Abstract group 1404.117: $\He_3:D_{26}$

• Label: 1404.117
• Order: \( 2^{2} \cdot 3^{3} \cdot 13 \)
• Exponent: \( 2 \cdot 3 \cdot 13 \)
• Nilpotent: no
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{2} \)
• $\card{Z(G)}$: \( 3 \)
• $\card{\Aut(G)}$: \( 2^{6} \cdot 3^{4} \cdot 13 \)
• $\card{\mathrm{Out}(G)}$: \( 2^{4} \cdot 3^{2} \)
• Perm deg.: $22$
• Trans deg.: $117$
• Rank: $3$

Group information

Description:	$\He_3:D_{26}$
Order:	\(1404\)\(\medspace = 2^{2} \cdot 3^{3} \cdot 13 \)
Exponent:	\(78\)\(\medspace = 2 \cdot 3 \cdot 13 \)
Automorphism group:	$F_{13}\times C_3^2:\GL(2,3)$, of order \(67392\)\(\medspace = 2^{6} \cdot 3^{4} \cdot 13 \)
Composition factors:	$C_2$ x 2, $C_3$ x 3, $C_{13}$
Derived length:	$3$

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

Group statistics

Order	1	2	3	6	13	26	39	78
Elements	1	139	26	590	12	108	312	216	1404
Conjugacy classes	1	3	6	10	6	6	36	12	80
Divisions	1	3	5	7	1	1	5	1	24
Autjugacy classes	1	3	2	4	1	1	2	1	15

Dimension	1	2	3	4	6	12	24	72
Irr. complex chars.	4	20	8	24	24	0	0	0	80
Irr. rational chars.	4	8	0	0	4	2	4	2	24

Minimal presentations

Permutation degree:	$22$
Transitive degree:	$117$
Rank:	$3$
Inequivalent generating triples:	$15876$

Minimal degrees of faithful linear representations

	Over $\mathbb{C}$	Over $\mathbb{R}$	Over $\mathbb{Q}$
Irreducible	6	12	72
Arbitrary	5	8	18

Constructions

Presentation:	$\langle a, b, c, d \mid a^{2}=b^{6}=c^{3}=d^{39}=[a,c]=[b,c]=[c,d]=1, b^{a}=b^{5}, d^{a}=cd^{14}, d^{b}=c^{2}d^{25} \rangle$
Permutation group:	Degree $22$ $\langle(11,12)(13,14)(15,16)(17,18)(19,20)(21,22), (2,4)(5,8)(7,9), (2,5,9)(4,7,8) \!\cdots\! \rangle$
Direct product:	$D_{13}$ $\, \times\, $ $(C_3^2:S_3)$
Semidirect product:	$\He_3$ $\,\rtimes\,$ $D_{26}$	$(C_3\times C_{39})$ $\,\rtimes\,$ $D_6$ (4)	$(C_3^2\times D_{13})$ $\,\rtimes\,$ $S_3$ (4)	$C_3^2$ $\,\rtimes\,$ $(S_3\times D_{13})$ (4)	all 9
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Non-split product:	$C_{39}$ . $(C_6:S_3)$	$C_3$ . $(C_{39}:D_6)$	$(C_3\times D_{13})$ . $(C_3:S_3)$		more information

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 1594 subgroups in 110 conjugacy classes, 25 normal (13 characteristic).

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

Special subgroups

Center:	$Z \simeq$ $C_3$	$G/Z \simeq$ $C_{39}:D_6$
Commutator:	$G' \simeq$ $C_{13}\times \He_3$	$G/G' \simeq$ $C_2^2$
Frattini:	$\Phi \simeq$ $C_3$	$G/\Phi \simeq$ $C_{39}:D_6$
Fitting:	$\operatorname{Fit} \simeq$ $C_{13}\times \He_3$	$G/\operatorname{Fit} \simeq$ $C_2^2$
Radical:	$R \simeq$ $\He_3:D_{26}$	$G/R \simeq$ $C_1$
Socle:	$\operatorname{soc} \simeq$ $C_{39}$	$G/\operatorname{soc} \simeq$ $C_6:S_3$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $C_2^2$
3-Sylow subgroup:	$P_{ 3 } \simeq$ $\He_3$
13-Sylow subgroup:	$P_{ 13 } \simeq$ $C_{13}$

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 1404: $\He_3:D_{26}$

Order 702: $C_3^2:D_{39}$, $\He_3:C_{26}$, $D_{13}\times \He_3$

Order 468: $C_{39}:D_6$ x 4

Order 351: $C_{13}\times \He_3$

Order 234: $C_3\times D_{39}$ x 4, $S_3\times C_{39}$ x 4, $C_3^2\times D_{13}$ x 4

Order 156: $S_3\times D_{13}$ x 4, $C_3\times D_{26}$

Order 117: $C_3\times C_{39}$ x 4

Order 108: $C_3^2:D_6$

Order 78: $C_3\times D_{13}$ x 6, $D_{39}$ x 4, $S_3\times C_{13}$ x 4, $C_{78}$

Order 54: $C_3^2:S_3$ x 2, $C_2\times \He_3$

Order 52: $D_{26}$

Order 39: $C_{39}$ x 5

Order 36: $C_6\times S_3$ x 4

Order 27: $\He_3$

Order 26: $D_{13}$ x 2, $C_{26}$

Order 18: $C_3\times S_3$ x 8, $C_3\times C_6$ x 4

Order 13: $C_{13}$

Order 12: $D_6$ x 4, $C_2\times C_6$

Order 9: $C_3^2$ x 4

Order 6: $S_3$ x 8, $C_6$ x 7

Order 4: $C_2^2$

Order 3: $C_3$ x 5

Order 2: $C_2$ x 3

Order 1: $C_1$

Classes of subgroups up to automorphism

Order 1404: $\He_3:D_{26}$

Order 702: $C_3^2:D_{39}$, $\He_3:C_{26}$, $D_{13}\times \He_3$

Order 468: $C_{39}:D_6$

Order 351: $C_{13}\times \He_3$

Order 234: $C_3\times D_{39}$, $S_3\times C_{39}$, $C_3^2\times D_{13}$

Order 156: $C_3\times D_{26}$, $S_3\times D_{13}$

Order 117: $C_3\times C_{39}$

Order 108: $C_3^2:D_6$

Order 78: $C_3\times D_{13}$ x 3, $C_{78}$, $D_{39}$, $S_3\times C_{13}$

Order 54: $C_3^2:S_3$ x 2, $C_2\times \He_3$

Order 52: $D_{26}$

Order 39: $C_{39}$ x 2

Order 36: $C_6\times S_3$

Order 27: $\He_3$

Order 26: $D_{13}$ x 2, $C_{26}$

Order 18: $C_3\times S_3$ x 2, $C_3\times C_6$

Order 13: $C_{13}$

Order 12: $C_2\times C_6$, $D_6$

Order 9: $C_3^2$

Order 6: $C_6$ x 4, $S_3$ x 2

Order 4: $C_2^2$

Order 3: $C_3$ x 2

Order 2: $C_2$ x 3

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 1404: $\He_3:D_{26}$ ($C_1$)

Order 702: $C_3^2:D_{39}$ ($C_2$), $\He_3:C_{26}$ ($C_2$), $D_{13}\times \He_3$ ($C_2$)

Order 351: $C_{13}\times \He_3$ ($C_2^2$)

Order 234: $C_3^2\times D_{13}$ ($S_3$) x 4

Order 117: $C_3\times C_{39}$ ($D_6$) x 4

Order 78: $C_3\times D_{13}$ ($C_3:S_3$)

Order 54: $C_3^2:S_3$ ($D_{13}$)

Order 39: $C_{39}$ ($C_6:S_3$)

Order 27: $\He_3$ ($D_{26}$)

Order 26: $D_{13}$ ($C_3^2:S_3$)

Order 13: $C_{13}$ ($C_3^2:D_6$)

Order 9: $C_3^2$ ($S_3\times D_{13}$) x 4

Order 3: $C_3$ ($C_{39}:D_6$)

Order 1: $C_1$ ($\He_3:D_{26}$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 1404: $\He_3:D_{26}$ ($C_1$)

Order 702: $C_3^2:D_{39}$ ($C_2$), $\He_3:C_{26}$ ($C_2$), $D_{13}\times \He_3$ ($C_2$)

Order 351: $C_{13}\times \He_3$ ($C_2^2$)

Order 234: $C_3^2\times D_{13}$ ($S_3$)

Order 117: $C_3\times C_{39}$ ($D_6$)

Order 78: $C_3\times D_{13}$ ($C_3:S_3$)

Order 54: $C_3^2:S_3$ ($D_{13}$)

Order 39: $C_{39}$ ($C_6:S_3$)

Order 27: $\He_3$ ($D_{26}$)

Order 26: $D_{13}$ ($C_3^2:S_3$)

Order 13: $C_{13}$ ($C_3^2:D_6$)

Order 9: $C_3^2$ ($S_3\times D_{13}$)

Order 3: $C_3$ ($C_{39}:D_6$)

Order 1: $C_1$ ($\He_3:D_{26}$)

Series

Derived series

$\He_3:D_{26}$

$\rhd$

$C_{13}\times \He_3$

$\rhd$

$C_3$

$\rhd$

$C_1$

Chief series

$\He_3:D_{26}$

$\rhd$

$\He_3:C_{26}$

$\rhd$

$C_{13}\times \He_3$

$\rhd$

$C_3\times C_{39}$

$\rhd$

$C_{39}$

$\rhd$

$C_{13}$

$\rhd$

$C_1$

Lower central series

$\He_3:D_{26}$

$\rhd$

$C_{13}\times \He_3$

Upper central series

$C_1$

$\lhd$

$C_3$

Character theory

Complex character table

See the $80 \times 80$ character table. Alternatively, you may search for characters of this group with desired properties.

Rational character table

See the $24 \times 24$ rational character table.