Abstract group 22472.a: $D_{106}:C_{106}$

• Label: 22472.a
• Order: \( 2^{3} \cdot 53^{2} \)
• Exponent: \( 2^{2} \cdot 53 \)
• Nilpotent: no
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{2} \cdot 53 \)
• $\card{Z(G)}$: \( 2 \cdot 53 \)
• $\card{\Aut(G)}$: \( 2^{6} \cdot 13^{2} \cdot 53 \)
• $\card{\mathrm{Out}(G)}$: \( 2^{4} \cdot 13^{2} \)
• Perm deg.: $110$
• Trans deg.: $212$
• Rank: $2$

Group information

Description:	$D_{106}:C_{106}$
Order:	\(22472\)\(\medspace = 2^{3} \cdot 53^{2} \)
Exponent:	\(212\)\(\medspace = 2^{2} \cdot 53 \)
Automorphism group:	$C_2\times C_{53}.C_{13}.C_{26}.C_2^3.C_2$, of order \(573248\)\(\medspace = 2^{6} \cdot 13^{2} \cdot 53 \)
Composition factors:	$C_2$ x 3, $C_{53}$ x 2
Derived length:	$2$

This group is nonabelian, supersolvable (hence solvable and monomial), and metabelian.

Group statistics

Order	1	2	4	53	106	212
Elements	1	109	106	2808	13936	5512	22472
Conjugacy classes	1	3	1	1430	4290	52	5777
Divisions	1	3	1	28	83	1	117
Autjugacy classes	1	3	1	3	7	1	16

Minimal presentations

Permutation degree:	$110$
Transitive degree:	$212$
Rank:	$2$
Inequivalent generating pairs:	not computed

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

Constructions

Groups of Lie type:	$\GOrthPlus(2,107)$
Presentation:	$\langle a, b, c \mid a^{2}=b^{106}=c^{106}=[a,c]=[b,c]=1, b^{a}=b^{105}c^{59} \rangle$
Permutation group:	Degree $110$ $\langle(1,2,4,7,11,15,19,23,27,31,35,39,43,47,51,50,46,42,38,34,30,26,22,18,14,10,3,5,8,12,16,20,24,28,32,36,40,44,48,52,53,49,45,41,37,33,29,25,21,17,13,9,6) \!\cdots\! \rangle$
Matrix group:	$\left\langle \left(\begin{array}{rr} 1 & 0 \\ 0 & 2 \end{array}\right), \left(\begin{array}{rr} 0 & 1 \\ 1 & 0 \end{array}\right), \left(\begin{array}{rr} 2 & 0 \\ 0 & 54 \end{array}\right) \right\rangle \subseteq \GL_{2}(\F_{107})$
Direct product:	$C_{53}$ $\, \times\, $ $(C_{53}:D_4)$
Semidirect product:	$D_{106}$ $\,\rtimes\,$ $C_{106}$	$C_{53}$ $\,\rtimes\,$ $(D_4\times C_{53})$	$(C_2\times C_{106})$ $\,\rtimes\,$ $D_{53}$	$(C_{53}:C_4)$ $\,\rtimes\,$ $C_{106}$	all 5
Trans. wreath product:	not computed
Non-split product:	$C_{106}$ . $D_{106}$	$C_{106}$ . $(C_2\times C_{106})$	$(C_{53}\times C_{106})$ . $C_2^2$	$C_2$ . $(C_{53}^2:C_2^2)$	more information

Elements of the group are displayed as matrices in $\GL_{2}(\F_{107})$.

Homology

Abelianization:	$C_{2} \times C_{106} \simeq C_{2}^{2} \times C_{53}$
Schur multiplier:	not computed
Commutator length:	not computed

Subgroups

There are 820 subgroups in 162 conjugacy classes, 18 normal, and all normal subgroups are characteristic.

Characteristic subgroups are shown in this color.

Special subgroups

Center:	$Z \simeq$ $C_{106}$	$G/Z \simeq$ $D_{106}$
Commutator:	$G' \simeq$ $C_{106}$	$G/G' \simeq$ $C_2\times C_{106}$
Frattini:	$\Phi \simeq$ $C_2$	$G/\Phi \simeq$ $C_{53}^2:C_2^2$
Fitting:	$\operatorname{Fit} \simeq$ $C_{106}^2$	$G/\operatorname{Fit} \simeq$ $C_2$
Radical:	$R \simeq$ $D_{106}:C_{106}$	$G/R \simeq$ $C_1$
Socle:	$\operatorname{soc} \simeq$ $C_{53}\times C_{106}$	$G/\operatorname{soc} \simeq$ $C_2^2$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $D_4$
53-Sylow subgroup:	$P_{ 53 } \simeq$ $C_{53}^2$

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

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Classes of subgroups up to conjugation

Order 22472: $D_{106}:C_{106}$

Order 11236: $C_{53}^2:C_2^2$, $C_{53}^2:C_4$, $C_{106}^2$

Order 5618: $C_{53}\times C_{106}$ x 2, $C_{53}^2:C_2$

Order 2809: $C_{53}^2$

Order 424: $C_{53}:D_4$, $D_4\times C_{53}$

Order 212: $C_2\times C_{106}$ x 29, $D_{106}$, $C_{212}$, $C_{53}:C_4$

Order 106: $C_{106}$ x 83, $D_{53}$

Order 53: $C_{53}$ x 28

Order 8: $D_4$

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

Order 2: $C_2$ x 3

Order 1: $C_1$

Classes of subgroups up to automorphism

Order 22472: $D_{106}:C_{106}$

Order 11236: $C_{53}^2:C_2^2$, $C_{53}^2:C_4$, $C_{106}^2$

Order 5618: $C_{53}\times C_{106}$ x 2, $C_{53}^2:C_2$

Order 2809: $C_{53}^2$

Order 424: $C_{53}:D_4$, $D_4\times C_{53}$

Order 212: $C_2\times C_{106}$ x 4, $D_{106}$, $C_{212}$, $C_{53}:C_4$

Order 106: $C_{106}$ x 7, $D_{53}$

Order 53: $C_{53}$ x 3

Order 8: $D_4$

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

Order 2: $C_2$ x 3

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 22472: $D_{106}:C_{106}$ ($C_1$)

Order 11236: $C_{53}^2:C_2^2$ ($C_2$), $C_{53}^2:C_4$ ($C_2$), $C_{106}^2$ ($C_2$)

Order 5618: $C_{53}\times C_{106}$ ($C_2^2$)

Order 2809: $C_{53}^2$ ($D_4$)

Order 424: $C_{53}:D_4$ ($C_{53}$)

Order 212: $C_2\times C_{106}$ ($C_{106}$), $C_2\times C_{106}$ ($D_{53}$), $D_{106}$ ($C_{106}$), $C_{53}:C_4$ ($C_{106}$)

Order 106: $C_{106}$ ($C_2\times C_{106}$), $C_{106}$ ($D_{106}$)

Order 53: $C_{53}$ ($C_{53}:D_4$), $C_{53}$ ($D_4\times C_{53}$)

Order 4: $C_2^2$ ($C_{53}^2:C_2$)

Order 2: $C_2$ ($C_{53}^2:C_2^2$)

Order 1: $C_1$ ($D_{106}:C_{106}$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 22472: $D_{106}:C_{106}$ ($C_1$)

Order 11236: $C_{53}^2:C_2^2$ ($C_2$), $C_{53}^2:C_4$ ($C_2$), $C_{106}^2$ ($C_2$)

Order 5618: $C_{53}\times C_{106}$ ($C_2^2$)

Order 2809: $C_{53}^2$ ($D_4$)

Order 424: $C_{53}:D_4$ ($C_{53}$)

Order 212: $C_2\times C_{106}$ ($C_{106}$), $C_2\times C_{106}$ ($D_{53}$), $D_{106}$ ($C_{106}$), $C_{53}:C_4$ ($C_{106}$)

Order 106: $C_{106}$ ($C_2\times C_{106}$), $C_{106}$ ($D_{106}$)

Order 53: $C_{53}$ ($C_{53}:D_4$), $C_{53}$ ($D_4\times C_{53}$)

Order 4: $C_2^2$ ($C_{53}^2:C_2$)

Order 2: $C_2$ ($C_{53}^2:C_2^2$)

Order 1: $C_1$ ($D_{106}:C_{106}$)

Series

Derived series

$D_{106}:C_{106}$

$\rhd$

$D_{106}:C_{106}$

$\rhd$

$C_{106}$

$\rhd$

$C_{106}$

$\rhd$

$C_1$

$\rhd$

$C_1$

Chief series

$D_{106}:C_{106}$

$\rhd$

$D_{106}:C_{106}$

$\rhd$

$C_{106}^2$

$\rhd$

$C_{106}^2$

$\rhd$

$C_{53}\times C_{106}$

$\rhd$

$C_{53}\times C_{106}$

$\rhd$

$C_{106}$

$\rhd$

$C_{106}$

$\rhd$

$C_{53}$

$\rhd$

$C_{53}$

$\rhd$

$C_1$

$\rhd$

$C_1$

Lower central series

$D_{106}:C_{106}$

$\rhd$

$D_{106}:C_{106}$

$\rhd$

$C_{106}$

$\rhd$

$C_{106}$

$\rhd$

$C_{53}$

$\rhd$

$C_{53}$

Upper central series

$C_1$

$\lhd$

$C_1$

$\lhd$

$C_{106}$

$\lhd$

$C_{106}$

$\lhd$

$C_2\times C_{106}$

$\lhd$

$C_2\times C_{106}$

Character theory

Complex character table

The $5777 \times 5777$ character table is not available for this group.

Rational character table

The $117 \times 117$ rational character table is not available for this group.