Abstract group 736.37: $C_{46}.C_4^2$

• Label: 736.37
• Order: \( 2^{5} \cdot 23 \)
• Exponent: \( 2^{2} \cdot 23 \)
• Nilpotent: no
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{4} \)
• $\card{Z(G)}$: \( 2^{3} \)
• $\card{\Aut(G)}$: \( 2^{8} \cdot 11 \cdot 23 \)
• $\card{\mathrm{Out}(G)}$: \( 2^{6} \cdot 11 \)
• Perm deg.: $35$
• Trans deg.: $736$
• Rank: $2$

Group information

Description:	$C_{46}.C_4^2$
Order:	\(736\)\(\medspace = 2^{5} \cdot 23 \)
Exponent:	\(92\)\(\medspace = 2^{2} \cdot 23 \)
Automorphism group:	$(C_2\times C_{46}).C_{22}.C_2^5$, of order \(64768\)\(\medspace = 2^{8} \cdot 11 \cdot 23 \)
Composition factors:	$C_2$ x 5, $C_{23}$
Derived length:	$2$

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

Group statistics

Order	1	2	4	23	46	92
Elements	1	7	376	22	154	176	736
Conjugacy classes	1	7	12	11	77	88	196
Divisions	1	7	6	1	7	2	24
Autjugacy classes	1	5	2	1	5	1	15

Dimension	1	2	22	44
Irr. complex chars.	16	180	0	0	196
Irr. rational chars.	4	10	4	6	24

Minimal presentations

Permutation degree:	$35$
Transitive degree:	$736$
Rank:	$2$
Inequivalent generating pairs:	$3$

Minimal degrees of faithful linear representations

	Over $\mathbb{C}$	Over $\mathbb{R}$	Over $\mathbb{Q}$
Irreducible	none	none	none
Arbitrary	4	6	28

Constructions

Presentation:	$\langle a, b, c \mid a^{4}=b^{2}=c^{92}=[a,b]=[b,c]=1, c^{a}=bc^{45} \rangle$
Permutation group:	Degree $35$ $\langle(2,3)(4,5)(6,7)(8,9)(10,11)(12,13)(14,15)(16,17)(18,19)(20,21)(22,23)(24,25,27,30) \!\cdots\! \rangle$
Direct product:	not isomorphic to a non-trivial direct product
Semidirect product:	$(C_2\times C_{92})$ $\,\rtimes\,$ $C_4$ (2)	$(C_{46}:C_4)$ $\,\rtimes\,$ $C_4$ (4)	$C_{23}$ $\,\rtimes\,$ $(C_2.C_4^2)$	$(C_2\times C_4)$ $\,\rtimes\,$ $(C_{23}:C_4)$ (2)	more information
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Non-split product:	$C_2^3$ . $D_{46}$	$C_2^2$ . $D_{92}$	$C_{46}$ . $C_4^2$	$(C_2\times C_{46})$ . $Q_8$	all 21

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

Homology

Abelianization:	$C_{4}^{2} $
Schur multiplier:	$C_{2}^{3}$
Commutator length:	$1$

Subgroups

There are 606 subgroups in 76 conjugacy classes, 45 normal (19 characteristic).

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

Special subgroups

Center:	$Z \simeq$ $C_2^3$	$G/Z \simeq$ $D_{46}$
Commutator:	$G' \simeq$ $C_{46}$	$G/G' \simeq$ $C_4^2$
Frattini:	$\Phi \simeq$ $C_2^3$	$G/\Phi \simeq$ $D_{46}$
Fitting:	$\operatorname{Fit} \simeq$ $C_2^2\times C_{92}$	$G/\operatorname{Fit} \simeq$ $C_2$
Radical:	$R \simeq$ $C_{46}.C_4^2$	$G/R \simeq$ $C_1$
Socle:	$\operatorname{soc} \simeq$ $C_2^2\times C_{46}$	$G/\operatorname{soc} \simeq$ $C_2^2$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $C_2.C_4^2$
23-Sylow subgroup:	$P_{ 23 } \simeq$ $C_{23}$

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 736: $C_{46}.C_4^2$

Order 368: $C_2^2.D_{46}$ x 2, $C_2^2\times C_{92}$

Order 184: $C_{46}:C_4$ x 8, $C_2\times C_{92}$ x 4, $C_2^2\times C_{46}$

Order 92: $C_2\times C_{46}$ x 7, $C_{23}:C_4$ x 4, $C_{92}$ x 2

Order 46: $C_{46}$ x 7

Order 32: $C_2.C_4^2$

Order 23: $C_{23}$

Order 16: $C_2^2\times C_4$ x 3

Order 8: $C_2\times C_4$ x 12, $C_2^3$

Order 4: $C_2^2$ x 7, $C_4$ x 6

Order 2: $C_2$ x 7

Order 1: $C_1$

Classes of subgroups up to automorphism

Order 736: $C_{46}.C_4^2$

Order 368: $C_2^2\times C_{92}$, $C_2^2.D_{46}$

Order 184: $C_{46}:C_4$ x 3, $C_2\times C_{92}$ x 2, $C_2^2\times C_{46}$

Order 92: $C_2\times C_{46}$ x 5, $C_{92}$, $C_{23}:C_4$

Order 46: $C_{46}$ x 5

Order 32: $C_2.C_4^2$

Order 23: $C_{23}$

Order 16: $C_2^2\times C_4$ x 2

Order 8: $C_2\times C_4$ x 5, $C_2^3$

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

Order 2: $C_2$ x 5

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 736: $C_{46}.C_4^2$ ($C_1$)

Order 368: $C_2^2.D_{46}$ ($C_2$) x 2, $C_2^2\times C_{92}$ ($C_2$)

Order 184: $C_{46}:C_4$ ($C_4$) x 4, $C_2\times C_{92}$ ($C_4$) x 2, $C_2^2\times C_{46}$ ($C_2^2$)

Order 92: $C_2\times C_{46}$ ($D_4$) x 3, $C_2\times C_{46}$ ($C_2\times C_4$) x 3, $C_2\times C_{46}$ ($Q_8$)

Order 46: $C_{46}$ ($C_4:C_4$) x 3, $C_{46}$ ($C_2^2:C_4$) x 3, $C_{46}$ ($C_4^2$)

Order 23: $C_{23}$ ($C_2.C_4^2$)

Order 16: $C_2^2\times C_4$ ($D_{23}$)

Order 8: $C_2\times C_4$ ($C_{23}:C_4$) x 2, $C_2^3$ ($D_{46}$)

Order 4: $C_2^2$ ($C_{23}:D_4$) x 2, $C_2^2$ ($C_4\times D_{23}$) x 2, $C_2^2$ ($C_{46}:C_4$), $C_2^2$ ($D_{92}$), $C_2^2$ ($C_{23}:Q_8$)

Order 2: $C_2$ ($D_{46}:C_4$) x 2, $C_2$ ($C_{46}.D_4$) x 2, $C_2$ ($C_{46}.D_4$), $C_2$ ($C_{92}:C_4$), $C_2$ ($C_{92}:C_4$)

Order 1: $C_1$ ($C_{46}.C_4^2$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 736: $C_{46}.C_4^2$ ($C_1$)

Order 368: $C_2^2\times C_{92}$ ($C_2$), $C_2^2.D_{46}$ ($C_2$)

Order 184: $C_2\times C_{92}$ ($C_4$), $C_{46}:C_4$ ($C_4$), $C_2^2\times C_{46}$ ($C_2^2$)

Order 92: $C_2\times C_{46}$ ($D_4$) x 2, $C_2\times C_{46}$ ($C_2\times C_4$) x 2, $C_2\times C_{46}$ ($Q_8$)

Order 46: $C_{46}$ ($C_4:C_4$) x 2, $C_{46}$ ($C_2^2:C_4$) x 2, $C_{46}$ ($C_4^2$)

Order 23: $C_{23}$ ($C_2.C_4^2$)

Order 16: $C_2^2\times C_4$ ($D_{23}$)

Order 8: $C_2^3$ ($D_{46}$), $C_2\times C_4$ ($C_{23}:C_4$)

Order 4: $C_2^2$ ($C_{23}:D_4$), $C_2^2$ ($C_{46}:C_4$), $C_2^2$ ($D_{92}$), $C_2^2$ ($C_4\times D_{23}$), $C_2^2$ ($C_{23}:Q_8$)

Order 2: $C_2$ ($C_{46}.D_4$), $C_2$ ($D_{46}:C_4$), $C_2$ ($C_{92}:C_4$), $C_2$ ($C_{46}.D_4$), $C_2$ ($C_{92}:C_4$)

Order 1: $C_1$ ($C_{46}.C_4^2$)

Series

Derived series

$C_{46}.C_4^2$

$\rhd$

$C_{46}$

$\rhd$

$C_1$

Chief series

$C_{46}.C_4^2$

$\rhd$

$C_2^2.D_{46}$

$\rhd$

$C_2^2\times C_{46}$

$\rhd$

$C_2\times C_{46}$

$\rhd$

$C_{46}$

$\rhd$

$C_{23}$

$\rhd$

$C_1$

Lower central series

$C_{46}.C_4^2$

$\rhd$

$C_{46}$

$\rhd$

$C_{23}$

Upper central series

$C_1$

$\lhd$

$C_2^3$

$\lhd$

$C_2^2\times C_4$

Supergroups

This group is a maximal subgroup of 93 larger groups in the database.

This group is a maximal quotient of 20 larger groups in the database.

Character theory

Complex character table

See the $196 \times 196$ 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 $24 \times 24$ rational character table.