Abstract group 12352.1674: $C_{772}:C_{16}$

• Label: 12352.1674
• Order: \( 2^{6} \cdot 193 \)
• Exponent: \( 2^{4} \cdot 193 \)
• Nilpotent: no
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{6} \)
• $\card{Z(G)}$: \( 2^{2} \)
• $\card{\Aut(G)}$: \( 2^{9} \cdot 3 \cdot 193 \)
• $\card{\mathrm{Out}(G)}$: \( 2^{5} \cdot 3 \)
• Perm deg.: $197$
• Trans deg.: $772$
• Rank: $2$

Group information

Description:	$C_{772}:C_{16}$
Order:	\(12352\)\(\medspace = 2^{6} \cdot 193 \)
Exponent:	\(3088\)\(\medspace = 2^{4} \cdot 193 \)
Automorphism group:	$C_{386}.C_{96}.C_2^3$, of order \(296448\)\(\medspace = 2^{9} \cdot 3 \cdot 193 \)
Composition factors:	$C_2$ x 6, $C_{193}$
Derived length:	$2$

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

Group statistics

Order	1	2	4	8	16	193	386	772
Elements	1	387	1932	3088	6176	192	192	384	12352
Conjugacy classes	1	3	12	16	32	12	12	24	112
Divisions	1	3	6	4	4	1	1	1	21
Autjugacy classes	1	3	8	8	8	1	1	1	31

Dimension	1	2	4	8	16	192	384
Irr. complex chars.	64	0	0	0	48	0	0	112
Irr. rational chars.	4	6	4	4	0	2	1	21

Minimal presentations

Permutation degree:	$197$
Transitive degree:	$772$
Rank:	$2$
Inequivalent generating pairs:	$192$

Minimal degrees of faithful linear representations

	Over $\mathbb{C}$	Over $\mathbb{R}$	Over $\mathbb{Q}$
Irreducible	16	32	384
Arbitrary	16	18	194

Constructions

Presentation:	$\langle a, b \mid a^{16}=b^{772}=1, b^{a}=b^{769} \rangle$
Permutation group:	Degree $197$ $\langle(2,4,5,16,6,17,14,13,3,11,12,31,10,19,9,8)(7,25,26,82,27,53,45,44,15,18,43,50,30,51,29,28) \!\cdots\! \rangle$
Matrix group:	$\left\langle \left(\begin{array}{rr} 1 & 1 \\ 0 & 1 \end{array}\right), \left(\begin{array}{rr} 43 & 0 \\ 0 & 190 \end{array}\right), \left(\begin{array}{rr} 112 & 0 \\ 0 & 81 \end{array}\right) \right\rangle \subseteq \GL_{2}(\F_{193})$
Direct product:	$C_4$ $\, \times\, $ $(C_{193}:C_{16})$
Semidirect product:	$C_{772}$ $\,\rtimes\,$ $C_{16}$	$(C_{193}:C_4)$ $\,\rtimes\,$ $C_{16}$	$(C_{193}:C_4)$ $\,\rtimes\,$ $C_{16}$ (2)	$C_{193}$ $\,\rtimes\,$ $(C_4\times C_{16})$	more information
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Non-split product:	$(C_4\times D_{193})$ . $C_8$	$D_{386}$ . $(C_2\times C_8)$	$D_{193}$ . $(C_4\times C_8)$	$(C_{772}:C_8)$ . $C_2$	all 17

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

Homology

Abelianization:	$C_{4} \times C_{16} $
Schur multiplier:	$C_{4}$
Commutator length:	$1$

Subgroups

There are 5050 subgroups in 58 conjugacy classes, 32 normal (20 characteristic).

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

Special subgroups

Center:	$Z \simeq$ $C_4$	$G/Z \simeq$ $C_{193}:C_{16}$
Commutator:	$G' \simeq$ $C_{193}$	$G/G' \simeq$ $C_4\times C_{16}$
Frattini:	$\Phi \simeq$ $C_2$	$G/\Phi \simeq$ $C_{386}:C_{16}$
Fitting:	$\operatorname{Fit} \simeq$ $C_{772}$	$G/\operatorname{Fit} \simeq$ $C_{16}$
Radical:	$R \simeq$ $C_{772}:C_{16}$	$G/R \simeq$ $C_1$
Socle:	$\operatorname{soc} \simeq$ $C_{386}$	$G/\operatorname{soc} \simeq$ $C_2\times C_{16}$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $C_4\times C_{16}$
193-Sylow subgroup:	$P_{ 193 } \simeq$ $C_{193}$

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 12352: $C_{772}:C_{16}$

Order 6176: $C_{386}:C_{16}$ x 2, $C_{772}:C_8$

Order 3088: $C_{193}:C_{16}$ x 4, $C_{386}:C_8$ x 2, $C_{772}:C_4$

Order 1544: $C_{193}:C_8$ x 4, $C_{386}:C_4$ x 2, $C_4\times D_{193}$

Order 772: $C_{193}:C_4$ x 4, $D_{386}$, $C_{772}$, $C_{193}:C_4$

Order 386: $D_{193}$ x 2, $C_{386}$

Order 193: $C_{193}$

Order 64: $C_4\times C_{16}$

Order 32: $C_2\times C_{16}$ x 2, $C_4\times C_8$

Order 16: $C_{16}$ x 4, $C_2\times C_8$ x 2, $C_4^2$

Order 8: $C_8$ x 4, $C_2\times C_4$ x 3

Order 4: $C_4$ x 6, $C_2^2$

Order 2: $C_2$ x 3

Order 1: $C_1$

Classes of subgroups up to automorphism

Order 12352: $C_{772}:C_{16}$

Order 6176: $C_{386}:C_{16}$, $C_{772}:C_8$

Order 3088: $C_{386}:C_8$ x 2, $C_{193}:C_{16}$, $C_{772}:C_4$

Order 1544: $C_{193}:C_8$ x 2, $C_{386}:C_4$ x 2, $C_4\times D_{193}$

Order 772: $C_{193}:C_4$ x 3, $D_{386}$, $C_{772}$, $C_{193}:C_4$

Order 386: $D_{193}$ x 2, $C_{386}$

Order 193: $C_{193}$

Order 64: $C_4\times C_{16}$

Order 32: $C_4\times C_8$, $C_2\times C_{16}$

Order 16: $C_2\times C_8$ x 2, $C_4^2$, $C_{16}$

Order 8: $C_2\times C_4$ x 3, $C_8$ x 2

Order 4: $C_4$ x 5, $C_2^2$

Order 2: $C_2$ x 3

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 12352: $C_{772}:C_{16}$ ($C_1$)

Order 6176: $C_{386}:C_{16}$ ($C_2$) x 2, $C_{772}:C_8$ ($C_2$)

Order 3088: $C_{193}:C_{16}$ ($C_4$) x 4, $C_{386}:C_8$ ($C_2^2$), $C_{386}:C_8$ ($C_4$), $C_{772}:C_4$ ($C_4$)

Order 1544: $C_{193}:C_8$ ($C_2\times C_4$) x 2, $C_{193}:C_8$ ($C_8$) x 2, $C_4\times D_{193}$ ($C_8$), $C_{386}:C_4$ ($C_2\times C_4$), $C_{386}:C_4$ ($C_8$)

Order 772: $C_{193}:C_4$ ($C_{16}$) x 2, $D_{386}$ ($C_2\times C_8$), $C_{193}:C_4$ ($C_2\times C_8$), $C_{193}:C_4$ ($C_4^2$), $C_{772}$ ($C_{16}$), $C_{193}:C_4$ ($C_{16}$)

Order 386: $C_{386}$ ($C_2\times C_{16}$), $D_{193}$ ($C_4\times C_8$), $D_{193}$ ($C_2\times C_{16}$)

Order 193: $C_{193}$ ($C_4\times C_{16}$)

Order 4: $C_4$ ($C_{193}:C_{16}$)

Order 2: $C_2$ ($C_{386}:C_{16}$)

Order 1: $C_1$ ($C_{772}:C_{16}$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 12352: $C_{772}:C_{16}$ ($C_1$)

Order 6176: $C_{386}:C_{16}$ ($C_2$), $C_{772}:C_8$ ($C_2$)

Order 3088: $C_{193}:C_{16}$ ($C_4$), $C_{386}:C_8$ ($C_2^2$), $C_{386}:C_8$ ($C_4$), $C_{772}:C_4$ ($C_4$)

Order 1544: $C_4\times D_{193}$ ($C_8$), $C_{193}:C_8$ ($C_2\times C_4$), $C_{193}:C_8$ ($C_8$), $C_{386}:C_4$ ($C_2\times C_4$), $C_{386}:C_4$ ($C_8$)

Order 772: $D_{386}$ ($C_2\times C_8$), $C_{193}:C_4$ ($C_2\times C_8$), $C_{193}:C_4$ ($C_4^2$), $C_{193}:C_4$ ($C_{16}$), $C_{772}$ ($C_{16}$), $C_{193}:C_4$ ($C_{16}$)

Order 386: $C_{386}$ ($C_2\times C_{16}$), $D_{193}$ ($C_4\times C_8$), $D_{193}$ ($C_2\times C_{16}$)

Order 193: $C_{193}$ ($C_4\times C_{16}$)

Order 4: $C_4$ ($C_{193}:C_{16}$)

Order 2: $C_2$ ($C_{386}:C_{16}$)

Order 1: $C_1$ ($C_{772}:C_{16}$)

Series

Derived series

$C_{772}:C_{16}$

$\rhd$

$C_{193}$

$\rhd$

$C_1$

Chief series

$C_{772}:C_{16}$

$\rhd$

$C_{386}:C_{16}$

$\rhd$

$C_{386}:C_8$

$\rhd$

$C_{193}:C_8$

$\rhd$

$C_{193}:C_4$

$\rhd$

$D_{193}$

$\rhd$

$C_{193}$

$\rhd$

$C_1$

Lower central series

$C_{772}:C_{16}$

$\rhd$

$C_{193}$

Upper central series

$C_1$

$\lhd$

$C_4$

Supergroups

This group is a maximal subgroup of 4 larger groups in the database.

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

Character theory

Complex character table

See the $112 \times 112$ character table. Alternatively, you may search for characters of this group with desired properties.

Rational character table

See the $21 \times 21$ rational character table.