Abstract group 966994.c: $C_{28441}:C_{34}$

• Label: 966994.c
• Order: \( 2 \cdot 7 \cdot 17^{2} \cdot 239 \)
• Exponent: \( 2 \cdot 7 \cdot 17 \cdot 239 \)
• Nilpotent: no
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2 \cdot 7 \cdot 17^{2} \)
• $\card{Z(G)}$: \( 2 \cdot 7 \cdot 17 \)
• $\card{\Aut(G)}$: \( 2^{6} \cdot 3 \cdot 7 \cdot 17^{2} \cdot 239 \)
• $\card{\mathrm{Out}(G)}$: \( 2^{6} \cdot 3 \cdot 7 \cdot 17 \)
• Perm deg.: $265$
• Trans deg.: $56882$
• Rank: $2$

Group information

Description:	$C_{28441}:C_{34}$
Order:	\(966994\)\(\medspace = 2 \cdot 7 \cdot 17^{2} \cdot 239 \)
Exponent:	\(56882\)\(\medspace = 2 \cdot 7 \cdot 17 \cdot 239 \)
Automorphism group:	$C_{4063}.C_{357}.C_8.C_2^3$, of order \(92831424\)\(\medspace = 2^{6} \cdot 3 \cdot 7 \cdot 17^{2} \cdot 239 \)
Composition factors:	$C_2$, $C_7$, $C_{17}$ x 2, $C_{239}$
Derived length:	$2$

This group is nonabelian, metacyclic (hence solvable, supersolvable, monomial, and metabelian), hyperelementary for $p = 17$, and an A-group.

Group statistics

Order	1	2	7	14	17	34	119	238	239	478	1673	3346	4063	8126	28441	56882
Elements	1	1	6	6	65024	65024	390144	390144	238	238	1428	1428	3808	3808	22848	22848	966994
Conjugacy classes	1	1	6	6	288	288	1728	1728	14	14	84	84	224	224	1344	1344	7378
Divisions	1	1	1	1	18	18	18	18	1	1	1	1	1	1	1	1	84
Autjugacy classes	1	1	1	1	17	17	17	17	1	1	1	1	1	1	1	1	80

Minimal presentations

Permutation degree:	$265$
Transitive degree:	$56882$
Rank:	$2$
Inequivalent generating pairs:	$6912$

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

Constructions

Presentation:	$\langle a, b \mid a^{34}=b^{28441}=1, b^{a}=b^{26657} \rangle$
Permutation group:	Degree $265$ $\langle(240,241,243,246,250,242,244,247,251,254,245,248,252,255,256,249,253)(257,258,259,260,261,262,263) \!\cdots\! \rangle$
Matrix group:	$\left\langle \left(\begin{array}{rr} 1 & 1 \\ 0 & 1 \end{array}\right), \left(\begin{array}{rr} 7 & 0 \\ 0 & 7 \end{array}\right), \left(\begin{array}{rr} 188 & 0 \\ 0 & 164 \end{array}\right) \right\rangle \subseteq \GL_{2}(\F_{239})$
Direct product:	$C_2$ $\, \times\, $ $C_7$ $\, \times\, $ $C_{17}$ $\, \times\, $ $(C_{239}:C_{17})$
Semidirect product:	$C_{56882}$ $\,\rtimes\,$ $C_{17}$	$C_{28441}$ $\,\rtimes\,$ $C_{34}$	$C_{8126}$ $\,\rtimes\,$ $C_{119}$	$C_{4063}$ $\,\rtimes\,$ $C_{238}$	all 8
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product

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

Homology

Abelianization:	$C_{17} \times C_{238} \simeq C_{2} \times C_{7} \times C_{17}^{2}$
Schur multiplier:	$C_{17}$
Commutator length:	not computed

Subgroups

There are 17296 subgroups in 160 conjugacy classes, 88 normal (20 characteristic).

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

Special subgroups

Center:	$Z \simeq$ $C_{238}$	$G/Z \simeq$ $C_{239}:C_{17}$
Commutator:	$G' \simeq$ $C_{239}$	$G/G' \simeq$ $C_{17}\times C_{238}$
Frattini:	$\Phi \simeq$ $C_1$	$G/\Phi \simeq$ $C_{28441}:C_{34}$
Fitting:	$\operatorname{Fit} \simeq$ $C_{56882}$	$G/\operatorname{Fit} \simeq$ $C_{17}$
Radical:	$R \simeq$ $C_{28441}:C_{34}$	$G/R \simeq$ $C_1$
Socle:	$\operatorname{soc} \simeq$ $C_{56882}$	$G/\operatorname{soc} \simeq$ $C_{17}$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $C_2$
7-Sylow subgroup:	$P_{ 7 } \simeq$ $C_7$
17-Sylow subgroup:	$P_{ 17 } \simeq$ $C_{17}^2$
239-Sylow subgroup:	$P_{ 239 } \simeq$ $C_{239}$

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

Normal subgroups

No diagram available: subgroups only stored up to automorphism

Normal subgroups up to automorphism

For the default diagram, subgroups are sorted vertically by the number of prime divisors (counted with multiplicity) in their orders.
To see subgroups sorted vertically by order instead, check this box.

Sorry, your browser does not support the subgroup diagram.

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

See a full page version of the diagram

See a full page version of the diagram

See a full page version of the diagram

See a full page version of the diagram

Classes of subgroups up to conjugation

Order 966994: $C_{28441}:C_{34}$

Order 483497: $C_{28441}:C_{17}$

Order 138142: $C_{4063}:C_{34}$

Order 69071: $C_{4063}:C_{17}$

Order 56882: $C_{1673}:C_{34}$ x 17, $C_{56882}$

Order 28441: $C_{1673}:C_{17}$ x 17, $C_{28441}$

Order 8126: $C_{239}:C_{34}$ x 17, $C_{8126}$

Order 4063: $C_{239}:C_{17}$ x 17, $C_{4063}$

Order 4046: $C_{17}\times C_{238}$

Order 3346: $C_{3346}$

Order 2023: $C_{17}\times C_{119}$

Order 1673: $C_{1673}$

Order 578: $C_{17}\times C_{34}$

Order 478: $C_{478}$

Order 289: $C_{17}^2$

Order 239: $C_{239}$

Order 238: $C_{238}$ x 18

Order 119: $C_{119}$ x 18

Order 34: $C_{34}$ x 18

Order 17: $C_{17}$ x 18

Order 14: $C_{14}$

Order 7: $C_7$

Order 2: $C_2$

Order 1: $C_1$

Classes of subgroups up to automorphism

Order 966994: $C_{28441}:C_{34}$

Order 483497: $C_{28441}:C_{17}$

Order 138142: $C_{4063}:C_{34}$

Order 69071: $C_{4063}:C_{17}$

Order 56882: $C_{1673}:C_{34}$, $C_{56882}$

Order 28441: $C_{28441}$, $C_{1673}:C_{17}$

Order 8126: $C_{8126}$, $C_{239}:C_{34}$

Order 4063: $C_{4063}$, $C_{239}:C_{17}$

Order 4046: $C_{17}\times C_{238}$

Order 3346: $C_{3346}$

Order 2023: $C_{17}\times C_{119}$

Order 1673: $C_{1673}$

Order 578: $C_{17}\times C_{34}$

Order 478: $C_{478}$

Order 289: $C_{17}^2$

Order 239: $C_{239}$

Order 238: $C_{238}$ x 2

Order 119: $C_{119}$ x 2

Order 34: $C_{34}$ x 2

Order 17: $C_{17}$ x 2

Order 14: $C_{14}$

Order 7: $C_7$

Order 2: $C_2$

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 966994: $C_{28441}:C_{34}$ ($C_1$)

Order 483497: $C_{28441}:C_{17}$ ($C_2$)

Order 138142: $C_{4063}:C_{34}$ ($C_7$)

Order 69071: $C_{4063}:C_{17}$ ($C_{14}$)

Order 56882: $C_{1673}:C_{34}$ ($C_{17}$) x 17, $C_{56882}$ ($C_{17}$)

Order 28441: $C_{1673}:C_{17}$ ($C_{34}$) x 17, $C_{28441}$ ($C_{34}$)

Order 8126: $C_{239}:C_{34}$ ($C_{119}$) x 17, $C_{8126}$ ($C_{119}$)

Order 4063: $C_{239}:C_{17}$ ($C_{238}$) x 17, $C_{4063}$ ($C_{238}$)

Order 3346: $C_{3346}$ ($C_{17}^2$)

Order 1673: $C_{1673}$ ($C_{17}\times C_{34}$)

Order 478: $C_{478}$ ($C_{17}\times C_{119}$)

Order 239: $C_{239}$ ($C_{17}\times C_{238}$)

Order 238: $C_{238}$ ($C_{239}:C_{17}$)

Order 119: $C_{119}$ ($C_{239}:C_{34}$)

Order 34: $C_{34}$ ($C_{1673}:C_{17}$)

Order 17: $C_{17}$ ($C_{1673}:C_{34}$)

Order 14: $C_{14}$ ($C_{4063}:C_{17}$)

Order 7: $C_7$ ($C_{4063}:C_{34}$)

Order 2: $C_2$ ($C_{28441}:C_{17}$)

Order 1: $C_1$ ($C_{28441}:C_{34}$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 966994: $C_{28441}:C_{34}$ ($C_1$)

Order 483497: $C_{28441}:C_{17}$ ($C_2$)

Order 138142: $C_{4063}:C_{34}$ ($C_7$)

Order 69071: $C_{4063}:C_{17}$ ($C_{14}$)

Order 56882: $C_{1673}:C_{34}$ ($C_{17}$), $C_{56882}$ ($C_{17}$)

Order 28441: $C_{28441}$ ($C_{34}$), $C_{1673}:C_{17}$ ($C_{34}$)

Order 8126: $C_{8126}$ ($C_{119}$), $C_{239}:C_{34}$ ($C_{119}$)

Order 4063: $C_{4063}$ ($C_{238}$), $C_{239}:C_{17}$ ($C_{238}$)

Order 3346: $C_{3346}$ ($C_{17}^2$)

Order 1673: $C_{1673}$ ($C_{17}\times C_{34}$)

Order 478: $C_{478}$ ($C_{17}\times C_{119}$)

Order 239: $C_{239}$ ($C_{17}\times C_{238}$)

Order 238: $C_{238}$ ($C_{239}:C_{17}$)

Order 119: $C_{119}$ ($C_{239}:C_{34}$)

Order 34: $C_{34}$ ($C_{1673}:C_{17}$)

Order 17: $C_{17}$ ($C_{1673}:C_{34}$)

Order 14: $C_{14}$ ($C_{4063}:C_{17}$)

Order 7: $C_7$ ($C_{4063}:C_{34}$)

Order 2: $C_2$ ($C_{28441}:C_{17}$)

Order 1: $C_1$ ($C_{28441}:C_{34}$)

Series

Derived series

$C_{28441}:C_{34}$

$\rhd$

$C_{28441}:C_{34}$

$\rhd$

$C_{239}$

$\rhd$

$C_{239}$

$\rhd$

$C_1$

$\rhd$

$C_1$

Chief series

$C_{28441}:C_{34}$

$\rhd$

$C_{28441}:C_{34}$

$\rhd$

$C_{28441}:C_{17}$

$\rhd$

$C_{28441}:C_{17}$

$\rhd$

$C_{28441}$

$\rhd$

$C_{28441}$

$\rhd$

$C_{4063}$

$\rhd$

$C_{4063}$

$\rhd$

$C_{239}$

$\rhd$

$C_{239}$

$\rhd$

$C_1$

$\rhd$

$C_1$

Lower central series

$C_{28441}:C_{34}$

$\rhd$

$C_{28441}:C_{34}$

$\rhd$

$C_{239}$

$\rhd$

$C_{239}$

Upper central series

$C_1$

$\lhd$

$C_1$

$\lhd$

$C_{238}$

$\lhd$

$C_{238}$

Supergroups

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

This group is a maximal quotient of 0 larger groups in the database.

Character theory

Complex character table

The $7378 \times 7378$ character table is not available for this group.

Rational character table

The $84 \times 84$ rational character table is not available for this group.