Abstract group 840.7: $C_{105}:C_8$

• Label: 840.7
• Order: \( 2^{3} \cdot 3 \cdot 5 \cdot 7 \)
• Exponent: \( 2^{3} \cdot 3 \cdot 5 \cdot 7 \)
• Nilpotent: no
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{3} \cdot 3 \)
• $\card{Z(G)}$: \( 2^{2} \cdot 3 \)
• $\card{\Aut(G)}$: \( 2^{6} \cdot 3 \cdot 5 \cdot 7 \)
• $\card{\mathrm{Out}(G)}$: \( 2^{5} \cdot 3 \)
• Perm deg.: $23$
• Trans deg.: $840$
• Rank: $2$

Group information

Description:	$C_{105}:C_8$
Order:	\(840\)\(\medspace = 2^{3} \cdot 3 \cdot 5 \cdot 7 \)
Exponent:	\(840\)\(\medspace = 2^{3} \cdot 3 \cdot 5 \cdot 7 \)
Automorphism group:	$C_2^3\times F_5\times F_7$, of order \(6720\)\(\medspace = 2^{6} \cdot 3 \cdot 5 \cdot 7 \)
Composition factors:	$C_2$ x 3, $C_3$, $C_5$, $C_7$
Derived length:	$2$

This group is nonabelian, a Z-group (hence solvable, supersolvable, monomial, metacyclic, metabelian, and an A-group), and hyperelementary for $p = 2$.

Group statistics

Order	1	2	3	4	5	6	7	8	10	12	14	15	20	21	24	28	30	35	42	60	70	84	105	140	210	420
Elements	1	1	2	2	4	2	6	140	4	4	6	8	8	12	280	12	8	24	12	16	24	24	48	48	48	96	840
Conjugacy classes	1	1	2	2	2	2	3	4	2	4	3	4	4	6	8	6	4	12	6	8	12	12	24	24	24	48	228
Divisions	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	26
Autjugacy classes	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	26

Dimension	1	2	4	6	8	12	16	24	48	96
Irr. complex chars.	24	204	0	0	0	0	0	0	0	0	228
Irr. rational chars.	2	3	4	2	4	3	1	3	3	1	26

Minimal presentations

Permutation degree:	$23$
Transitive degree:	$840$
Rank:	$2$
Inequivalent generating pairs:	$48$

Minimal degrees of faithful linear representations

	Over $\mathbb{C}$	Over $\mathbb{R}$	Over $\mathbb{Q}$
Irreducible	2	4	96
Arbitrary	2	4	16

Constructions

Presentation:	$\langle a, b \mid a^{8}=b^{105}=1, b^{a}=b^{34} \rangle$
Permutation group:	Degree $23$ $\langle(2,3)(4,5)(6,7)(12,13)(14,15)(16,17,18,20,19,21,22,23), (8,9,10), (16,18,19,22) \!\cdots\! \rangle$
Matrix group:	$\left\langle \left(\begin{array}{rr} 98 & 316 \\ 191 & 251 \end{array}\right), \left(\begin{array}{rr} 104 & 129 \\ 239 & 104 \end{array}\right) \right\rangle \subseteq \GL_{2}(\F_{349})$
Direct product:	$C_3$ $\, \times\, $ $(C_{35}:C_8)$
Semidirect product:	$C_{105}$ $\,\rtimes\,$ $C_8$	$C_{35}$ $\,\rtimes\,$ $C_{24}$	$C_{21}$ $\,\rtimes\,$ $(C_5:C_8)$	$C_{15}$ $\,\rtimes\,$ $(C_7:C_8)$	all 6
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Non-split product:	$C_{84}$ . $D_5$	$C_{60}$ . $D_7$	$C_{12}$ . $D_{35}$	$C_{420}$ . $C_2$	all 16

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

Homology

Abelianization:	$C_{24} \simeq C_{8} \times C_{3}$
Schur multiplier:	$C_1$
Commutator length:	$1$

Subgroups

There are 120 subgroups in 32 conjugacy classes, 26 normal, and all normal subgroups are characteristic.

Characteristic subgroups are shown in this color.

Special subgroups

Center:	$Z \simeq$ $C_{12}$	$G/Z \simeq$ $D_{35}$
Commutator:	$G' \simeq$ $C_{35}$	$G/G' \simeq$ $C_{24}$
Frattini:	$\Phi \simeq$ $C_4$	$G/\Phi \simeq$ $C_3\times D_{35}$
Fitting:	$\operatorname{Fit} \simeq$ $C_{420}$	$G/\operatorname{Fit} \simeq$ $C_2$
Radical:	$R \simeq$ $C_{105}:C_8$	$G/R \simeq$ $C_1$
Socle:	$\operatorname{soc} \simeq$ $C_{210}$	$G/\operatorname{soc} \simeq$ $C_4$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $C_8$
3-Sylow subgroup:	$P_{ 3 } \simeq$ $C_3$
5-Sylow subgroup:	$P_{ 5 } \simeq$ $C_5$
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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Classes of subgroups up to conjugation

Order 840: $C_{105}:C_8$

Order 420: $C_{420}$

Order 280: $C_{35}:C_8$

Order 210: $C_{210}$

Order 168: $C_7:C_{24}$

Order 140: $C_{140}$

Order 120: $C_5:C_{24}$

Order 105: $C_{105}$

Order 84: $C_{84}$

Order 70: $C_{70}$

Order 60: $C_{60}$

Order 56: $C_7:C_8$

Order 42: $C_{42}$

Order 40: $C_5:C_8$

Order 35: $C_{35}$

Order 30: $C_{30}$

Order 28: $C_{28}$

Order 24: $C_{24}$

Order 21: $C_{21}$

Order 20: $C_{20}$

Order 15: $C_{15}$

Order 14: $C_{14}$

Order 12: $C_{12}$

Order 10: $C_{10}$

Order 8: $C_8$

Order 7: $C_7$

Order 6: $C_6$

Order 5: $C_5$

Order 4: $C_4$

Order 3: $C_3$

Order 2: $C_2$

Order 1: $C_1$

Classes of subgroups up to automorphism

Order 840: $C_{105}:C_8$

Order 420: $C_{420}$

Order 280: $C_{35}:C_8$

Order 210: $C_{210}$

Order 168: $C_7:C_{24}$

Order 140: $C_{140}$

Order 120: $C_5:C_{24}$

Order 105: $C_{105}$

Order 84: $C_{84}$

Order 70: $C_{70}$

Order 60: $C_{60}$

Order 56: $C_7:C_8$

Order 42: $C_{42}$

Order 40: $C_5:C_8$

Order 35: $C_{35}$

Order 30: $C_{30}$

Order 28: $C_{28}$

Order 24: $C_{24}$

Order 21: $C_{21}$

Order 20: $C_{20}$

Order 15: $C_{15}$

Order 14: $C_{14}$

Order 12: $C_{12}$

Order 10: $C_{10}$

Order 8: $C_8$

Order 7: $C_7$

Order 6: $C_6$

Order 5: $C_5$

Order 4: $C_4$

Order 3: $C_3$

Order 2: $C_2$

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 840: $C_{105}:C_8$ ($C_1$)

Order 420: $C_{420}$ ($C_2$)

Order 280: $C_{35}:C_8$ ($C_3$)

Order 210: $C_{210}$ ($C_4$)

Order 140: $C_{140}$ ($C_6$)

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

Order 84: $C_{84}$ ($D_5$)

Order 70: $C_{70}$ ($C_{12}$)

Order 60: $C_{60}$ ($D_7$)

Order 42: $C_{42}$ ($C_5:C_4$)

Order 35: $C_{35}$ ($C_{24}$)

Order 30: $C_{30}$ ($C_7:C_4$)

Order 28: $C_{28}$ ($C_3\times D_5$)

Order 21: $C_{21}$ ($C_5:C_8$)

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

Order 15: $C_{15}$ ($C_7:C_8$)

Order 14: $C_{14}$ ($C_5:C_{12}$)

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

Order 10: $C_{10}$ ($C_7:C_{12}$)

Order 7: $C_7$ ($C_5:C_{24}$)

Order 6: $C_6$ ($C_{35}:C_4$)

Order 5: $C_5$ ($C_7:C_{24}$)

Order 4: $C_4$ ($C_3\times D_{35}$)

Order 3: $C_3$ ($C_{35}:C_8$)

Order 2: $C_2$ ($C_{105}:C_4$)

Order 1: $C_1$ ($C_{105}:C_8$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 840: $C_{105}:C_8$ ($C_1$)

Order 420: $C_{420}$ ($C_2$)

Order 280: $C_{35}:C_8$ ($C_3$)

Order 210: $C_{210}$ ($C_4$)

Order 140: $C_{140}$ ($C_6$)

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

Order 84: $C_{84}$ ($D_5$)

Order 70: $C_{70}$ ($C_{12}$)

Order 60: $C_{60}$ ($D_7$)

Order 42: $C_{42}$ ($C_5:C_4$)

Order 35: $C_{35}$ ($C_{24}$)

Order 30: $C_{30}$ ($C_7:C_4$)

Order 28: $C_{28}$ ($C_3\times D_5$)

Order 21: $C_{21}$ ($C_5:C_8$)

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

Order 15: $C_{15}$ ($C_7:C_8$)

Order 14: $C_{14}$ ($C_5:C_{12}$)

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

Order 10: $C_{10}$ ($C_7:C_{12}$)

Order 7: $C_7$ ($C_5:C_{24}$)

Order 6: $C_6$ ($C_{35}:C_4$)

Order 5: $C_5$ ($C_7:C_{24}$)

Order 4: $C_4$ ($C_3\times D_{35}$)

Order 3: $C_3$ ($C_{35}:C_8$)

Order 2: $C_2$ ($C_{105}:C_4$)

Order 1: $C_1$ ($C_{105}:C_8$)

Series

Derived series

$C_{105}:C_8$

$\rhd$

$C_{35}$

$\rhd$

$C_1$

Chief series

$C_{105}:C_8$

$\rhd$

$C_{420}$

$\rhd$

$C_{210}$

$\rhd$

$C_{105}$

$\rhd$

$C_{35}$

$\rhd$

$C_7$

$\rhd$

$C_1$

Lower central series

$C_{105}:C_8$

$\rhd$

$C_{35}$

Upper central series

$C_1$

$\lhd$

$C_{12}$

Supergroups

This group is a maximal subgroup of 37 larger groups in the database.

This group is a maximal quotient of 7 larger groups in the database.

Character theory

Complex character table

See the $228 \times 228$ 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 $26 \times 26$ rational character table.