Abstract group 37056.b: $C_{772}:C_{48}$

• Label: 37056.b
• Order: \( 2^{6} \cdot 3 \cdot 193 \)
• Exponent: \( 2^{4} \cdot 3 \cdot 193 \)
• Nilpotent: no
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{6} \cdot 3 \)
• $\card{Z(G)}$: \( 2^{2} \)
• $\card{\Aut(G)}$: \( 2^{9} \cdot 3 \cdot 193 \)
• $\card{\mathrm{Out}(G)}$: \( 2^{5} \)
• Perm deg.: $197$
• Trans deg.: $772$
• Rank: $2$

Group information

Description:	$C_{772}:C_{48}$
Order:	\(37056\)\(\medspace = 2^{6} \cdot 3 \cdot 193 \)
Exponent:	\(9264\)\(\medspace = 2^{4} \cdot 3 \cdot 193 \)
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, $C_3$, $C_{193}$
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	6	8	12	16	24	48	193	386	772
Elements	1	387	386	1932	1158	3088	4632	6176	6176	12352	192	192	384	37056
Conjugacy classes	1	3	2	12	6	16	24	32	32	64	4	4	8	208
Divisions	1	3	1	6	3	4	6	4	4	4	1	1	1	39
Autjugacy classes	1	3	2	8	6	8	16	8	16	16	1	1	1	87

Dimension	1	2	4	8	16	48	192	384
Irr. complex chars.	192	0	0	0	0	16	0	0	208
Irr. rational chars.	4	10	10	8	4	0	2	1	39

Minimal presentations

Permutation degree:	$197$
Transitive degree:	$772$
Rank:	$2$
Inequivalent generating pairs:	$1536$

Minimal degrees of faithful linear representations

	Over $\mathbb{C}$	Over $\mathbb{R}$	Over $\mathbb{Q}$
Irreducible	48	96	384
Arbitrary	48	50	194

Constructions

Presentation:	$\langle a, b \mid a^{48}=b^{772}=1, b^{a}=b^{21} \rangle$
Permutation group:	Degree $197$ $\langle(2,4,14,56,27,92,55,62,26,6,16,58,79,93,47,13,39,77,25,33,78,20,45,11,3,9,35,89,40,103,34,72,42,10,37,111,141,98,31,8,18,60,41,53,140,51,29,7) \!\cdots\! \rangle$
Matrix group:	$\left\langle \left(\begin{array}{rr} 1 & 1 \\ 0 & 1 \end{array}\right), \left(\begin{array}{rr} 46 & 0 \\ 0 & 186 \end{array}\right), \left(\begin{array}{rr} 49 & 0 \\ 0 & 130 \end{array}\right) \right\rangle \subseteq \GL_{2}(\F_{193})$
Direct product:	$C_4$ $\, \times\, $ $(C_{193}:C_{48})$
Semidirect product:	$C_{772}$ $\,\rtimes\,$ $C_{48}$	$(C_{772}:C_{16})$ $\,\rtimes\,$ $C_3$	$(C_{193}:C_4)$ $\,\rtimes\,$ $C_{48}$	$(C_{193}:C_4)$ $\,\rtimes\,$ $C_{48}$ (2)	all 10
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Non-split product:	$(C_{772}:C_8)$ . $C_6$	$(C_{772}:C_6)$ . $C_8$	$(C_4\times D_{193})$ . $C_{24}$	$D_{386}$ . $(C_2\times C_{24})$	all 33
Aut. group:	$\Aut(C_2^5.C_2^8:C_5)$

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

Homology

Abelianization:	$C_{4} \times C_{48} \simeq C_{4} \times C_{16} \times C_{3}$
Schur multiplier:	$C_{4}$
Commutator length:	$1$

Subgroups

There are 10676 subgroups in 116 conjugacy classes, 61 normal (37 characteristic).

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

Special subgroups

Center:	$Z \simeq$ $C_4$	$G/Z \simeq$ $C_{193}:C_{48}$
Commutator:	$G' \simeq$ $C_{193}$	$G/G' \simeq$ $C_4\times C_{48}$
Frattini:	$\Phi \simeq$ $C_2$	$G/\Phi \simeq$ $C_{386}:C_{48}$
Fitting:	$\operatorname{Fit} \simeq$ $C_{772}$	$G/\operatorname{Fit} \simeq$ $C_{48}$
Radical:	$R \simeq$ $C_{772}:C_{48}$	$G/R \simeq$ $C_1$
Socle:	$\operatorname{soc} \simeq$ $C_{386}$	$G/\operatorname{soc} \simeq$ $C_2\times C_{48}$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $C_4\times C_{16}$
3-Sylow subgroup:	$P_{ 3 } \simeq$ $C_3$
193-Sylow subgroup:	$P_{ 193 } \simeq$ $C_{193}$

Hi

Subgroup diagram and profile

Classes of subgroups up to conjugation

There are too many subgroups to show. See a full page version of the diagram.

Classes of subgroups up to automorphism

There are too many subgroups to show. See a full page version of the diagram.

Normal subgroups

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 37056: $C_{772}:C_{48}$

Order 18528: $C_{386}:C_{48}$ x 2, $C_{772}:C_{24}$

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

Order 9264: $C_{193}:C_{48}$ x 4, $C_{386}:C_{24}$ x 2, $C_{772}:C_{12}$

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

Order 4632: $C_{193}:C_{24}$ x 4, $C_{386}:C_{12}$ x 2, $C_{772}:C_6$

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

Order 2316: $C_{193}:C_{12}$ x 4, $C_{193}:C_{12}$, $C_{386}:C_6$, $C_{193}:C_{12}$

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

Order 1158: $C_{193}:C_6$ x 2, $C_{193}:C_6$

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

Order 579: $C_{193}:C_3$

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

Order 193: $C_{193}$

Order 192: $C_4\times C_{48}$

Order 96: $C_2\times C_{48}$ x 2, $C_4\times C_{24}$

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

Order 48: $C_{48}$ x 4, $C_2\times C_{24}$ x 2, $C_4\times C_{12}$

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

Order 24: $C_{24}$ x 4, $C_2\times C_{12}$ x 3

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

Order 12: $C_{12}$ x 6, $C_2\times C_6$

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

Order 6: $C_6$ x 3

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

Order 3: $C_3$

Order 2: $C_2$ x 3

Order 1: $C_1$

Classes of subgroups up to automorphism

Order 37056: $C_{772}:C_{48}$

Order 18528: $C_{772}:C_{24}$, $C_{386}:C_{48}$

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

Order 9264: $C_{386}:C_{24}$ x 2, $C_{772}:C_{12}$, $C_{193}:C_{48}$

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

Order 4632: $C_{193}:C_{24}$ x 2, $C_{386}:C_{12}$ x 2, $C_{772}:C_6$

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

Order 2316: $C_{193}:C_{12}$ x 3, $C_{193}:C_{12}$, $C_{386}:C_6$, $C_{193}:C_{12}$

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

Order 1158: $C_{193}:C_6$ x 2, $C_{193}:C_6$

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

Order 579: $C_{193}:C_3$

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

Order 193: $C_{193}$

Order 192: $C_4\times C_{48}$

Order 96: $C_2\times C_{48}$, $C_4\times C_{24}$

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

Order 48: $C_2\times C_{24}$ x 2, $C_4\times C_{12}$, $C_{48}$

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

Order 24: $C_2\times C_{12}$ x 3, $C_{24}$ x 2

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

Order 12: $C_{12}$ x 5, $C_2\times C_6$

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

Order 6: $C_6$ x 3

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

Order 3: $C_3$

Order 2: $C_2$ x 3

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 37056: $C_{772}:C_{48}$ ($C_1$)

Order 18528: $C_{386}:C_{48}$ ($C_2$) x 2, $C_{772}:C_{24}$ ($C_2$)

Order 12352: $C_{772}:C_{16}$ ($C_3$)

Order 9264: $C_{193}:C_{48}$ ($C_4$) x 4, $C_{386}:C_{24}$ ($C_2^2$), $C_{386}:C_{24}$ ($C_4$), $C_{772}:C_{12}$ ($C_4$)

Order 6176: $C_{386}:C_{16}$ ($C_6$) x 2, $C_{772}:C_8$ ($C_6$)

Order 4632: $C_{193}:C_{24}$ ($C_2\times C_4$) x 2, $C_{193}:C_{24}$ ($C_8$) x 2, $C_{772}:C_6$ ($C_8$), $C_{386}:C_{12}$ ($C_2\times C_4$), $C_{386}:C_{12}$ ($C_8$)

Order 3088: $C_{193}:C_{16}$ ($C_{12}$) x 4, $C_{386}:C_8$ ($C_2\times C_6$), $C_{386}:C_8$ ($C_{12}$), $C_{772}:C_4$ ($C_{12}$)

Order 2316: $C_{193}:C_{12}$ ($C_{16}$) x 2, $C_{193}:C_{12}$ ($C_{16}$), $C_{386}:C_6$ ($C_2\times C_8$), $C_{193}:C_{12}$ ($C_{16}$), $C_{193}:C_{12}$ ($C_2\times C_8$), $C_{193}:C_{12}$ ($C_4^2$)

Order 1544: $C_{193}:C_8$ ($C_2\times C_{12}$) x 2, $C_{193}:C_8$ ($C_{24}$) x 2, $C_4\times D_{193}$ ($C_{24}$), $C_{386}:C_4$ ($C_2\times C_{12}$), $C_{386}:C_4$ ($C_{24}$)

Order 1158: $C_{193}:C_6$ ($C_2\times C_{16}$), $C_{193}:C_6$ ($C_4\times C_8$), $C_{193}:C_6$ ($C_2\times C_{16}$)

Order 772: $C_{193}:C_4$ ($C_{48}$) x 2, $D_{386}$ ($C_2\times C_{24}$), $C_{193}:C_4$ ($C_2\times C_{24}$), $C_{193}:C_4$ ($C_4\times C_{12}$), $C_{772}$ ($C_{48}$), $C_{193}:C_4$ ($C_{48}$)

Order 579: $C_{193}:C_3$ ($C_4\times C_{16}$)

Order 386: $C_{386}$ ($C_2\times C_{48}$), $D_{193}$ ($C_2\times C_{48}$), $D_{193}$ ($C_4\times C_{24}$)

Order 193: $C_{193}$ ($C_4\times C_{48}$)

Order 4: $C_4$ ($C_{193}:C_{48}$)

Order 2: $C_2$ ($C_{386}:C_{48}$)

Order 1: $C_1$ ($C_{772}:C_{48}$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 37056: $C_{772}:C_{48}$ ($C_1$)

Order 18528: $C_{772}:C_{24}$ ($C_2$), $C_{386}:C_{48}$ ($C_2$)

Order 12352: $C_{772}:C_{16}$ ($C_3$)

Order 9264: $C_{386}:C_{24}$ ($C_2^2$), $C_{386}:C_{24}$ ($C_4$), $C_{772}:C_{12}$ ($C_4$), $C_{193}:C_{48}$ ($C_4$)

Order 6176: $C_{386}:C_{16}$ ($C_6$), $C_{772}:C_8$ ($C_6$)

Order 4632: $C_{772}:C_6$ ($C_8$), $C_{193}:C_{24}$ ($C_2\times C_4$), $C_{193}:C_{24}$ ($C_8$), $C_{386}:C_{12}$ ($C_2\times C_4$), $C_{386}:C_{12}$ ($C_8$)

Order 3088: $C_{193}:C_{16}$ ($C_{12}$), $C_{386}:C_8$ ($C_2\times C_6$), $C_{386}:C_8$ ($C_{12}$), $C_{772}:C_4$ ($C_{12}$)

Order 2316: $C_{193}:C_{12}$ ($C_{16}$), $C_{386}:C_6$ ($C_2\times C_8$), $C_{193}:C_{12}$ ($C_{16}$), $C_{193}:C_{12}$ ($C_2\times C_8$), $C_{193}:C_{12}$ ($C_4^2$), $C_{193}:C_{12}$ ($C_{16}$)

Order 1544: $C_4\times D_{193}$ ($C_{24}$), $C_{193}:C_8$ ($C_2\times C_{12}$), $C_{193}:C_8$ ($C_{24}$), $C_{386}:C_4$ ($C_2\times C_{12}$), $C_{386}:C_4$ ($C_{24}$)

Order 1158: $C_{193}:C_6$ ($C_2\times C_{16}$), $C_{193}:C_6$ ($C_4\times C_8$), $C_{193}:C_6$ ($C_2\times C_{16}$)

Order 772: $D_{386}$ ($C_2\times C_{24}$), $C_{193}:C_4$ ($C_2\times C_{24}$), $C_{193}:C_4$ ($C_4\times C_{12}$), $C_{193}:C_4$ ($C_{48}$), $C_{772}$ ($C_{48}$), $C_{193}:C_4$ ($C_{48}$)

Order 579: $C_{193}:C_3$ ($C_4\times C_{16}$)

Order 386: $C_{386}$ ($C_2\times C_{48}$), $D_{193}$ ($C_2\times C_{48}$), $D_{193}$ ($C_4\times C_{24}$)

Order 193: $C_{193}$ ($C_4\times C_{48}$)

Order 4: $C_4$ ($C_{193}:C_{48}$)

Order 2: $C_2$ ($C_{386}:C_{48}$)

Order 1: $C_1$ ($C_{772}:C_{48}$)

Series

Derived series

$C_{772}:C_{48}$

$\rhd$

$C_{193}$

$\rhd$

$C_1$

Chief series

$C_{772}:C_{48}$

$\rhd$

$C_{772}:C_{24}$

$\rhd$

$C_{772}:C_{12}$

$\rhd$

$C_{772}:C_6$

$\rhd$

$C_{193}:C_{12}$

$\rhd$

$C_{772}$

$\rhd$

$C_{386}$

$\rhd$

$C_{193}$

$\rhd$

$C_1$

Lower central series

$C_{772}:C_{48}$

$\rhd$

$C_{193}$

Upper central series

$C_1$

$\lhd$

$C_4$

Supergroups

This group is a maximal subgroup of 3 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 $208 \times 208$ 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 $39 \times 39$ rational character table.