Abstract group 29400.c: $C_{420}.C_{70}$

• Label: 29400.c
• Order: \( 2^{3} \cdot 3 \cdot 5^{2} \cdot 7^{2} \)
• Exponent: \( 2^{2} \cdot 3 \cdot 5 \cdot 7 \)
• Nilpotent: no
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{3} \cdot 5 \cdot 7 \)
• $\card{Z(G)}$: \( 2^{2} \cdot 5 \cdot 7 \)
• $\card{\Aut(G)}$: \( 2^{9} \cdot 3^{3} \cdot 5 \cdot 7 \)
• $\card{\mathrm{Out}(G)}$: \( 2^{8} \cdot 3^{2} \)
• Perm deg.: $31$
• Trans deg.: $840$
• Rank: $2$

Group information

Description:	$C_{420}.C_{70}$
Order:	\(29400\)\(\medspace = 2^{3} \cdot 3 \cdot 5^{2} \cdot 7^{2} \)
Exponent:	\(420\)\(\medspace = 2^{2} \cdot 3 \cdot 5 \cdot 7 \)
Automorphism group:	$C_2^3\times C_{12}\times F_5\times S_3\times F_7$, of order \(483840\)\(\medspace = 2^{9} \cdot 3^{3} \cdot 5 \cdot 7 \)
Composition factors:	$C_2$ x 3, $C_3$, $C_5$ x 2, $C_7$ x 2
Derived length:	$2$

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

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	211	2	212	24	2	48	864	4	1308	48	888	96	1356	48	1152	96	96	6192	192	2304	7344	2304	4608	29400
Conjugacy classes	1	3	1	4	14	1	27	22	2	39	24	36	48	66	24	588	48	48	636	96	1152	1224	1152	2304	7560
Divisions	1	3	1	2	4	1	5	6	1	7	4	5	5	6	4	26	5	4	28	5	26	27	26	26	228
Autjugacy classes	1	2	1	2	3	1	3	4	1	4	3	4	3	4	3	9	3	3	10	3	9	10	9	9	104

Minimal presentations

Permutation degree:	$31$
Transitive degree:	$840$
Rank:	$2$
Inequivalent generating pairs:	not computed

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

Constructions

Presentation:	$\langle a, b \mid b^{420}=1, a^{70}=b^{210}, b^{a}=b^{209} \rangle$
Permutation group:	Degree $31$ $\langle(2,3)(4,5)(7,8)(10,11)(12,13)(14,15), (1,2,4,5,3)(6,7,8)(9,10,12,14,15,13,11) \!\cdots\! \rangle$
Matrix group:	$\left\langle \left(\begin{array}{rr} 4 & 0 \\ 0 & 316 \end{array}\right), \left(\begin{array}{rr} 0 & 1 \\ 1 & 0 \end{array}\right), \left(\begin{array}{rr} 2 & 0 \\ 0 & 32 \end{array}\right) \right\rangle \subseteq \GL_{2}(\F_{421})$
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:	$D_{210}$ . $C_{70}$	$C_{420}$ . $D_{35}$	$C_{210}$ . $D_{70}$	$C_{70}$ . $D_{210}$	all 103

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

Homology

Abelianization:	$C_{2} \times C_{140} \simeq C_{2} \times C_{4} \times C_{5} \times C_{7}$
Schur multiplier:	$C_{2}$
Commutator length:	not computed

Subgroups

There are 4320 subgroups in 448 conjugacy classes, 116 normal (108 characteristic).

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

Special subgroups

Center:	$Z \simeq$ $C_{140}$	$G/Z \simeq$ $D_{105}$
Commutator:	$G' \simeq$ $C_{105}$	$G/G' \simeq$ $C_2\times C_{140}$
Frattini:	$\Phi \simeq$ $C_2$	$G/\Phi \simeq$ $C_{35}\times D_{210}$
Fitting:	$\operatorname{Fit} \simeq$ $C_{35}\times C_{420}$	$G/\operatorname{Fit} \simeq$ $C_2$
Radical:	$R \simeq$ $C_{420}.C_{70}$	$G/R \simeq$ $C_1$
Socle:	$\operatorname{soc} \simeq$ $C_{35}\times C_{210}$	$G/\operatorname{soc} \simeq$ $C_2^2$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $C_2\times C_4$
3-Sylow subgroup:	$P_{ 3 } \simeq$ $C_3$
5-Sylow subgroup:	$P_{ 5 } \simeq$ $C_5^2$
7-Sylow subgroup:	$P_{ 7 } \simeq$ $C_7^2$

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 29400: $C_{420}.C_{70}$

Order 14700: $C_{35}\times C_{420}$, $C_{105}:C_{140}$, $C_{35}\times D_{210}$

Order 9800: $D_{35}\times C_{140}$

Order 7350: $C_{35}\times D_{105}$ x 2, $C_{35}\times C_{210}$

Order 5880: $D_{21}\times C_{140}$, $C_{28}\times D_{105}$

Order 4900: $C_{35}\times C_{140}$, $C_{35}\times D_{70}$, $C_{35}:C_{140}$

Order 4200: $D_{15}\times C_{140}$, $C_{20}\times D_{105}$

Order 3675: $C_{35}\times C_{105}$

Order 2940: $C_7\times C_{420}$ x 4, $C_{105}:C_{28}$, $C_{21}:C_{140}$, $C_{35}\times D_{42}$, $C_7\times D_{210}$

Order 2450: $C_{35}\times D_{35}$ x 2, $C_{35}\times C_{70}$

Order 2100: $C_5\times C_{420}$ x 5, $C_{35}\times D_{30}$, $C_{105}:C_{20}$, $C_{15}:C_{140}$, $C_5\times D_{210}$

Order 1960: $D_7\times C_{140}$, $C_{28}\times D_{35}$

Order 1470: $C_7\times C_{210}$ x 4, $C_7\times D_{105}$ x 2, $C_{35}\times D_{21}$ x 2

Order 1400: $C_{20}\times D_{35}$, $D_5\times C_{140}$

Order 1225: $C_{35}^2$

Order 1176: $C_{28}\times D_{21}$

Order 1050: $C_5\times C_{210}$ x 5, $C_5\times D_{105}$ x 2, $D_{15}\times C_{35}$ x 2

Order 980: $C_7\times C_{140}$ x 4, $C_7\times D_{70}$, $D_7\times C_{70}$, $C_{35}:C_{28}$, $C_7:C_{140}$

Order 840: $C_4\times D_{105}$, $C_{28}\times D_{15}$, $C_{20}\times D_{21}$, $S_3\times C_{140}$

Order 735: $C_7\times C_{105}$ x 4

Order 700: $C_5\times C_{140}$ x 5, $C_5\times D_{70}$, $D_5\times C_{70}$, $C_{35}:C_{20}$, $C_5:C_{140}$

Order 600: $C_{20}\times D_{15}$

Order 588: $C_7\times D_{42}$, $C_7\times C_{84}$, $C_{21}:C_{28}$

Order 525: $C_5\times C_{105}$ x 5

Order 490: $C_7\times C_{70}$ x 4, $C_7\times D_{35}$ x 2, $D_7\times C_{35}$ x 2

Order 420: $C_{420}$ x 26, $C_{105}:C_4$, $C_3:C_{140}$, $D_{210}$, $C_7\times D_{30}$, $C_5\times D_{42}$, $S_3\times C_{70}$, $C_{105}:C_4$, $C_{15}:C_{28}$

Order 392: $D_7\times C_{28}$

Order 350: $C_5\times C_{70}$ x 5, $C_5\times D_{35}$ x 2, $D_5\times C_{35}$ x 2

Order 300: $C_5\times D_{30}$, $C_5\times C_{60}$, $C_{15}:C_{20}$

Order 294: $C_7\times D_{21}$ x 2, $C_7\times C_{42}$

Order 280: $C_2\times C_{140}$, $C_4\times D_{35}$, $D_5\times C_{28}$, $D_7\times C_{20}$

Order 245: $C_7\times C_{35}$ x 4

Order 210: $C_{210}$ x 26, $C_5\times D_{21}$ x 2, $S_3\times C_{35}$ x 2, $D_{105}$ x 2, $C_7\times D_{15}$ x 2

Order 200: $D_5\times C_{20}$

Order 196: $C_7\times C_{28}$, $C_7:C_{28}$, $C_7\times D_{14}$

Order 175: $C_5\times C_{35}$ x 5

Order 168: $C_4\times D_{21}$, $S_3\times C_{28}$

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

Order 147: $C_7\times C_{21}$

Order 140: $C_{140}$ x 27, $C_7\times D_{10}$, $C_5\times D_{14}$, $C_{35}:C_4$, $C_7:C_{20}$, $C_2\times C_{70}$, $D_{70}$, $C_5:C_{28}$

Order 120: $C_4\times D_{15}$, $S_3\times C_{20}$

Order 105: $C_{105}$ x 26

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

Order 98: $C_7\times D_7$ x 2, $C_7\times C_{14}$

Order 84: $C_{84}$ x 5, $C_{21}:C_4$, $C_3:C_{28}$, $D_{42}$, $S_3\times C_{14}$

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

Order 70: $C_{70}$ x 28, $D_{35}$ x 2, $C_5\times D_7$ x 2, $C_7\times D_5$ x 2

Order 60: $C_{60}$ x 4, $C_{15}:C_4$, $D_{30}$, $S_3\times C_{10}$, $C_3:C_{20}$

Order 56: $C_2\times C_{28}$, $C_4\times D_7$

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

Order 49: $C_7^2$

Order 42: $C_{42}$ x 5, $D_{21}$ x 2, $S_3\times C_7$ x 2

Order 40: $C_2\times C_{20}$, $C_4\times D_5$

Order 35: $C_{35}$ x 26

Order 30: $C_{30}$ x 4, $D_{15}$ x 2, $C_5\times S_3$ x 2

Order 28: $C_{28}$ x 6, $C_2\times C_{14}$, $D_{14}$, $C_7:C_4$

Order 25: $C_5^2$

Order 24: $C_4\times S_3$

Order 21: $C_{21}$ x 5

Order 20: $C_{20}$ x 5, $C_2\times C_{10}$, $D_{10}$, $C_5:C_4$

Order 15: $C_{15}$ x 4

Order 14: $C_{14}$ x 7, $D_7$ x 2

Order 12: $D_6$, $C_{12}$, $C_3:C_4$

Order 10: $C_{10}$ x 6, $D_5$ x 2

Order 8: $C_2\times C_4$

Order 7: $C_7$ x 5

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

Order 5: $C_5$ x 4

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

Order 3: $C_3$

Order 2: $C_2$ x 3

Order 1: $C_1$

Classes of subgroups up to automorphism

Order 29400: $C_{420}.C_{70}$

Order 14700: $C_{105}:C_{140}$

Order 9800: $D_{35}\times C_{140}$

Order 3675: $C_{35}\times C_{105}$

Order 1225: $C_{35}^2$

Order 1176: $C_{28}\times D_{21}$

Order 600: $C_{20}\times D_{15}$

Order 392: $D_7\times C_{28}$

Order 200: $D_5\times C_{20}$

Order 147: $C_7\times C_{21}$

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

Order 49: $C_7^2$

Order 25: $C_5^2$

Order 24: $C_4\times S_3$

Order 8: $C_2\times C_4$

Order 3: $C_3$

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 29400: $C_{420}.C_{70}$ ($C_1$)

Order 14700: $C_{35}\times C_{420}$ ($C_2$), $C_{105}:C_{140}$ ($C_2$), $C_{35}\times D_{210}$ ($C_2$)

Order 7350: $C_{35}\times D_{105}$ ($C_4$) x 2, $C_{35}\times C_{210}$ ($C_2^2$)

Order 5880: $C_{28}\times D_{105}$ ($C_5$)

Order 4900: $C_{35}\times C_{140}$ ($S_3$)

Order 4200: $C_{20}\times D_{105}$ ($C_7$)

Order 3675: $C_{35}\times C_{105}$ ($C_2\times C_4$)

Order 2940: $C_7\times C_{420}$ ($C_{10}$), $C_7\times C_{420}$ ($D_5$), $C_{105}:C_{28}$ ($C_{10}$), $C_7\times D_{210}$ ($C_{10}$)

Order 2450: $C_{35}\times C_{70}$ ($D_6$)

Order 2100: $C_5\times C_{420}$ ($C_{14}$), $C_5\times C_{420}$ ($D_7$), $C_{105}:C_{20}$ ($C_{14}$), $C_5\times D_{210}$ ($C_{14}$)

Order 1470: $C_7\times D_{105}$ ($C_{20}$) x 2, $C_7\times C_{210}$ ($C_2\times C_{10}$), $C_7\times C_{210}$ ($D_{10}$)

Order 1225: $C_{35}^2$ ($C_4\times S_3$)

Order 1050: $C_5\times D_{105}$ ($C_{28}$) x 2, $C_5\times C_{210}$ ($C_2\times C_{14}$), $C_5\times C_{210}$ ($D_{14}$)

Order 980: $C_7\times C_{140}$ ($D_{15}$), $C_7\times C_{140}$ ($C_5\times S_3$)

Order 840: $C_4\times D_{105}$ ($C_{35}$)

Order 735: $C_7\times C_{105}$ ($C_2\times C_{20}$), $C_7\times C_{105}$ ($C_4\times D_5$)

Order 700: $C_5\times C_{140}$ ($D_{21}$), $C_5\times C_{140}$ ($S_3\times C_7$)

Order 588: $C_7\times C_{84}$ ($C_5\times D_5$)

Order 525: $C_5\times C_{105}$ ($C_2\times C_{28}$), $C_5\times C_{105}$ ($C_4\times D_7$)

Order 490: $C_7\times C_{70}$ ($D_{30}$), $C_7\times C_{70}$ ($S_3\times C_{10}$)

Order 420: $D_{210}$ ($C_{70}$), $C_{420}$ ($C_{70}$), $C_{420}$ ($D_{35}$), $C_{420}$ ($C_5\times D_7$), $C_{420}$ ($C_7\times D_5$), $C_{105}:C_4$ ($C_{70}$)

Order 350: $C_5\times C_{70}$ ($D_{42}$), $C_5\times C_{70}$ ($S_3\times C_{14}$)

Order 300: $C_5\times C_{60}$ ($C_7\times D_7$)

Order 294: $C_7\times C_{42}$ ($C_5\times D_{10}$)

Order 245: $C_7\times C_{35}$ ($C_4\times D_{15}$), $C_7\times C_{35}$ ($S_3\times C_{20}$)

Order 210: $D_{105}$ ($C_{140}$) x 2, $C_{210}$ ($C_7\times D_{10}$), $C_{210}$ ($C_5\times D_{14}$), $C_{210}$ ($C_2\times C_{70}$), $C_{210}$ ($D_{70}$)

Order 196: $C_7\times C_{28}$ ($C_5\times D_{15}$)

Order 175: $C_5\times C_{35}$ ($C_4\times D_{21}$), $C_5\times C_{35}$ ($S_3\times C_{28}$)

Order 150: $C_5\times C_{30}$ ($C_7\times D_{14}$)

Order 147: $C_7\times C_{21}$ ($D_5\times C_{20}$)

Order 140: $C_{140}$ ($C_5\times D_{21}$), $C_{140}$ ($S_3\times C_{35}$), $C_{140}$ ($D_{105}$), $C_{140}$ ($C_7\times D_{15}$)

Order 105: $C_{105}$ ($C_2\times C_{140}$), $C_{105}$ ($C_4\times D_{35}$), $C_{105}$ ($D_5\times C_{28}$), $C_{105}$ ($D_7\times C_{20}$)

Order 100: $C_5\times C_{20}$ ($C_7\times D_{21}$)

Order 98: $C_7\times C_{14}$ ($C_5\times D_{30}$)

Order 84: $C_{84}$ ($C_5\times D_{35}$), $C_{84}$ ($D_5\times C_{35}$)

Order 75: $C_5\times C_{15}$ ($D_7\times C_{28}$)

Order 70: $C_{70}$ ($D_{210}$), $C_{70}$ ($C_7\times D_{30}$), $C_{70}$ ($C_5\times D_{42}$), $C_{70}$ ($S_3\times C_{70}$)

Order 60: $C_{60}$ ($C_7\times D_{35}$), $C_{60}$ ($D_7\times C_{35}$)

Order 50: $C_5\times C_{10}$ ($C_7\times D_{42}$)

Order 49: $C_7^2$ ($C_{20}\times D_{15}$)

Order 42: $C_{42}$ ($C_5\times D_{70}$), $C_{42}$ ($D_5\times C_{70}$)

Order 35: $C_{35}$ ($C_4\times D_{105}$), $C_{35}$ ($C_{28}\times D_{15}$), $C_{35}$ ($C_{20}\times D_{21}$), $C_{35}$ ($S_3\times C_{140}$)

Order 30: $C_{30}$ ($C_7\times D_{70}$), $C_{30}$ ($D_7\times C_{70}$)

Order 28: $C_{28}$ ($C_5\times D_{105}$), $C_{28}$ ($D_{15}\times C_{35}$)

Order 25: $C_5^2$ ($C_{28}\times D_{21}$)

Order 21: $C_{21}$ ($C_{20}\times D_{35}$), $C_{21}$ ($D_5\times C_{140}$)

Order 20: $C_{20}$ ($C_7\times D_{105}$), $C_{20}$ ($C_{35}\times D_{21}$)

Order 15: $C_{15}$ ($D_7\times C_{140}$), $C_{15}$ ($C_{28}\times D_{35}$)

Order 14: $C_{14}$ ($C_{35}\times D_{30}$), $C_{14}$ ($C_5\times D_{210}$)

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

Order 10: $C_{10}$ ($C_{35}\times D_{42}$), $C_{10}$ ($C_7\times D_{210}$)

Order 7: $C_7$ ($D_{15}\times C_{140}$), $C_7$ ($C_{20}\times D_{105}$)

Order 6: $C_6$ ($C_{35}\times D_{70}$)

Order 5: $C_5$ ($D_{21}\times C_{140}$), $C_5$ ($C_{28}\times D_{105}$)

Order 4: $C_4$ ($C_{35}\times D_{105}$)

Order 3: $C_3$ ($D_{35}\times C_{140}$)

Order 2: $C_2$ ($C_{35}\times D_{210}$)

Order 1: $C_1$ ($C_{420}.C_{70}$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 29400: $C_{420}.C_{70}$ ($C_1$)

Order 14700: $C_{105}:C_{140}$ ($C_2$)

Order 3675: $C_{35}\times C_{105}$ ($C_2\times C_4$)

Order 1225: $C_{35}^2$ ($C_4\times S_3$)

Order 147: $C_7\times C_{21}$ ($D_5\times C_{20}$)

Order 75: $C_5\times C_{15}$ ($D_7\times C_{28}$)

Order 49: $C_7^2$ ($C_{20}\times D_{15}$)

Order 25: $C_5^2$ ($C_{28}\times D_{21}$)

Order 3: $C_3$ ($D_{35}\times C_{140}$)

Order 1: $C_1$ ($C_{420}.C_{70}$)

Series

Derived series

$C_{420}.C_{70}$

$\rhd$

$C_{105}$

$\rhd$

$C_1$

Chief series

$C_{420}.C_{70}$

$\rhd$

$C_{35}\times C_{420}$

$\rhd$

$C_7\times C_{420}$

$\rhd$

$C_{420}$

$\rhd$

$C_{210}$

$\rhd$

$C_{105}$

$\rhd$

$C_{35}$

$\rhd$

$C_7$

$\rhd$

$C_1$

Lower central series

$C_{420}.C_{70}$

$\rhd$

$C_{105}$

Upper central series

$C_1$

$\lhd$

$C_{140}$

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 $7560 \times 7560$ character table is not available for this group.

Rational character table

The $228 \times 228$ rational character table is not available for this group.