Abstract group 4200.o: $C_{35}:C_{120}$

• Label: 4200.o
• Order: \( 2^{3} \cdot 3 \cdot 5^{2} \cdot 7 \)
• Exponent: \( 2^{3} \cdot 3 \cdot 5 \cdot 7 \)
• Nilpotent: no
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{3} \cdot 3 \cdot 5 \)
• $\card{Z(G)}$: \( 2^{2} \cdot 3 \cdot 5 \)
• $\card{\Aut(G)}$: \( 2^{8} \cdot 3 \cdot 5 \cdot 7 \)
• $\card{\mathrm{Out}(G)}$: \( 2^{7} \cdot 3 \)
• Perm deg.: $28$
• Trans deg.: $840$
• Rank: $2$

Group information

Description:	$C_{35}:C_{120}$
Order:	\(4200\)\(\medspace = 2^{3} \cdot 3 \cdot 5^{2} \cdot 7 \)
Exponent:	\(840\)\(\medspace = 2^{3} \cdot 3 \cdot 5 \cdot 7 \)
Automorphism group:	$C_2^3\times C_4\times F_5\times F_7$, of order \(26880\)\(\medspace = 2^{8} \cdot 3 \cdot 5 \cdot 7 \)
Composition factors:	$C_2$ x 3, $C_3$, $C_5$ x 2, $C_7$
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	8	10	12	14	15	20	21	24	28	30	35	40	42	60	70	84	105	120	140	210	420
Elements	1	1	2	2	24	2	6	140	24	4	6	48	48	12	280	12	48	144	560	12	96	144	24	288	1120	288	288	576	4200
Conjugacy classes	1	1	2	2	14	2	3	4	14	4	3	28	28	6	8	6	28	72	16	6	56	72	12	144	32	144	144	288	1140
Divisions	1	1	1	1	4	1	1	1	4	1	1	4	4	1	1	1	4	4	1	1	4	4	1	4	1	4	4	4	64
Autjugacy classes	1	1	1	1	3	1	1	1	3	1	1	3	3	1	1	1	3	3	1	1	3	3	1	3	1	3	3	3	52

Dimension	1	2	4	6	8	12	16	24	32	48	96	192
Irr. complex chars.	120	1020	0	0	0	0	0	0	0	0	0	0	1140
Irr. rational chars.	2	3	6	2	11	3	9	5	3	10	8	2	64

Minimal presentations

Permutation degree:	$28$
Transitive degree:	$840$
Rank:	$2$
Inequivalent generating pairs:	$288$

Minimal degrees of faithful linear representations

	Over $\mathbb{C}$	Over $\mathbb{R}$	Over $\mathbb{Q}$
Irreducible	2	not computed	192
Arbitrary	not computed	not computed	not computed

Constructions

Presentation:	$\langle a, b \mid a^{120}=b^{35}=1, b^{a}=b^{34} \rangle$
Permutation group:	Degree $28$ $\langle(1,2,4,3,5)(6,7,8)(9,10,11,13)(12,14,15,16)(17,18,20,22,23,21,19)(24,25,27,28,26) \!\cdots\! \rangle$
Matrix group:	$\left\langle \left(\begin{array}{rr} 2 & 0 \\ 0 & 193 \end{array}\right), \left(\begin{array}{rr} 64 & 0 \\ 0 & 125 \end{array}\right), \left(\begin{array}{rr} 0 & 128 \\ 1 & 0 \end{array}\right) \right\rangle \subseteq \GL_{2}(\F_{421})$
Direct product:	$C_3$ $\, \times\, $ $C_5$ $\, \times\, $ $(C_{35}:C_8)$
Semidirect product:	$C_{105}$ $\,\rtimes\,$ $C_{40}$	$C_{35}$ $\,\rtimes\,$ $C_{120}$	$(C_5\times C_{105})$ $\,\rtimes\,$ $C_8$	$(C_5\times C_{35})$ $\,\rtimes\,$ $C_{24}$	all 12
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Non-split product:	$C_{420}$ . $D_5$	$C_{60}$ . $D_{35}$	$C_{70}$ . $C_{60}$	$C_{420}$ . $C_{10}$	all 32

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

Homology

Abelianization:	$C_{120} \simeq C_{8} \times C_{3} \times C_{5}$
Schur multiplier:	$C_1$
Commutator length:	$1$

Subgroups

There are 288 subgroups in 88 conjugacy classes, 52 normal, and all normal subgroups are characteristic.

Characteristic subgroups are shown in this color.

Special subgroups

Center:	$Z \simeq$ $C_{60}$	$G/Z \simeq$ $D_{35}$
Commutator:	$G' \simeq$ $C_{35}$	$G/G' \simeq$ $C_{120}$
Frattini:	$\Phi \simeq$ $C_4$	$G/\Phi \simeq$ $C_{15}\times D_{35}$
Fitting:	$\operatorname{Fit} \simeq$ $C_5\times C_{420}$	$G/\operatorname{Fit} \simeq$ $C_2$
Radical:	$R \simeq$ $C_{35}:C_{120}$	$G/R \simeq$ $C_1$
Socle:	$\operatorname{soc} \simeq$ $C_5\times C_{210}$	$G/\operatorname{soc} \simeq$ $C_4$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $C_8$
3-Sylow subgroup:	$P_{ 3 } \simeq$ $C_3$
5-Sylow subgroup:	$P_{ 5 } \simeq$ $C_5^2$
7-Sylow subgroup:	$P_{ 7 } \simeq$ $C_7$

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

Click on a subgroup in the diagram to see information about it.

Classes of subgroups up to conjugation

Order 4200: $C_{35}:C_{120}$

Order 2100: $C_5\times C_{420}$

Order 1400: $C_{35}:C_{40}$

Order 1050: $C_5\times C_{210}$

Order 840: $C_{105}:C_8$, $C_7:C_{120}$

Order 700: $C_5\times C_{140}$

Order 600: $C_5:C_{120}$

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

Order 420: $C_{420}$ x 4

Order 350: $C_5\times C_{70}$

Order 300: $C_5\times C_{60}$

Order 280: $C_{35}:C_8$, $C_7:C_{40}$

Order 210: $C_{210}$ x 4

Order 200: $C_5:C_{40}$

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

Order 168: $C_7:C_{24}$

Order 150: $C_5\times C_{30}$

Order 140: $C_{140}$ x 4

Order 120: $C_{120}$, $C_5:C_{24}$

Order 105: $C_{105}$ x 4

Order 100: $C_5\times C_{20}$

Order 84: $C_{84}$

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

Order 70: $C_{70}$ x 4

Order 60: $C_{60}$ x 4

Order 56: $C_7:C_8$

Order 50: $C_5\times C_{10}$

Order 42: $C_{42}$

Order 40: $C_{40}$, $C_5:C_8$

Order 35: $C_{35}$ x 4

Order 30: $C_{30}$ x 4

Order 28: $C_{28}$

Order 25: $C_5^2$

Order 24: $C_{24}$

Order 21: $C_{21}$

Order 20: $C_{20}$ x 4

Order 15: $C_{15}$ x 4

Order 14: $C_{14}$

Order 12: $C_{12}$

Order 10: $C_{10}$ x 4

Order 8: $C_8$

Order 7: $C_7$

Order 6: $C_6$

Order 5: $C_5$ x 4

Order 4: $C_4$

Order 3: $C_3$

Order 2: $C_2$

Order 1: $C_1$

Classes of subgroups up to automorphism

Order 4200: $C_{35}:C_{120}$

Order 2100: $C_5\times C_{420}$

Order 1400: $C_{35}:C_{40}$

Order 1050: $C_5\times C_{210}$

Order 840: $C_{105}:C_8$, $C_7:C_{120}$

Order 700: $C_5\times C_{140}$

Order 600: $C_5:C_{120}$

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

Order 420: $C_{420}$ x 3

Order 350: $C_5\times C_{70}$

Order 300: $C_5\times C_{60}$

Order 280: $C_{35}:C_8$, $C_7:C_{40}$

Order 210: $C_{210}$ x 3

Order 200: $C_5:C_{40}$

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

Order 168: $C_7:C_{24}$

Order 150: $C_5\times C_{30}$

Order 140: $C_{140}$ x 3

Order 120: $C_{120}$, $C_5:C_{24}$

Order 105: $C_{105}$ x 3

Order 100: $C_5\times C_{20}$

Order 84: $C_{84}$

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

Order 70: $C_{70}$ x 3

Order 60: $C_{60}$ x 3

Order 56: $C_7:C_8$

Order 50: $C_5\times C_{10}$

Order 42: $C_{42}$

Order 40: $C_{40}$, $C_5:C_8$

Order 35: $C_{35}$ x 3

Order 30: $C_{30}$ x 3

Order 28: $C_{28}$

Order 25: $C_5^2$

Order 24: $C_{24}$

Order 21: $C_{21}$

Order 20: $C_{20}$ x 3

Order 15: $C_{15}$ x 3

Order 14: $C_{14}$

Order 12: $C_{12}$

Order 10: $C_{10}$ x 3

Order 8: $C_8$

Order 7: $C_7$

Order 6: $C_6$

Order 5: $C_5$ x 3

Order 4: $C_4$

Order 3: $C_3$

Order 2: $C_2$

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 4200: $C_{35}:C_{120}$ ($C_1$)

Order 2100: $C_5\times C_{420}$ ($C_2$)

Order 1400: $C_{35}:C_{40}$ ($C_3$)

Order 1050: $C_5\times C_{210}$ ($C_4$)

Order 840: $C_{105}:C_8$ ($C_5$)

Order 700: $C_5\times C_{140}$ ($C_6$)

Order 525: $C_5\times C_{105}$ ($C_8$)

Order 420: $C_{420}$ ($C_{10}$), $C_{420}$ ($D_5$)

Order 350: $C_5\times C_{70}$ ($C_{12}$)

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

Order 280: $C_{35}:C_8$ ($C_{15}$)

Order 210: $C_{210}$ ($C_{20}$), $C_{210}$ ($C_5:C_4$)

Order 175: $C_5\times C_{35}$ ($C_{24}$)

Order 150: $C_5\times C_{30}$ ($C_7:C_4$)

Order 140: $C_{140}$ ($C_{30}$), $C_{140}$ ($C_3\times D_5$)

Order 105: $C_{105}$ ($C_{40}$), $C_{105}$ ($C_5:C_8$)

Order 100: $C_5\times C_{20}$ ($C_3\times D_7$)

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

Order 75: $C_5\times C_{15}$ ($C_7:C_8$)

Order 70: $C_{70}$ ($C_{60}$), $C_{70}$ ($C_5:C_{12}$)

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

Order 50: $C_5\times C_{10}$ ($C_7:C_{12}$)

Order 42: $C_{42}$ ($C_5:C_{20}$)

Order 35: $C_{35}$ ($C_{120}$), $C_{35}$ ($C_5:C_{24}$)

Order 30: $C_{30}$ ($C_{35}:C_4$), $C_{30}$ ($C_7:C_{20}$)

Order 28: $C_{28}$ ($D_5\times C_{15}$)

Order 25: $C_5^2$ ($C_7:C_{24}$)

Order 21: $C_{21}$ ($C_5:C_{40}$)

Order 20: $C_{20}$ ($C_3\times D_{35}$), $C_{20}$ ($D_7\times C_{15}$)

Order 15: $C_{15}$ ($C_{35}:C_8$), $C_{15}$ ($C_7:C_{40}$)

Order 14: $C_{14}$ ($C_5:C_{60}$)

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

Order 10: $C_{10}$ ($C_{105}:C_4$), $C_{10}$ ($C_7:C_{60}$)

Order 7: $C_7$ ($C_5:C_{120}$)

Order 6: $C_6$ ($C_{35}:C_{20}$)

Order 5: $C_5$ ($C_{105}:C_8$), $C_5$ ($C_7:C_{120}$)

Order 4: $C_4$ ($C_{15}\times D_{35}$)

Order 3: $C_3$ ($C_{35}:C_{40}$)

Order 2: $C_2$ ($C_{35}:C_{60}$)

Order 1: $C_1$ ($C_{35}:C_{120}$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 4200: $C_{35}:C_{120}$ ($C_1$)

Order 2100: $C_5\times C_{420}$ ($C_2$)

Order 1400: $C_{35}:C_{40}$ ($C_3$)

Order 1050: $C_5\times C_{210}$ ($C_4$)

Order 840: $C_{105}:C_8$ ($C_5$)

Order 700: $C_5\times C_{140}$ ($C_6$)

Order 525: $C_5\times C_{105}$ ($C_8$)

Order 420: $C_{420}$ ($C_{10}$), $C_{420}$ ($D_5$)

Order 350: $C_5\times C_{70}$ ($C_{12}$)

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

Order 280: $C_{35}:C_8$ ($C_{15}$)

Order 210: $C_{210}$ ($C_{20}$), $C_{210}$ ($C_5:C_4$)

Order 175: $C_5\times C_{35}$ ($C_{24}$)

Order 150: $C_5\times C_{30}$ ($C_7:C_4$)

Order 140: $C_{140}$ ($C_{30}$), $C_{140}$ ($C_3\times D_5$)

Order 105: $C_{105}$ ($C_{40}$), $C_{105}$ ($C_5:C_8$)

Order 100: $C_5\times C_{20}$ ($C_3\times D_7$)

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

Order 75: $C_5\times C_{15}$ ($C_7:C_8$)

Order 70: $C_{70}$ ($C_{60}$), $C_{70}$ ($C_5:C_{12}$)

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

Order 50: $C_5\times C_{10}$ ($C_7:C_{12}$)

Order 42: $C_{42}$ ($C_5:C_{20}$)

Order 35: $C_{35}$ ($C_{120}$), $C_{35}$ ($C_5:C_{24}$)

Order 30: $C_{30}$ ($C_{35}:C_4$), $C_{30}$ ($C_7:C_{20}$)

Order 28: $C_{28}$ ($D_5\times C_{15}$)

Order 25: $C_5^2$ ($C_7:C_{24}$)

Order 21: $C_{21}$ ($C_5:C_{40}$)

Order 20: $C_{20}$ ($C_3\times D_{35}$), $C_{20}$ ($D_7\times C_{15}$)

Order 15: $C_{15}$ ($C_{35}:C_8$), $C_{15}$ ($C_7:C_{40}$)

Order 14: $C_{14}$ ($C_5:C_{60}$)

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

Order 10: $C_{10}$ ($C_{105}:C_4$), $C_{10}$ ($C_7:C_{60}$)

Order 7: $C_7$ ($C_5:C_{120}$)

Order 6: $C_6$ ($C_{35}:C_{20}$)

Order 5: $C_5$ ($C_{105}:C_8$), $C_5$ ($C_7:C_{120}$)

Order 4: $C_4$ ($C_{15}\times D_{35}$)

Order 3: $C_3$ ($C_{35}:C_{40}$)

Order 2: $C_2$ ($C_{35}:C_{60}$)

Order 1: $C_1$ ($C_{35}:C_{120}$)

Series

Derived series

$C_{35}:C_{120}$

$\rhd$

$C_{35}$

$\rhd$

$C_1$

Chief series

$C_{35}:C_{120}$

$\rhd$

$C_5\times C_{420}$

$\rhd$

$C_5\times C_{210}$

$\rhd$

$C_5\times C_{105}$

$\rhd$

$C_5\times C_{35}$

$\rhd$

$C_{35}$

$\rhd$

$C_7$

$\rhd$

$C_1$

Lower central series

$C_{35}:C_{120}$

$\rhd$

$C_{35}$

Upper central series

$C_1$

$\lhd$

$C_{60}$

Supergroups

This group is a maximal subgroup of 3 larger groups in the database.

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

Character theory

Complex character table

The $1140 \times 1140$ character table is not available for this group.

Rational character table

See the $64 \times 64$ rational character table.