Abstract group 428.3: $D_{214}$

• Label: 428.3
• Order: \( 2^{2} \cdot 107 \)
• Exponent: \( 2 \cdot 107 \)
• Nilpotent: no
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{2} \)
• $\card{Z(G)}$: \( 2 \)
• $\card{\Aut(G)}$: \( 2^{2} \cdot 53 \cdot 107 \)
• $\card{\mathrm{Out}(G)}$: \( 2 \cdot 53 \)
• Perm deg.: $109$
• Trans deg.: $214$
• Rank: $2$

Group information

Description:	$D_{214}$
Order:	\(428\)\(\medspace = 2^{2} \cdot 107 \)
Exponent:	\(214\)\(\medspace = 2 \cdot 107 \)
Automorphism group:	$C_2\times F_{107}$, of order \(22684\)\(\medspace = 2^{2} \cdot 53 \cdot 107 \)
Composition factors:	$C_2$ x 2, $C_{107}$
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	107	214
Elements	1	215	106	106	428
Conjugacy classes	1	3	53	53	110
Divisions	1	3	1	1	6
Autjugacy classes	1	2	1	1	5

Dimension	1	2	106
Irr. complex chars.	4	106	0	110
Irr. rational chars.	4	0	2	6

Minimal presentations

Permutation degree:	$109$
Transitive degree:	$214$
Rank:	$2$
Inequivalent generating pairs:	$3$

Minimal degrees of faithful linear representations

	Over $\mathbb{C}$	Over $\mathbb{R}$	Over $\mathbb{Q}$
Irreducible	2	2	106
Arbitrary	2	2	106

Constructions

Presentation:	$\langle a, b \mid a^{2}=b^{214}=1, b^{a}=b^{213} \rangle$
Permutation group:	Degree $109$ $\langle(2,5)(3,11)(4,6)(7,17)(8,22)(9,13)(10,16)(12,15)(14,27)(18,33)(19,38)(20,23) \!\cdots\! \rangle$
Matrix group:	$\left\langle \left(\begin{array}{rr} 1 & 1 \\ 0 & 1 \end{array}\right), \left(\begin{array}{rr} 1 & 0 \\ 0 & 106 \end{array}\right), \left(\begin{array}{rr} 106 & 0 \\ 0 & 106 \end{array}\right) \right\rangle \subseteq \GL_{2}(\F_{107})$
Direct product:	$C_2$ $\, \times\, $ $D_{107}$
Semidirect product:	$C_{214}$ $\,\rtimes\,$ $C_2$	$C_{107}$ $\,\rtimes\,$ $C_2^2$			more information
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product

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

Homology

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

Subgroups

There are 328 subgroups in 10 conjugacy classes, 7 normal (5 characteristic).

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

Special subgroups

Center:	$Z \simeq$ $C_2$	$G/Z \simeq$ $D_{107}$
Commutator:	$G' \simeq$ $C_{107}$	$G/G' \simeq$ $C_2^2$
Frattini:	$\Phi \simeq$ $C_1$	$G/\Phi \simeq$ $D_{214}$
Fitting:	$\operatorname{Fit} \simeq$ $C_{214}$	$G/\operatorname{Fit} \simeq$ $C_2$
Radical:	$R \simeq$ $D_{214}$	$G/R \simeq$ $C_1$
Socle:	$\operatorname{soc} \simeq$ $C_{214}$	$G/\operatorname{soc} \simeq$ $C_2$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $C_2^2$
107-Sylow subgroup:	$P_{ 107 } \simeq$ $C_{107}$

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 428: $D_{214}$

Order 214: $D_{107}$ x 2, $C_{214}$

Order 107: $C_{107}$

Order 4: $C_2^2$

Order 2: $C_2$ x 3

Order 1: $C_1$

Classes of subgroups up to automorphism

Order 428: $D_{214}$

Order 214: $C_{214}$, $D_{107}$

Order 107: $C_{107}$

Order 4: $C_2^2$

Order 2: $C_2$ x 2

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 428: $D_{214}$ ($C_1$)

Order 214: $D_{107}$ ($C_2$) x 2, $C_{214}$ ($C_2$)

Order 107: $C_{107}$ ($C_2^2$)

Order 2: $C_2$ ($D_{107}$)

Order 1: $C_1$ ($D_{214}$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 428: $D_{214}$ ($C_1$)

Order 214: $C_{214}$ ($C_2$), $D_{107}$ ($C_2$)

Order 107: $C_{107}$ ($C_2^2$)

Order 2: $C_2$ ($D_{107}$)

Order 1: $C_1$ ($D_{214}$)

Series

Derived series	$D_{214}$	$\rhd$	$C_{107}$	$\rhd$	$C_1$
Chief series	$D_{214}$	$\rhd$	$D_{107}$	$\rhd$	$C_{107}$	$\rhd$	$C_1$
Lower central series	$D_{214}$	$\rhd$	$C_{107}$
Upper central series	$C_1$	$\lhd$	$C_2$

Supergroups

This group is a maximal subgroup of 9 larger groups in the database.

This group is a maximal quotient of 10 larger groups in the database.

Character theory

Complex character table

See the $110 \times 110$ character table. Alternatively, you may search for characters of this group with desired properties.

Rational character table

		1A	2A	2B	2C	107A	214A
	Size	1	1	107	107	106	106
	2 P	1A	1A	1A	1A	107A	107A
	107 P	1A	2A	2B	2C	107A	214A
428.3.1a		$1$	$1$	$1$	$1$	$1$	$1$
428.3.1b		$1$	$-1$	$-1$	$1$	$1$	$-1$
428.3.1c		$1$	$-1$	$1$	$-1$	$1$	$-1$
428.3.1d		$1$	$1$	$-1$	$-1$	$1$	$1$
428.3.2a		$106$	$106$	$0$	$0$	$-1$	$-1$
428.3.2b		$106$	$-106$	$0$	$0$	$-1$	$1$