Abstract group 56862.a: $C_3^7:C_{26}$

• Label: 56862.a
• Order: \( 2 \cdot 3^{7} \cdot 13 \)
• Exponent: \( 2 \cdot 3 \cdot 13 \)
• Nilpotent: no
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2 \cdot 13 \)
• $\card{Z(G)}$: \( 1 \)
• $\card{\Aut(G)}$: \( 2^{4} \cdot 3^{8} \cdot 13^{2} \)
• $\card{\mathrm{Out}(G)}$: \( 2^{3} \cdot 3 \cdot 13 \)
• Perm deg.: $39$
• Trans deg.: $39$
• Rank: $2$

Group information

Description:	$C_3^7:C_{26}$
Order:	\(56862\)\(\medspace = 2 \cdot 3^{7} \cdot 13 \)
Exponent:	\(78\)\(\medspace = 2 \cdot 3 \cdot 13 \)
Automorphism group:	$C_3^6.C_{13}.C_{78}.C_6.C_2^2$, of order \(17740944\)\(\medspace = 2^{4} \cdot 3^{8} \cdot 13^{2} \)
Composition factors:	$C_2$, $C_3$ x 7, $C_{13}$
Derived length:	$2$

This group is nonabelian, metabelian (hence solvable), and an A-group. Whether it is monomial has not been computed.

Group statistics

Order	1	2	3	13	26	39
Elements	1	2187	2186	8748	26244	17496	56862
Conjugacy classes	1	1	85	12	12	12	123
Divisions	1	1	85	1	1	1	90
Autjugacy classes	1	1	5	2	2	2	13

Dimension	1	2	12	24	26
Irr. complex chars.	26	13	0	0	84	123
Irr. rational chars.	2	1	2	1	84	90

Minimal presentations

Permutation degree:	$39$
Transitive degree:	$39$
Rank:	$2$
Inequivalent generating pairs:	not computed

Minimal degrees of faithful linear representations

	Over $\mathbb{C}$	Over $\mathbb{R}$	Over $\mathbb{Q}$
Irreducible	26	26	26
Arbitrary	not computed	not computed	not computed

Constructions

Presentation:	${\langle a, b, c, d, e, f, g, h \mid a^{26}=b^{3}=c^{3}=d^{3}=e^{3}=f^{3}= \!\cdots\! \rangle}$
Permutation group:	Degree $39$ $\langle(1,29,18,6,32,19,7,36,22,12,38,27,14)(2,28,16,5,33,21,8,35,23,11,39,26,15,3,30,17,4,31,20,9,34,24,10,37,25,13) \!\cdots\! \rangle$
Transitive group:	39T82				more information
Direct product:	not isomorphic to a non-trivial direct product
Semidirect product:	not computed
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Possibly split product:	$C_3^4$ . $F_{27}$	$(C_3^6:C_{13})$ . $S_3$	$C_3^6$ . $(S_3\times C_{13})$	$C_3^3$ . $(C_3:F_{27})$	all 5

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

Homology

Abelianization:	$C_{26} \simeq C_{2} \times C_{13}$
Schur multiplier:	$C_{3}^{3}$
Commutator length:	$1$

Subgroups

There are 12 normal subgroups (8 characteristic).

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

Special subgroups

Center:	$Z \simeq$ $C_1$	$G/Z \simeq$ $C_3^7:C_{26}$
Commutator:	$G' \simeq$ $C_3^7$	$G/G' \simeq$ $C_{26}$
Frattini:	$\Phi \simeq$ $C_1$	$G/\Phi \simeq$ $C_3^7:C_{26}$
Fitting:	$\operatorname{Fit} \simeq$ $C_3^7$	$G/\operatorname{Fit} \simeq$ $C_{26}$
Radical:	$R \simeq$ $C_3^7:C_{26}$	$G/R \simeq$ $C_1$
Socle:	$\operatorname{soc} \simeq$ $C_3^7$	$G/\operatorname{soc} \simeq$ $C_{26}$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $C_2$
3-Sylow subgroup:	$P_{ 3 } \simeq$ $C_3^7$
13-Sylow subgroup:	$P_{ 13 } \simeq$ $C_{13}$

Hi

Subgroup diagram and profile

Classes of subgroups up to conjugation

No diagram available: inclusions not computed

Classes of subgroups up to automorphism

No diagram available: inclusions not computed

Normal subgroups

No diagram available: inclusions not computed

Normal subgroups up to automorphism

No diagram available: inclusions not computed

Classes of subgroups up to conjugation

All subgroups of index up to 234 (order at least 243) are shown, as well as all normal subgroups of any index.

Order 56862: $C_3^7:C_{26}$

Order 28431: $C_3^7:C_{13}$

Order 18954: $C_3^3.F_{27}$

Order 9477: $C_3^6:C_{13}$

Order 4374: $C_3^7.C_2$

Order 2187: $C_3^7$

Order 2106: $C_3:F_{27}$ x 2

Order 1458: $C_3^5:S_3$ x 85

Order 1053: $C_3^3:C_{39}$ x 2

Order 729: $C_3^6$ x 85

Order 702: $F_{27}$ x 2

Order 486: $C_3^4:S_3$ x 7651

Order 351: $C_3^3:C_{13}$ x 2

Order 243: $C_3^5$ x 7651

Order 81: $C_3^4$ x 2

Order 27: $C_3^3$ x 2

Order 13: $C_{13}$

Order 3: $C_3$

Order 2: $C_2$

Order 1: $C_1$

Classes of subgroups up to automorphism

All subgroups of index up to 234 (order at least 243) are shown, as well as all normal subgroups of any index.

Order 56862: $C_3^7:C_{26}$

Order 28431: $C_3^7:C_{13}$

Order 18954: $C_3^3.F_{27}$

Order 9477: $C_3^6:C_{13}$

Order 4374: $C_3^7.C_2$

Order 2187: $C_3^7$

Order 2106: $C_3:F_{27}$

Order 1458: $C_3^5:S_3$ x 5

Order 1053: $C_3^3:C_{39}$

Order 729: $C_3^6$ x 5

Order 702: $F_{27}$

Order 486: $C_3^4:S_3$ x 43

Order 351: $C_3^3:C_{13}$

Order 243: $C_3^5$ x 43

Order 81: $C_3^4$

Order 27: $C_3^3$

Order 13: $C_{13}$

Order 3: $C_3$

Order 2: $C_2$

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 56862: $C_3^7:C_{26}$ ($C_1$)

Order 28431: $C_3^7:C_{13}$ ($C_2$)

Order 9477: $C_3^6:C_{13}$ ($S_3$)

Order 4374: $C_3^7.C_2$ ($C_{13}$)

Order 2187: $C_3^7$ ($C_{26}$)

Order 729: $C_3^6$ ($S_3\times C_{13}$)

Order 81: $C_3^4$ ($F_{27}$) x 2

Order 27: $C_3^3$ ($C_3:F_{27}$) x 2

Order 3: $C_3$ ($C_3^3.F_{27}$)

Order 1: $C_1$ ($C_3^7:C_{26}$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 56862: $C_3^7:C_{26}$ ($C_1$)

Order 28431: $C_3^7:C_{13}$ ($C_2$)

Order 9477: $C_3^6:C_{13}$ ($S_3$)

Order 4374: $C_3^7.C_2$ ($C_{13}$)

Order 2187: $C_3^7$ ($C_{26}$)

Order 729: $C_3^6$ ($S_3\times C_{13}$)

Order 81: $C_3^4$ ($F_{27}$)

Order 27: $C_3^3$ ($C_3:F_{27}$)

Order 3: $C_3$ ($C_3^3.F_{27}$)

Order 1: $C_1$ ($C_3^7:C_{26}$)

Series

Derived series

$C_3^7:C_{26}$

$\rhd$

$C_3^7$

$\rhd$

$C_1$

Chief series

$C_3^7:C_{26}$

$\rhd$

$C_3^7:C_{13}$

$\rhd$

$C_3^7$

$\rhd$

$C_3^4$

$\rhd$

$C_3$

$\rhd$

$C_1$

Lower central series

$C_3^7:C_{26}$

$\rhd$

$C_3^7$

Upper central series

$C_1$

Supergroups

This group is a maximal subgroup of 3 larger groups in the database.

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

Character theory

Complex character table

See the $123 \times 123$ character table. Alternatively, you may search for characters of this group with desired properties.

Rational character table

See the $90 \times 90$ rational character table.