Abstract group 25992.bj: $D_{19}^2:C_{18}$

• Label: 25992.bj
• Order: \( 2^{3} \cdot 3^{2} \cdot 19^{2} \)
• Exponent: \( 2^{2} \cdot 3^{2} \cdot 19 \)
• Nilpotent: no
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{2} \cdot 3^{2} \)
• $\card{Z(G)}$: \( 1 \)
• $\card{\Aut(G)}$: \( 2^{4} \cdot 3^{2} \cdot 19^{2} \)
• $\card{\mathrm{Out}(G)}$: \( 2 \)
• Perm deg.: $38$
• Trans deg.: $38$
• Rank: $2$

Group information

Description:	$D_{19}^2:C_{18}$
Order:	\(25992\)\(\medspace = 2^{3} \cdot 3^{2} \cdot 19^{2} \)
Exponent:	\(684\)\(\medspace = 2^{2} \cdot 3^{2} \cdot 19 \)
Automorphism group:	$C_{19}^2.C_{36}.C_2^2$, of order \(51984\)\(\medspace = 2^{4} \cdot 3^{2} \cdot 19^{2} \)
Composition factors:	$C_2$ x 3, $C_3$ x 2, $C_{19}$ x 2
Derived length:	$3$

This group is nonabelian and monomial (hence solvable).

Group statistics

Order	1	2	3	4	6	9	12	18	19	36	38
Elements	1	437	722	722	3610	2166	1444	10830	360	4332	1368	25992
Conjugacy classes	1	3	2	1	6	6	2	18	6	6	2	53
Divisions	1	3	1	1	3	1	1	3	6	1	2	23
Autjugacy classes	1	2	2	1	4	6	2	12	3	6	1	40

Dimension	1	2	4	6	12	36	72
Irr. complex chars.	36	9	0	0	0	4	4	53
Irr. rational chars.	4	5	1	4	1	4	4	23

Minimal presentations

Permutation degree:	$38$
Transitive degree:	$38$
Rank:	$2$
Inequivalent generating pairs:	$4320$

Minimal degrees of faithful linear representations

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

Constructions

Presentation:	${\langle a, b, c, d \mid a^{18}=b^{4}=c^{19}=d^{19}=[c,d]=1, b^{a}=b^{3}c^{13}d^{17}, c^{a}= \!\cdots\! \rangle}$
Permutation group:	Degree $38$ $\langle(1,2)(3,19)(4,18)(5,17)(6,16)(7,15)(8,14)(9,13)(10,12)(20,25,30,35,21,26,31,36,22,27,32,37,23,28,33,38,24,29,34) \!\cdots\! \rangle$
Transitive group:	38T36				more information
Direct product:	not isomorphic to a non-trivial direct product
Semidirect product:	$D_{19}^2$ $\,\rtimes\,$ $C_{18}$ (2)	$(D_{19}\wr C_2)$ $\,\rtimes\,$ $C_9$	$(C_{19}^2:C_9)$ $\,\rtimes\,$ $D_4$	$C_{19}^2$ $\,\rtimes\,$ $(D_4\times C_9)$	all 6
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Non-split product:	$(D_{19}^2:C_6)$ . $C_3$	$(C_{19}^2:C_{12})$ . $C_6$	$(C_{19}:F_{19})$ . $C_2^2$	$(C_{19}^2:C_3)$ . $(C_3\times D_4)$	all 7

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

Homology

Abelianization:	$C_{2} \times C_{18} \simeq C_{2}^{2} \times C_{9}$
Schur multiplier:	$C_{2}$
Commutator length:	$1$

Subgroups

There are 11664 subgroups in 102 conjugacy classes, 19 normal (13 characteristic).

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

Special subgroups

Center:	$Z \simeq$ $C_1$	$G/Z \simeq$ $D_{19}^2:C_{18}$
Commutator:	$G' \simeq$ $C_{19}:D_{19}$	$G/G' \simeq$ $C_2\times C_{18}$
Frattini:	$\Phi \simeq$ $C_1$	$G/\Phi \simeq$ $D_{19}^2:C_{18}$
Fitting:	$\operatorname{Fit} \simeq$ $C_{19}^2$	$G/\operatorname{Fit} \simeq$ $D_4\times C_9$
Radical:	$R \simeq$ $D_{19}^2:C_{18}$	$G/R \simeq$ $C_1$
Socle:	$\operatorname{soc} \simeq$ $C_{19}^2$	$G/\operatorname{soc} \simeq$ $D_4\times C_9$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $D_4$
3-Sylow subgroup:	$P_{ 3 } \simeq$ $C_9$
19-Sylow subgroup:	$P_{ 19 } \simeq$ $C_{19}^2$

Hi

Subgroup diagram and profile

Classes of subgroups up to conjugation

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Classes of subgroups up to automorphism

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Normal subgroups

Normal subgroups up to automorphism

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Classes of subgroups up to conjugation

Order 25992: $D_{19}^2:C_{18}$

Order 12996: $C_{19}^2:(C_2\times C_{18})$ x 2, $C_{19}^2:C_{36}$

Order 8664: $D_{19}^2:C_6$

Order 6498: $C_{19}^2:C_{18}$ x 2, $C_{19}:F_{19}$

Order 4332: $C_{19}^2:(C_2\times C_6)$ x 2, $C_{19}^2:C_{12}$

Order 3249: $C_{19}^2:C_9$

Order 2888: $D_{19}\wr C_2$

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

Order 1444: $D_{19}^2$ x 2, $C_{19}^2:C_4$

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

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

Order 684: $C_2\times F_{19}$ x 2

Order 361: $C_{19}^2$

Order 342: $F_{19}$ x 8, $C_{19}:C_{18}$ x 2

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

Order 171: $C_{19}:C_9$ x 6

Order 114: $C_{19}:C_6$ x 8, $C_{19}:C_6$ x 2

Order 76: $D_{38}$ x 2

Order 72: $D_4\times C_9$

Order 57: $C_{19}:C_3$ x 6

Order 38: $D_{19}$ x 8, $C_{38}$ x 2

Order 36: $C_2\times C_{18}$ x 2, $C_{36}$

Order 24: $C_3\times D_4$

Order 19: $C_{19}$ x 6

Order 18: $C_{18}$ x 3

Order 12: $C_2\times C_6$ x 2, $C_{12}$

Order 9: $C_9$

Order 8: $D_4$

Order 6: $C_6$ x 3

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

Order 3: $C_3$

Order 2: $C_2$ x 3

Order 1: $C_1$

Classes of subgroups up to automorphism

Order 25992: $D_{19}^2:C_{18}$

Order 12996: $C_{19}^2:(C_2\times C_{18})$, $C_{19}^2:C_{36}$

Order 8664: $D_{19}^2:C_6$

Order 6498: $C_{19}^2:C_{18}$, $C_{19}:F_{19}$

Order 4332: $C_{19}^2:(C_2\times C_6)$, $C_{19}^2:C_{12}$

Order 3249: $C_{19}^2:C_9$

Order 2888: $D_{19}\wr C_2$

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

Order 1444: $D_{19}^2$, $C_{19}^2:C_4$

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

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

Order 684: $C_2\times F_{19}$

Order 361: $C_{19}^2$

Order 342: $F_{19}$ x 4, $C_{19}:C_{18}$

Order 228: $C_{38}:C_6$

Order 171: $C_{19}:C_9$ x 3

Order 114: $C_{19}:C_6$ x 4, $C_{19}:C_6$

Order 76: $D_{38}$

Order 72: $D_4\times C_9$

Order 57: $C_{19}:C_3$ x 3

Order 38: $D_{19}$ x 4, $C_{38}$

Order 36: $C_2\times C_{18}$, $C_{36}$

Order 24: $C_3\times D_4$

Order 19: $C_{19}$ x 3

Order 18: $C_{18}$ x 2

Order 12: $C_2\times C_6$, $C_{12}$

Order 9: $C_9$

Order 8: $D_4$

Order 6: $C_6$ x 2

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

Order 3: $C_3$

Order 2: $C_2$ x 2

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 25992: $D_{19}^2:C_{18}$ ($C_1$)

Order 12996: $C_{19}^2:(C_2\times C_{18})$ ($C_2$) x 2, $C_{19}^2:C_{36}$ ($C_2$)

Order 8664: $D_{19}^2:C_6$ ($C_3$)

Order 6498: $C_{19}:F_{19}$ ($C_2^2$)

Order 4332: $C_{19}^2:(C_2\times C_6)$ ($C_6$) x 2, $C_{19}^2:C_{12}$ ($C_6$)

Order 3249: $C_{19}^2:C_9$ ($D_4$)

Order 2888: $D_{19}\wr C_2$ ($C_9$)

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

Order 1444: $D_{19}^2$ ($C_{18}$) x 2, $C_{19}^2:C_4$ ($C_{18}$)

Order 1083: $C_{19}^2:C_3$ ($C_3\times D_4$)

Order 722: $C_{19}:D_{19}$ ($C_2\times C_{18}$)

Order 361: $C_{19}^2$ ($D_4\times C_9$)

Order 1: $C_1$ ($D_{19}^2:C_{18}$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 25992: $D_{19}^2:C_{18}$ ($C_1$)

Order 12996: $C_{19}^2:(C_2\times C_{18})$ ($C_2$), $C_{19}^2:C_{36}$ ($C_2$)

Order 8664: $D_{19}^2:C_6$ ($C_3$)

Order 6498: $C_{19}:F_{19}$ ($C_2^2$)

Order 4332: $C_{19}^2:(C_2\times C_6)$ ($C_6$), $C_{19}^2:C_{12}$ ($C_6$)

Order 3249: $C_{19}^2:C_9$ ($D_4$)

Order 2888: $D_{19}\wr C_2$ ($C_9$)

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

Order 1444: $D_{19}^2$ ($C_{18}$), $C_{19}^2:C_4$ ($C_{18}$)

Order 1083: $C_{19}^2:C_3$ ($C_3\times D_4$)

Order 722: $C_{19}:D_{19}$ ($C_2\times C_{18}$)

Order 361: $C_{19}^2$ ($D_4\times C_9$)

Order 1: $C_1$ ($D_{19}^2:C_{18}$)

Series

Derived series

$D_{19}^2:C_{18}$

$\rhd$

$C_{19}:D_{19}$

$\rhd$

$C_{19}^2$

$\rhd$

$C_1$

Chief series

$D_{19}^2:C_{18}$

$\rhd$

$C_{19}^2:C_{36}$

$\rhd$

$C_{19}^2:C_{12}$

$\rhd$

$C_{19}^2:C_4$

$\rhd$

$C_{19}:D_{19}$

$\rhd$

$C_{19}^2$

$\rhd$

$C_1$

Lower central series

$D_{19}^2:C_{18}$

$\rhd$

$C_{19}:D_{19}$

$\rhd$

$C_{19}^2$

Upper central series

$C_1$

Supergroups

This group is a maximal subgroup of 1 larger groups in the database.

This group is a maximal quotient of 0 larger groups in the database.

Character theory

Complex character table

See the $53 \times 53$ character table. Alternatively, you may search for characters of this group with desired properties.

Rational character table

See the $23 \times 23$ rational character table.