Abstract group 4332.p: $D_{19}^2:C_3$

• Label: 4332.p
• Order: \( 2^{2} \cdot 3 \cdot 19^{2} \)
• Exponent: \( 2 \cdot 3 \cdot 19 \)
• Nilpotent: no
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{2} \cdot 3 \)
• $\card{Z(G)}$: \( 1 \)
• $\card{\Aut(G)}$: \( 2^{3} \cdot 3^{4} \cdot 19^{2} \)
• $\card{\mathrm{Out}(G)}$: \( 2 \cdot 3^{3} \)
• Perm deg.: $38$
• Trans deg.: $38$
• Rank: $2$

Group information

Description:	$D_{19}^2:C_3$
Order:	\(4332\)\(\medspace = 2^{2} \cdot 3 \cdot 19^{2} \)
Exponent:	\(114\)\(\medspace = 2 \cdot 3 \cdot 19 \)
Automorphism group:	$F_{19}\wr C_2$, of order \(233928\)\(\medspace = 2^{3} \cdot 3^{4} \cdot 19^{2} \)
Composition factors:	$C_2$ x 2, $C_3$, $C_{19}$ x 2
Derived length:	$2$

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

Group statistics

Order	1	2	3	6	19	38
Elements	1	399	722	2166	360	684	4332
Conjugacy classes	1	3	2	6	33	6	51
Divisions	1	3	1	3	11	2	21
Autjugacy classes	1	2	2	4	2	1	12

Dimension	1	2	6	12	18	36
Irr. complex chars.	12	0	12	27	0	0	51
Irr. rational chars.	4	4	0	0	4	9	21

Minimal presentations

Permutation degree:	$38$
Transitive degree:	$38$
Rank:	$2$
Inequivalent generating pairs:	$24$

Minimal degrees of faithful linear representations

	Over $\mathbb{C}$	Over $\mathbb{R}$	Over $\mathbb{Q}$
Irreducible	12	12	36
Arbitrary	12	12	36

Constructions

Presentation:	$\langle a, b, c \mid a^{6}=b^{38}=c^{19}=1, b^{a}=b^{27}c^{4}, c^{a}=c^{11}, c^{b}=c^{18} \rangle$
Permutation group:	Degree $38$ $\langle(1,26,16,34,10,27)(2,24,8,31,17,32)(3,22,19,28,5,37)(4,20,11,25,12,23)(6,35,14,38,7,33) \!\cdots\! \rangle$
Transitive group:	38T19				more information
Direct product:	not isomorphic to a non-trivial direct product
Semidirect product:	$D_{19}^2$ $\,\rtimes\,$ $C_3$	$(C_{19}\times D_{19})$ $\,\rtimes\,$ $C_6$ (2)	$(C_{19}:D_{19})$ $\,\rtimes\,$ $C_6$	$D_{19}$ $\,\rtimes\,$ $(C_{19}:C_6)$ (2)	all 9
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:	$C_{2} \times C_{6} \simeq C_{2}^{2} \times C_{3}$
Schur multiplier:	$C_{2}$
Commutator length:	$1$

Subgroups

There are 3928 subgroups in 76 conjugacy classes, 15 normal (7 characteristic).

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

Special subgroups

Center:	$Z \simeq$ $C_1$	$G/Z \simeq$ $D_{19}^2:C_3$
Commutator:	$G' \simeq$ $C_{19}^2$	$G/G' \simeq$ $C_2\times C_6$
Frattini:	$\Phi \simeq$ $C_1$	$G/\Phi \simeq$ $D_{19}^2:C_3$
Fitting:	$\operatorname{Fit} \simeq$ $C_{19}^2$	$G/\operatorname{Fit} \simeq$ $C_2\times C_6$
Radical:	$R \simeq$ $D_{19}^2:C_3$	$G/R \simeq$ $C_1$
Socle:	$\operatorname{soc} \simeq$ $C_{19}^2$	$G/\operatorname{soc} \simeq$ $C_2\times C_6$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $C_2^2$
3-Sylow subgroup:	$P_{ 3 } \simeq$ $C_3$
19-Sylow subgroup:	$P_{ 19 } \simeq$ $C_{19}^2$

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 4332: $D_{19}^2:C_3$

Order 2166: $C_{19}^2:C_6$ x 2, $C_{19}^2:C_6$

Order 1444: $D_{19}^2$

Order 1083: $C_{19}^2:C_3$

Order 722: $C_{19}\times D_{19}$ x 2, $C_{19}:D_{19}$

Order 361: $C_{19}^2$

Order 228: $C_{38}:C_6$ x 2

Order 114: $C_{19}:C_6$ x 13, $C_{19}:C_6$ x 2

Order 76: $D_{38}$ x 2

Order 57: $C_{19}:C_3$ x 11

Order 38: $D_{19}$ x 13, $C_{38}$ x 2

Order 19: $C_{19}$ x 11

Order 12: $C_2\times C_6$

Order 6: $C_6$ x 3

Order 4: $C_2^2$

Order 3: $C_3$

Order 2: $C_2$ x 3

Order 1: $C_1$

Classes of subgroups up to automorphism

Order 4332: $D_{19}^2:C_3$

Order 2166: $C_{19}^2:C_6$, $C_{19}^2:C_6$

Order 1444: $D_{19}^2$

Order 1083: $C_{19}^2:C_3$

Order 722: $C_{19}:D_{19}$, $C_{19}\times D_{19}$

Order 361: $C_{19}^2$

Order 228: $C_{38}:C_6$

Order 114: $C_{19}:C_6$ x 3, $C_{19}:C_6$

Order 76: $D_{38}$

Order 57: $C_{19}:C_3$ x 2

Order 38: $D_{19}$ x 3, $C_{38}$

Order 19: $C_{19}$ x 2

Order 12: $C_2\times C_6$

Order 6: $C_6$ x 2

Order 4: $C_2^2$

Order 3: $C_3$

Order 2: $C_2$ x 2

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 4332: $D_{19}^2:C_3$ ($C_1$)

Order 2166: $C_{19}^2:C_6$ ($C_2$) x 2, $C_{19}^2:C_6$ ($C_2$)

Order 1444: $D_{19}^2$ ($C_3$)

Order 1083: $C_{19}^2:C_3$ ($C_2^2$)

Order 722: $C_{19}\times D_{19}$ ($C_6$) x 2, $C_{19}:D_{19}$ ($C_6$)

Order 361: $C_{19}^2$ ($C_2\times C_6$)

Order 38: $D_{19}$ ($C_{19}:C_6$) x 2

Order 19: $C_{19}$ ($C_{38}:C_6$) x 2

Order 1: $C_1$ ($D_{19}^2:C_3$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 4332: $D_{19}^2:C_3$ ($C_1$)

Order 2166: $C_{19}^2:C_6$ ($C_2$), $C_{19}^2:C_6$ ($C_2$)

Order 1444: $D_{19}^2$ ($C_3$)

Order 1083: $C_{19}^2:C_3$ ($C_2^2$)

Order 722: $C_{19}:D_{19}$ ($C_6$), $C_{19}\times D_{19}$ ($C_6$)

Order 361: $C_{19}^2$ ($C_2\times C_6$)

Order 38: $D_{19}$ ($C_{19}:C_6$)

Order 19: $C_{19}$ ($C_{38}:C_6$)

Order 1: $C_1$ ($D_{19}^2:C_3$)

Series

Derived series

$D_{19}^2:C_3$

$\rhd$

$C_{19}^2$

$\rhd$

$C_1$

Chief series

$D_{19}^2:C_3$

$\rhd$

$C_{19}^2:C_6$

$\rhd$

$C_{19}^2:C_3$

$\rhd$

$C_{19}^2$

$\rhd$

$C_{19}$

$\rhd$

$C_{19}$

$\rhd$

$C_1$

Lower central series

$D_{19}^2:C_3$

$\rhd$

$C_{19}^2$

Upper central series

$C_1$

Character theory

Complex character table

See the $51 \times 51$ character table. Alternatively, you may search for characters of this group with desired properties.

Rational character table

See the $21 \times 21$ rational character table.