Abstract group 352.22: $C_{44}.D_4$

• Label: 352.22
• Order: \( 2^{5} \cdot 11 \)
• Exponent: \( 2^{3} \cdot 11 \)
• Nilpotent: no
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{3} \)
• $\card{Z(G)}$: \( 2^{2} \)
• $\card{\Aut(G)}$: \( 2^{7} \cdot 5 \cdot 11 \)
• $\card{\mathrm{Out}(G)}$: \( 2^{4} \cdot 5 \)
• Perm deg.: $23$
• Trans deg.: $352$
• Rank: $2$

Group information

Description:	$C_{44}.D_4$
Order:	\(352\)\(\medspace = 2^{5} \cdot 11 \)
Exponent:	\(88\)\(\medspace = 2^{3} \cdot 11 \)
Automorphism group:	$C_{22}.(C_2^5\times C_{10})$, of order \(7040\)\(\medspace = 2^{7} \cdot 5 \cdot 11 \)
Composition factors:	$C_2$ x 5, $C_{11}$
Derived length:	$2$

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

Group statistics

Order	1	2	4	8	11	22	44	88
Elements	1	3	180	8	10	30	40	80	352
Conjugacy classes	1	3	6	4	5	15	20	40	94
Divisions	1	3	5	1	1	3	2	1	17
Autjugacy classes	1	3	4	1	1	3	2	1	16

Dimension	1	2	4	10	20	40
Irr. complex chars.	8	86	0	0	0	0	94
Irr. rational chars.	4	4	2	2	3	2	17

Minimal presentations

Permutation degree:	$23$
Transitive degree:	$352$
Rank:	$2$
Inequivalent generating pairs:	$6$

Minimal degrees of faithful linear representations

	Over $\mathbb{C}$	Over $\mathbb{R}$	Over $\mathbb{Q}$
Irreducible	none	none	none
Arbitrary	3	6	16

Constructions

Presentation:	$\langle a, b, c \mid b^{2}=c^{88}=[a,b]=[b,c]=1, a^{2}=bc^{44}, c^{a}=bc^{87} \rangle$
Permutation group:	Degree $23$ $\langle(2,5)(3,7)(6,8)(9,10,11,12)(14,15)(16,17)(18,19)(20,21)(22,23), (1,2,3,5,4,6,7,8) \!\cdots\! \rangle$
Matrix group:	$\left\langle \left(\begin{array}{rr} 60 & 5 \\ 53 & 55 \end{array}\right), \left(\begin{array}{rr} 1 & 42 \\ 45 & 28 \end{array}\right), \left(\begin{array}{rr} 47 & 0 \\ 0 & 47 \end{array}\right), \left(\begin{array}{rr} 42 & 28 \\ 20 & 27 \end{array}\right), \left(\begin{array}{rr} 22 & 0 \\ 0 & 22 \end{array}\right), \left(\begin{array}{rr} 1 & 23 \\ 23 & 47 \end{array}\right) \right\rangle \subseteq \GL_{2}(\Z/69\Z)$
Direct product:	not isomorphic to a non-trivial direct product
Semidirect product:	$(C_{11}:Q_8)$ $\,\rtimes\,$ $C_4$ (2)	$C_{11}$ $\,\rtimes\,$ $(Q_8:C_4)$			more information
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Non-split product:	$C_{44}$ . $D_4$	$C_{22}$ . $Q_{16}$	$C_2^2$ . $D_{44}$	$C_{22}$ . $\SD_{16}$	all 18

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

Homology

Abelianization:	$C_{2} \times C_{4} $
Schur multiplier:	$C_{2}$
Commutator length:	$1$

Subgroups

There are 250 subgroups in 42 conjugacy classes, 23 normal (21 characteristic).

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

Special subgroups

Center:	$Z \simeq$ $C_2^2$	$G/Z \simeq$ $D_{44}$
Commutator:	$G' \simeq$ $C_{44}$	$G/G' \simeq$ $C_2\times C_4$
Frattini:	$\Phi \simeq$ $C_2\times C_4$	$G/\Phi \simeq$ $D_{22}$
Fitting:	$\operatorname{Fit} \simeq$ $C_2\times C_{88}$	$G/\operatorname{Fit} \simeq$ $C_2$
Radical:	$R \simeq$ $C_{44}.D_4$	$G/R \simeq$ $C_1$
Socle:	$\operatorname{soc} \simeq$ $C_2\times C_{22}$	$G/\operatorname{soc} \simeq$ $D_4$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $Q_8:C_4$
11-Sylow subgroup:	$P_{ 11 } \simeq$ $C_{11}$

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 352: $C_{44}.D_4$

Order 176: $C_{22}:Q_8$, $C_2\times C_{88}$, $C_{44}:C_4$

Order 88: $C_{11}:Q_8$ x 3, $C_{22}:C_4$ x 2, $C_2\times C_{44}$, $C_{88}$

Order 44: $C_{11}:C_4$ x 3, $C_{44}$ x 2, $C_2\times C_{22}$

Order 32: $Q_8:C_4$

Order 22: $C_{22}$ x 3

Order 16: $C_2\times C_8$, $C_4:C_4$, $C_2\times Q_8$

Order 11: $C_{11}$

Order 8: $Q_8$ x 3, $C_2\times C_4$ x 3, $C_8$

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

Order 2: $C_2$ x 3

Order 1: $C_1$

Classes of subgroups up to automorphism

Order 352: $C_{44}.D_4$

Order 176: $C_{22}:Q_8$, $C_2\times C_{88}$, $C_{44}:C_4$

Order 88: $C_{22}:C_4$ x 2, $C_{11}:Q_8$ x 2, $C_2\times C_{44}$, $C_{88}$

Order 44: $C_{44}$ x 2, $C_{11}:C_4$ x 2, $C_2\times C_{22}$

Order 32: $Q_8:C_4$

Order 22: $C_{22}$ x 3

Order 16: $C_2\times C_8$, $C_4:C_4$, $C_2\times Q_8$

Order 11: $C_{11}$

Order 8: $C_2\times C_4$ x 3, $Q_8$ x 2, $C_8$

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

Order 2: $C_2$ x 3

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 352: $C_{44}.D_4$ ($C_1$)

Order 176: $C_{22}:Q_8$ ($C_2$), $C_2\times C_{88}$ ($C_2$), $C_{44}:C_4$ ($C_2$)

Order 88: $C_{11}:Q_8$ ($C_4$) x 2, $C_2\times C_{44}$ ($C_2^2$)

Order 44: $C_2\times C_{22}$ ($D_4$), $C_{44}$ ($D_4$), $C_{44}$ ($C_2\times C_4$)

Order 22: $C_{22}$ ($Q_{16}$), $C_{22}$ ($\SD_{16}$), $C_{22}$ ($C_2^2:C_4$)

Order 16: $C_2\times C_8$ ($D_{11}$)

Order 11: $C_{11}$ ($Q_8:C_4$)

Order 8: $C_2\times C_4$ ($D_{22}$)

Order 4: $C_2^2$ ($D_{44}$), $C_4$ ($C_{11}:D_4$), $C_4$ ($C_4\times D_{11}$)

Order 2: $C_2$ ($C_{11}:Q_{16}$), $C_2$ ($C_{88}:C_2$), $C_2$ ($D_{22}:C_4$)

Order 1: $C_1$ ($C_{44}.D_4$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 352: $C_{44}.D_4$ ($C_1$)

Order 176: $C_{22}:Q_8$ ($C_2$), $C_2\times C_{88}$ ($C_2$), $C_{44}:C_4$ ($C_2$)

Order 88: $C_2\times C_{44}$ ($C_2^2$), $C_{11}:Q_8$ ($C_4$)

Order 44: $C_2\times C_{22}$ ($D_4$), $C_{44}$ ($D_4$), $C_{44}$ ($C_2\times C_4$)

Order 22: $C_{22}$ ($Q_{16}$), $C_{22}$ ($\SD_{16}$), $C_{22}$ ($C_2^2:C_4$)

Order 16: $C_2\times C_8$ ($D_{11}$)

Order 11: $C_{11}$ ($Q_8:C_4$)

Order 8: $C_2\times C_4$ ($D_{22}$)

Order 4: $C_2^2$ ($D_{44}$), $C_4$ ($C_{11}:D_4$), $C_4$ ($C_4\times D_{11}$)

Order 2: $C_2$ ($C_{11}:Q_{16}$), $C_2$ ($C_{88}:C_2$), $C_2$ ($D_{22}:C_4$)

Order 1: $C_1$ ($C_{44}.D_4$)

Series

Derived series

$C_{44}.D_4$

$\rhd$

$C_{44}$

$\rhd$

$C_1$

Chief series

$C_{44}.D_4$

$\rhd$

$C_{44}:C_4$

$\rhd$

$C_2\times C_{44}$

$\rhd$

$C_2\times C_{22}$

$\rhd$

$C_{22}$

$\rhd$

$C_{11}$

$\rhd$

$C_1$

Lower central series

$C_{44}.D_4$

$\rhd$

$C_{44}$

$\rhd$

$C_{22}$

$\rhd$

$C_{11}$

Upper central series

$C_1$

$\lhd$

$C_2^2$

$\lhd$

$C_2\times C_4$

$\lhd$

$C_2\times C_8$

Supergroups

This group is a maximal subgroup of 72 larger groups in the database.

This group is a maximal quotient of 15 larger groups in the database.

Character theory

Complex character table

See the $94 \times 94$ character table. Alternatively, you may search for characters of this group with desired properties.

Rational character table

See the $17 \times 17$ rational character table.