Abstract group 77976.p: $D_{19}^2:(S_3\times C_9)$

• Label: 77976.p
• Order: \( 2^{3} \cdot 3^{3} \cdot 19^{2} \)
• Exponent: \( 2^{2} \cdot 3^{2} \cdot 19 \)
• Nilpotent: no
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{2} \cdot 3^{2} \)
• $\card{Z(G)}$: \( 1 \)
• $\card{\Aut(G)}$: \( 2^{3} \cdot 3^{3} \cdot 19^{2} \)
• $\card{\mathrm{Out}(G)}$: \( 1 \)
• Perm deg.: $38$
• Trans deg.: $38$
• Rank: $2$

Group information

Description:	$D_{19}^2:(S_3\times C_9)$
Order:	\(77976\)\(\medspace = 2^{3} \cdot 3^{3} \cdot 19^{2} \)
Exponent:	\(684\)\(\medspace = 2^{2} \cdot 3^{2} \cdot 19 \)
Automorphism group:	$D_{19}^2:(S_3\times C_9)$, of order \(77976\)\(\medspace = 2^{3} \cdot 3^{3} \cdot 19^{2} \)
Composition factors:	$C_2$ x 3, $C_3$ x 3, $C_{19}$ 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	57	114
Elements	1	513	1520	2166	11628	6498	4332	32490	360	12996	2736	1368	1368	77976
Conjugacy classes	1	3	5	1	15	12	2	36	3	6	2	2	2	90
Divisions	1	3	3	1	8	2	1	6	3	1	2	1	1	33
Autjugacy classes	1	3	5	1	15	12	2	36	3	6	2	2	2	90

Dimension	1	2	4	6	12	36	72	108	216
Irr. complex chars.	36	45	0	0	0	6	0	2	1	90
Irr. rational chars.	4	7	6	4	5	2	2	2	1	33

Minimal presentations

Permutation degree:	$38$
Transitive degree:	$38$
Rank:	$2$
Inequivalent generating pairs:	$17280$

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:	${\langle a, b, c, d \mid a^{18}=b^{6}=c^{38}=d^{19}=1, b^{a}=b^{5}c^{34}d^{18}, c^{a}= \!\cdots\! \rangle}$
Permutation group:	Degree $38$ $\langle(1,29,19,24,14,37,8,26,16,28,18,38,9,31,2,34,5,30)(3,20,10,36,7,21,11,22,12,27,17,33,4,25,15,23,13,32) \!\cdots\! \rangle$
Transitive group:	38T44				more information
Direct product:	not isomorphic to a non-trivial direct product
Semidirect product:	$(D_{19}^2:S_3)$ $\,\rtimes\,$ $C_9$	$D_{19}^2$ $\,\rtimes\,$ $(S_3\times C_9)$	$C_{19}^2$ $\,\rtimes\,$ $(D_6:C_{18})$	$(C_{19}^2:C_3:C_{36})$ $\,\rtimes\,$ $C_2$	all 7
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Non-split product:	$(C_{19}:F_{19})$ . $D_6$	$(C_{19}^2:C_6^2)$ . $C_6$ (3)	$(D_{19}^2:(C_3\times S_3))$ . $C_3$	$(C_{19}:D_{19})$ . $(S_3\times C_{18})$	all 11
Aut. group:	$\Aut(C_{19}^2:D_6)$	$\Aut(C_{19}^2:C_3:C_4)$	$\Aut(D_{19}^2:S_3)$	$\Aut(C_{19}^2:C_3:C_{12})$	all 8

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 39758 subgroups in 187 conjugacy classes, 28 normal, and all normal subgroups are characteristic.

Characteristic subgroups are shown in this color.

Special subgroups

Center:	$Z \simeq$ $C_1$	$G/Z \simeq$ $D_{19}^2:(S_3\times C_9)$
Commutator:	$G' \simeq$ $C_{19}^2:C_6$	$G/G' \simeq$ $C_2\times C_{18}$
Frattini:	$\Phi \simeq$ $C_1$	$G/\Phi \simeq$ $D_{19}^2:(S_3\times C_9)$
Fitting:	$\operatorname{Fit} \simeq$ $C_{19}^2$	$G/\operatorname{Fit} \simeq$ $D_6:C_{18}$
Radical:	$R \simeq$ $D_{19}^2:(S_3\times C_9)$	$G/R \simeq$ $C_1$
Socle:	$\operatorname{soc} \simeq$ $C_{19}^2$	$G/\operatorname{soc} \simeq$ $D_6:C_{18}$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $D_4$
3-Sylow subgroup:	$P_{ 3 } \simeq$ $C_3\times C_9$
19-Sylow subgroup:	$P_{ 19 } \simeq$ $C_{19}^2$

Hi

Subgroup diagram and profile

Classes of subgroups up to conjugation

There are too many subgroups to show. See a full page version of the diagram.

Classes of subgroups up to automorphism

There are too many subgroups to show. See a full page version of the diagram.

Normal subgroups

Normal subgroups up to automorphism

For the default diagram, subgroups are sorted vertically by the number of prime divisors (counted with multiplicity) in their orders.
To see subgroups sorted vertically by order instead, check this box.

Sorry, your browser does not support the subgroup diagram.

Subgroup information Click on a subgroup in the diagram to see information about it.

See a full page version of the diagram

See a full page version of the diagram

See a full page version of the diagram

See a full page version of the diagram

Classes of subgroups up to conjugation

Order 77976: $D_{19}^2:(S_3\times C_9)$

Order 38988: $C_{19}^2:(C_6\times C_{18})$, $C_{19}^2:(S_3\times C_{18})$, $C_{19}^2:C_3:C_{36}$

Order 25992: $D_{19}^2:C_{18}$, $D_{19}^2:(C_3\times S_3)$

Order 19494: $C_{19}^2:(C_3\times C_{18})$, $C_{19}^2:(C_3\times C_{18})$, $C_{19}^2:(S_3\times C_9)$

Order 12996: $C_{19}^2:(C_2\times C_{18})$ x 3, $C_{19}^2:(C_6\times S_3)$, $C_{19}^2:(C_3\times C_3:C_4)$, $C_{19}^2:C_6^2$, $C_{19}^2:C_{36}$

Order 9747: $C_{19}^2:(C_3\times C_9)$

Order 8664: $D_{19}^2:C_6$, $D_{19}^2:S_3$

Order 6498: $C_{19}^2:C_{18}$ x 2, $C_{19}^2:C_{18}$ x 2, $C_{19}^2:(C_3\times C_6)$, $C_{19}^2:C_{18}$, $C_{19}^2:(C_3\times C_6)$, $C_{19}^2:(C_3\times S_3)$, $C_{19}:F_{19}$

Order 4332: $C_{19}^2:(C_2\times C_6)$ x 2, $C_{19}^2:(C_3:C_4)$, $C_{19}^2:D_6$, $C_{19}^2:(C_2\times C_6)$, $C_{19}^2:(C_2\times C_6)$, $C_{19}^2:C_{12}$

Order 3249: $C_{19}^2:C_9$, $(C_{19}:C_3)^2$, $C_{19}^2:C_9$

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

Order 2166: $C_{19}^2:C_6$ x 2, $C_{19}^2:C_6$, $C_{19}^2:C_6$, $C_{19}^2:C_6$, $C_{19}\times D_{19}:C_3$, $D_{19}\times C_{19}:C_3$, $C_{19}^2:S_3$, $C_{19}^2:C_6$

Order 2052: $C_6\times F_{19}$

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

Order 1083: $C_{19}^2:C_3$, $C_{19}^2:C_3$, $C_{19}:C_{57}$

Order 1026: $C_3\times F_{19}$ x 2, $C_{114}:C_9$

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

Order 684: $C_2\times F_{19}$ x 4, $C_{114}:C_6$

Order 513: $C_{57}:C_9$

Order 361: $C_{19}^2$

Order 342: $F_{19}$ x 9, $C_{19}:C_{18}$ x 4, $C_{57}:C_6$ x 2, $C_{114}:C_3$

Order 228: $C_{38}:C_6$ x 4, $C_3\times D_{38}$

Order 216: $D_6:C_{18}$

Order 171: $C_{19}:C_9$ x 5, $C_{57}:C_3$

Order 114: $C_{19}:C_6$ x 9, $C_{19}:C_6$ x 4, $C_3\times D_{19}$ x 2, $C_{114}$

Order 108: $C_3:C_{36}$, $C_6\times C_{18}$, $S_3\times C_{18}$

Order 76: $D_{38}$ x 2

Order 72: $C_6\wr C_2$, $D_4\times C_9$

Order 57: $C_{19}:C_3$ x 5, $C_{57}$

Order 54: $C_3\times C_{18}$ x 2, $S_3\times C_9$

Order 38: $D_{19}$ x 5, $C_{38}$ x 2

Order 36: $C_2\times C_{18}$ x 3, $C_3:C_{12}$, $C_{36}$, $C_6^2$, $C_6\times S_3$

Order 27: $C_3\times C_9$

Order 24: $C_3:D_4$, $C_3\times D_4$

Order 19: $C_{19}$ x 3

Order 18: $C_{18}$ x 6, $C_3\times C_6$ x 2, $C_3\times S_3$

Order 12: $C_2\times C_6$ x 4, $D_6$, $C_{12}$, $C_3:C_4$

Order 9: $C_9$ x 2, $C_3^2$

Order 8: $D_4$

Order 6: $C_6$ x 8, $S_3$

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

Order 3: $C_3$ x 3

Order 2: $C_2$ x 3

Order 1: $C_1$

Classes of subgroups up to automorphism

Order 77976: $D_{19}^2:(S_3\times C_9)$

Order 38988: $C_{19}^2:(C_6\times C_{18})$, $C_{19}^2:(S_3\times C_{18})$, $C_{19}^2:C_3:C_{36}$

Order 25992: $D_{19}^2:C_{18}$, $D_{19}^2:(C_3\times S_3)$

Order 19494: $C_{19}^2:(C_3\times C_{18})$, $C_{19}^2:(C_3\times C_{18})$, $C_{19}^2:(S_3\times C_9)$

Order 12996: $C_{19}^2:(C_2\times C_{18})$ x 3, $C_{19}^2:(C_6\times S_3)$, $C_{19}^2:(C_3\times C_3:C_4)$, $C_{19}^2:C_6^2$, $C_{19}^2:C_{36}$

Order 9747: $C_{19}^2:(C_3\times C_9)$

Order 8664: $D_{19}^2:C_6$, $D_{19}^2:S_3$

Order 6498: $C_{19}^2:C_{18}$ x 2, $C_{19}^2:C_{18}$ x 2, $C_{19}^2:(C_3\times C_6)$, $C_{19}^2:C_{18}$, $C_{19}^2:(C_3\times C_6)$, $C_{19}^2:(C_3\times S_3)$, $C_{19}:F_{19}$

Order 4332: $C_{19}^2:(C_2\times C_6)$ x 2, $C_{19}^2:(C_3:C_4)$, $C_{19}^2:D_6$, $C_{19}^2:(C_2\times C_6)$, $C_{19}^2:(C_2\times C_6)$, $C_{19}^2:C_{12}$

Order 3249: $C_{19}^2:C_9$, $(C_{19}:C_3)^2$, $C_{19}^2:C_9$

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

Order 2166: $C_{19}^2:C_6$ x 2, $C_{19}^2:C_6$, $C_{19}^2:C_6$, $C_{19}^2:C_6$, $C_{19}\times D_{19}:C_3$, $D_{19}\times C_{19}:C_3$, $C_{19}^2:S_3$, $C_{19}^2:C_6$

Order 2052: $C_6\times F_{19}$

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

Order 1083: $C_{19}^2:C_3$, $C_{19}^2:C_3$, $C_{19}:C_{57}$

Order 1026: $C_3\times F_{19}$ x 2, $C_{114}:C_9$

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

Order 684: $C_2\times F_{19}$ x 4, $C_{114}:C_6$

Order 513: $C_{57}:C_9$

Order 361: $C_{19}^2$

Order 342: $F_{19}$ x 9, $C_{19}:C_{18}$ x 4, $C_{57}:C_6$ x 2, $C_{114}:C_3$

Order 228: $C_{38}:C_6$ x 4, $C_3\times D_{38}$

Order 216: $D_6:C_{18}$

Order 171: $C_{19}:C_9$ x 5, $C_{57}:C_3$

Order 114: $C_{19}:C_6$ x 9, $C_{19}:C_6$ x 4, $C_3\times D_{19}$ x 2, $C_{114}$

Order 108: $C_3:C_{36}$, $C_6\times C_{18}$, $S_3\times C_{18}$

Order 76: $D_{38}$ x 2

Order 72: $C_6\wr C_2$, $D_4\times C_9$

Order 57: $C_{19}:C_3$ x 5, $C_{57}$

Order 54: $C_3\times C_{18}$ x 2, $S_3\times C_9$

Order 38: $D_{19}$ x 5, $C_{38}$ x 2

Order 36: $C_2\times C_{18}$ x 3, $C_3:C_{12}$, $C_{36}$, $C_6^2$, $C_6\times S_3$

Order 27: $C_3\times C_9$

Order 24: $C_3:D_4$, $C_3\times D_4$

Order 19: $C_{19}$ x 3

Order 18: $C_{18}$ x 6, $C_3\times C_6$ x 2, $C_3\times S_3$

Order 12: $C_2\times C_6$ x 4, $D_6$, $C_{12}$, $C_3:C_4$

Order 9: $C_9$ x 2, $C_3^2$

Order 8: $D_4$

Order 6: $C_6$ x 8, $S_3$

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

Order 3: $C_3$ x 3

Order 2: $C_2$ x 3

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 77976: $D_{19}^2:(S_3\times C_9)$ ($C_1$)

Order 38988: $C_{19}^2:(C_6\times C_{18})$ ($C_2$), $C_{19}^2:(S_3\times C_{18})$ ($C_2$), $C_{19}^2:C_3:C_{36}$ ($C_2$)

Order 25992: $D_{19}^2:(C_3\times S_3)$ ($C_3$)

Order 19494: $C_{19}^2:(C_3\times C_{18})$ ($C_2^2$)

Order 12996: $C_{19}^2:(C_6\times S_3)$ ($C_6$), $C_{19}^2:(C_3\times C_3:C_4)$ ($C_6$), $C_{19}^2:(C_2\times C_{18})$ ($S_3$), $C_{19}^2:C_6^2$ ($C_6$)

Order 9747: $C_{19}^2:(C_3\times C_9)$ ($D_4$)

Order 8664: $D_{19}^2:S_3$ ($C_9$)

Order 6498: $C_{19}^2:(C_3\times C_6)$ ($C_2\times C_6$), $C_{19}:F_{19}$ ($D_6$)

Order 4332: $C_{19}^2:(C_2\times C_6)$ ($C_3\times S_3$), $C_{19}^2:(C_3:C_4)$ ($C_{18}$), $C_{19}^2:D_6$ ($C_{18}$), $C_{19}^2:(C_2\times C_6)$ ($C_{18}$)

Order 3249: $C_{19}^2:C_9$ ($C_3:D_4$), $(C_{19}:C_3)^2$ ($C_3\times D_4$)

Order 2166: $C_{19}^2:C_6$ ($C_6\times S_3$), $C_{19}^2:C_6$ ($C_2\times C_{18}$)

Order 1444: $D_{19}^2$ ($S_3\times C_9$)

Order 1083: $C_{19}^2:C_3$ ($D_4\times C_9$), $C_{19}^2:C_3$ ($C_6\wr C_2$)

Order 722: $C_{19}:D_{19}$ ($S_3\times C_{18}$)

Order 361: $C_{19}^2$ ($D_6:C_{18}$)

Order 1: $C_1$ ($D_{19}^2:(S_3\times C_9)$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 77976: $D_{19}^2:(S_3\times C_9)$ ($C_1$)

Order 38988: $C_{19}^2:(C_6\times C_{18})$ ($C_2$), $C_{19}^2:(S_3\times C_{18})$ ($C_2$), $C_{19}^2:C_3:C_{36}$ ($C_2$)

Order 25992: $D_{19}^2:(C_3\times S_3)$ ($C_3$)

Order 19494: $C_{19}^2:(C_3\times C_{18})$ ($C_2^2$)

Order 12996: $C_{19}^2:(C_6\times S_3)$ ($C_6$), $C_{19}^2:(C_3\times C_3:C_4)$ ($C_6$), $C_{19}^2:(C_2\times C_{18})$ ($S_3$), $C_{19}^2:C_6^2$ ($C_6$)

Order 9747: $C_{19}^2:(C_3\times C_9)$ ($D_4$)

Order 8664: $D_{19}^2:S_3$ ($C_9$)

Order 6498: $C_{19}^2:(C_3\times C_6)$ ($C_2\times C_6$), $C_{19}:F_{19}$ ($D_6$)

Order 4332: $C_{19}^2:(C_2\times C_6)$ ($C_3\times S_3$), $C_{19}^2:(C_3:C_4)$ ($C_{18}$), $C_{19}^2:D_6$ ($C_{18}$), $C_{19}^2:(C_2\times C_6)$ ($C_{18}$)

Order 3249: $C_{19}^2:C_9$ ($C_3:D_4$), $(C_{19}:C_3)^2$ ($C_3\times D_4$)

Order 2166: $C_{19}^2:C_6$ ($C_6\times S_3$), $C_{19}^2:C_6$ ($C_2\times C_{18}$)

Order 1444: $D_{19}^2$ ($S_3\times C_9$)

Order 1083: $C_{19}^2:C_3$ ($D_4\times C_9$), $C_{19}^2:C_3$ ($C_6\wr C_2$)

Order 722: $C_{19}:D_{19}$ ($S_3\times C_{18}$)

Order 361: $C_{19}^2$ ($D_6:C_{18}$)

Order 1: $C_1$ ($D_{19}^2:(S_3\times C_9)$)

Series

Derived series

$D_{19}^2:(S_3\times C_9)$

$\rhd$

$C_{19}^2:C_6$

$\rhd$

$C_{19}^2$

$\rhd$

$C_1$

Chief series

$D_{19}^2:(S_3\times C_9)$

$\rhd$

$C_{19}^2:(S_3\times C_{18})$

$\rhd$

$C_{19}^2:(C_3\times C_{18})$

$\rhd$

$C_{19}^2:(C_3\times C_6)$

$\rhd$

$(C_{19}:C_3)^2$

$\rhd$

$C_{19}^2:C_3$

$\rhd$

$C_{19}^2$

$\rhd$

$C_1$

Lower central series

$D_{19}^2:(S_3\times C_9)$

$\rhd$

$C_{19}^2:C_6$

$\rhd$

$C_{19}^2:C_3$

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 $90 \times 90$ character table. Alternatively, you may search for characters of this group with desired properties.

Rational character table

See the $33 \times 33$ rational character table.