($C_2\times D_{42}$), $C_5\times C_{30}$ ($C_6\times D_{14}$)

Order 140: $C_{140}$ ($C_3\times D_{30}$) x 2, $C_{140}$ ($S_3\times C_{30}$) x 2, $C_2\times C_{70}$ ($C_3\times D_{30}$), $C_2\times C_{70}$ ($S_3\times C_{30}$)

Order 126: $C_3\times C_{42}$ ($C_{10}\times D_{10}$)

Order 120: $C_2\times C_{60}$ ($C_5\times D_{21}$), $C_2\times C_{60}$ ($C_3\times D_{35}$), $C_2\times C_{60}$ ($D_7\times C_{15}$), $C_2\times C_{60}$ ($D_{105}$)

Order 105: $C_{105}$ ($D_4:C_{30}$), $C_{105}$ ($D_{60}:C_2$), $C_{105}$ ($D_{12}:C_{10}$), $C_{105}$ ($D_{20}:C_6$)

Order 100: $C_5\times C_{20}$ ($C_3\times D_{42}$) x 2, $C_{10}^2$ ($C_3\times D_{42}$)

Order 90: $C_3\times C_{30}$ ($C_2\times D_{70}$), $C_3\times C_{30}$ ($C_{10}\times D_{14}$)

Order 84: $C_{84}$ ($C_5\times D_{30}$) x 2, $C_{84}$ ($C_{15}\times D_{10}$) x 2, $C_2\times C_{42}$ ($C_5\times D_{30}$), $C_2\times C_{42}$ ($C_{15}\times D_{10}$)

Order 75: $C_5\times C_{15}$ ($D_{84}:C_2$), $C_5\times C_{15}$ ($D_{28}:C_6$)

Order 72: $C_6\times C_{12}$ ($C_5\times D_{35}$)

Order 70: $C_{70}$ ($C_6\times D_{30}$), $C_{70}$ ($D_6\times C_{30}$)

Order 63: $C_3\times C_{21}$ ($D_{20}:C_{10}$)

Order 60: $C_{60}$ ($D_{210}$) x 2, $C_{60}$ ($C_5\times D_{42}$) x 2, $C_{60}$ ($C_3\times D_{70}$) x 2, $C_{60}$ ($C_{15}\times D_{14}$) x 2, $C_2\times C_{30}$ ($D_{210}$), $C_2\times C_{30}$ ($C_5\times D_{42}$), $C_2\times C_{30}$ ($C_3\times D_{70}$), $C_2\times C_{30}$ ($C_{15}\times D_{14}$)

Order 56: $C_2\times C_{28}$ ($C_{15}\times D_{15}$)

Order 50: $C_5\times C_{10}$ ($C_6\times D_{42}$)

Order 45: $C_3\times C_{15}$ ($D_{140}:C_2$), $C_3\times C_{15}$ ($D_{28}:C_{10}$)

Order 42: $C_{42}$ ($C_{10}\times D_{30}$), $C_{42}$ ($D_{10}\times C_{30}$)

Order 40: $C_2\times C_{20}$ ($C_3\times D_{105}$), $C_2\times C_{20}$ ($C_{15}\times D_{21}$)

Order 36: $C_3\times C_{12}$ ($C_5\times D_{70}$) x 2, $C_6^2$ ($C_5\times D_{70}$)

Order 35: $C_{35}$ ($D_{60}:C_6$), $C_{35}$ ($D_{12}:C_{30}$)

Order 30: $C_{30}$ ($C_2\times D_{210}$), $C_{30}$ ($C_{10}\times D_{42}$), $C_{30}$ ($C_6\times D_{70}$), $C_{30}$ ($D_{14}\times C_{30}$)

Order 28: $C_{28}$ ($C_{15}\times D_{30}$) x 2, $C_2\times C_{14}$ ($C_{15}\times D_{30}$)

Order 25: $C_5^2$ ($D_{84}:C_6$)

Order 24: $C_2\times C_{12}$ ($C_5\times D_{105}$), $C_2\times C_{12}$ ($C_{15}\times D_{35}$)

Order 21: $C_{21}$ ($D_{60}:C_{10}$), $C_{21}$ ($D_{20}:C_{30}$)

Order 20: $C_{20}$ ($C_3\times D_{210}$) x 2, $C_{20}$ ($C_{15}\times D_{42}$) x 2, $C_2\times C_{10}$ ($C_3\times D_{210}$), $C_2\times C_{10}$ ($C_{15}\times D_{42}$)

Order 18: $C_3\times C_6$ ($C_{10}\times D_{70}$)

Order 15: $C_{15}$ ($D_{420}:C_2$), $C_{15}$ ($D_{84}:C_{10}$), $C_{15}$ ($D_{140}:C_6$), $C_{15}$ ($D_{28}:C_{30}$)

Order 14: $C_{14}$ ($C_{30}\times D_{30}$)

Order 12: $C_{12}$ ($C_{15}\times D_{70}$) x 2, $C_{12}$ ($C_5\times D_{210}$) x 2, $C_2\times C_6$ ($C_{15}\times D_{70}$), $C_2\times C_6$ ($C_5\times D_{210}$)

Order 10: $C_{10}$ ($C_7:(C_2\times C_{30}\times S_3)$), $C_{10}$ ($C_7:(C_2\times C_6\times D_{15})$)

Order 9: $C_3^2$ ($D_{140}:C_{10}$)

Order 8: $C_2\times C_4$ ($C_{15}\times D_{105}$)

Order 7: $C_7$ ($D_{60}:C_{30}$)

Order 6: $C_6$ ($C_7:(C_2^2\times C_5\times D_{15})$), $C_6$ ($C_7:(C_2\times C_6\times C_5\times D_5)$)

Order 5: $C_5$ ($D_{420}:C_6$), $C_5$ ($D_{84}:C_{30}$)

Order 4: $C_4$ ($C_{15}\times D_{210}$) x 2, $C_2^2$ ($C_{15}\times D_{210}$)

Order 3: $C_3$ ($D_{140}:C_{30}$), $C_3$ ($D_{420}:C_{10}$)

Order 2: $C_2$ ($C_{35}.(C_{30}\times S_3).C_2$)

Order 1: $C_1$ ($C_{60}.D_{210}$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 25200: $C_{60}.D_{210}$ ($C_1$)

Order 12600: $C_{60}\times D_{105}$ ($C_2$)

Order 1575: $C_{15}\times C_{105}$ ($D_4:C_2$)

Order 225: $C_{15}^2$ ($D_{28}:C_2$)

Order 175: $C_5\times C_{35}$ ($D_{12}:C_6$)

Order 63: $C_3\times C_{21}$ ($D_{20}:C_{10}$)

Order 25: $C_5^2$ ($D_{84}:C_6$)

Order 9: $C_3^2$ ($D_{140}:C_{10}$)

Order 7: $C_7$ ($D_{60}:C_{30}$)

Order 1: $C_1$ ($C_{60}.D_{210}$)

Series

Derived series

$C_{60}.D_{210}$

$\rhd$

$C_{210}$

$\rhd$

$C_1$

Chief series

$C_{60}.D_{210}$

$\rhd$

$C_{30}\times C_{420}$

$\rhd$

$C_{30}\times C_{210}$

$\rhd$

$C_{15}\times C_{210}$

$\rhd$

$C_5\times C_{210}$

$\rhd$

$C_{210}$

$\rhd$

$C_{105}$

$\rhd$

$C_{35}$

$\rhd$

$C_7$

$\rhd$

$C_1$

Lower central series

$C_{60}.D_{210}$

$\rhd$

$C_{210}$

$\rhd$

$C_{105}$

Upper central series

$C_1$

$\lhd$

$C_{60}$

$\lhd$

$C_2\times C_{60}$

Supergroups

This group is a maximal subgroup of 2 larger groups in the database.

This group is a maximal quotient of 1 larger groups in the database.

Character theory

Complex character table

The $6390 \times 6390$ character table is not available for this group.

Rational character table

The $262 \times 262$ rational character table is not available for this group.