Abstract group 1404.99: $(C_3\times C_{78}):C_6$

• Label: 1404.99
• Order: \( 2^{2} \cdot 3^{3} \cdot 13 \)
• Exponent: \( 2 \cdot 3 \cdot 13 \)
• Nilpotent: no
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{2} \cdot 3 \)
• $\card{Z(G)}$: \( 2 \)
• $\card{\Aut(G)}$: \( 2^{4} \cdot 3^{4} \cdot 13 \)
• $\card{\mathrm{Out}(G)}$: \( 2^{3} \cdot 3 \)
• Perm deg.: $24$
• Trans deg.: $234$
• Rank: $2$

Group information

Description:	$(C_3\times C_{78}):C_6$
Order:	\(1404\)\(\medspace = 2^{2} \cdot 3^{3} \cdot 13 \)
Exponent:	\(78\)\(\medspace = 2 \cdot 3 \cdot 13 \)
Automorphism group:	$C_{39}.(C_6\times C_{12}\times S_3)$, of order \(16848\)\(\medspace = 2^{4} \cdot 3^{4} \cdot 13 \)
Composition factors:	$C_2$ x 2, $C_3$ x 3, $C_{13}$
Derived length:	$2$

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

Group statistics

Order	1	2	3	6	13	26	39	78
Elements	1	235	242	710	12	12	96	96	1404
Conjugacy classes	1	3	6	10	2	2	16	16	56
Divisions	1	3	4	6	1	1	2	2	20
Autjugacy classes	1	2	6	8	1	1	2	2	23

Dimension	1	2	4	6	12	24	72
Irr. complex chars.	12	6	0	38	0	0	0	56
Irr. rational chars.	4	6	2	2	2	2	2	20

Minimal presentations

Permutation degree:	$24$
Transitive degree:	$234$
Rank:	$2$
Inequivalent generating pairs:	$24$

Minimal degrees of faithful linear representations

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

Constructions

Presentation:	$\langle a, b, c \mid a^{6}=b^{6}=c^{39}=[b,c]=1, b^{a}=b^{5}c^{13}, c^{a}=c^{17} \rangle$
Permutation group:	Degree $24$ $\langle(2,4)(3,6)(5,7)(8,9)(13,14)(15,17)(16,20)(18,19)(21,22)(23,24), (10,11) \!\cdots\! \rangle$
Matrix group:	$\left\langle \left(\begin{array}{rr} 64 & 0 \\ 0 & 64 \end{array}\right), \left(\begin{array}{rr} 55 & 21 \\ 0 & 79 \end{array}\right), \left(\begin{array}{rr} 1 & 52 \\ 78 & 79 \end{array}\right), \left(\begin{array}{rr} 25 & 77 \\ 39 & 92 \end{array}\right), \left(\begin{array}{rr} 1 & 9 \\ 0 & 1 \end{array}\right), \left(\begin{array}{rr} 1 & 39 \\ 0 & 1 \end{array}\right) \right\rangle \subseteq \GL_{2}(\Z/117\Z)$
Direct product:	$C_2$ $\, \times\, $ $((C_3\times C_{39}):C_6)$
Semidirect product:	$(C_{39}:C_6)$ $\,\rtimes\,$ $S_3$	$(C_3:D_{78})$ $\,\rtimes\,$ $C_3$	$(C_3:D_{39})$ $\,\rtimes\,$ $C_6$ (2)	$(C_{39}:C_3)$ $\,\rtimes\,$ $D_6$	all 12
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Non-split product:	$C_{78}$ . $(C_3\times S_3)$	$C_{39}$ . $(C_6\times S_3)$	$C_6$ . $(C_{39}:C_6)$	$C_3$ . $(C_{78}:C_6)$	more information

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:	$C_{6}$
Commutator length:	$1$

Subgroups

There are 1858 subgroups in 78 conjugacy classes, 22 normal (18 characteristic).

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

Special subgroups

Center:	$Z \simeq$ $C_2$	$G/Z \simeq$ $(C_3\times C_{39}):C_6$
Commutator:	$G' \simeq$ $C_3\times C_{39}$	$G/G' \simeq$ $C_2\times C_6$
Frattini:	$\Phi \simeq$ $C_3$	$G/\Phi \simeq$ $C_{78}:C_6$
Fitting:	$\operatorname{Fit} \simeq$ $C_3\times C_{78}$	$G/\operatorname{Fit} \simeq$ $C_6$
Radical:	$R \simeq$ $(C_3\times C_{78}):C_6$	$G/R \simeq$ $C_1$
Socle:	$\operatorname{soc} \simeq$ $C_{78}$	$G/\operatorname{soc} \simeq$ $C_3\times S_3$
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

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 1404: $(C_3\times C_{78}):C_6$

Order 702: $(C_3\times C_{39}):C_6$ x 2, $C_{26}:\He_3$

Order 468: $C_3:D_{78}$, $C_{78}:C_6$

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

Order 234: $C_{39}:C_6$ x 2, $C_3:D_{39}$ x 2, $C_{39}:C_6$ x 2, $C_3\times C_{78}$

Order 156: $D_{78}$ x 2, $C_{26}:C_6$

Order 117: $C_{39}:C_3$ x 2, $C_3\times C_{39}$

Order 108: $C_3^2:D_6$

Order 78: $D_{39}$ x 4, $C_{78}$ x 2, $C_{13}:C_6$ x 2, $C_{13}:C_6$ x 2

Order 54: $C_3^2:C_6$ x 2, $C_2\times \He_3$

Order 52: $D_{26}$

Order 39: $C_{39}$ x 2, $C_{13}:C_3$ x 2

Order 36: $C_6:S_3$, $C_6\times S_3$

Order 27: $\He_3$

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

Order 18: $C_3\times C_6$ x 3, $C_3:S_3$ x 2, $C_3\times S_3$ x 2

Order 13: $C_{13}$

Order 12: $D_6$ x 2, $C_2\times C_6$

Order 9: $C_3^2$ x 3

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

Order 4: $C_2^2$

Order 3: $C_3$ x 4

Order 2: $C_2$ x 3

Order 1: $C_1$

Classes of subgroups up to automorphism

Order 1404: $(C_3\times C_{78}):C_6$

Order 702: $C_{26}:\He_3$, $(C_3\times C_{39}):C_6$

Order 468: $C_3:D_{78}$, $C_{78}:C_6$

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

Order 234: $C_{39}:C_6$ x 2, $C_{39}:C_6$, $C_3\times C_{78}$, $C_3:D_{39}$

Order 156: $D_{78}$ x 2, $C_{26}:C_6$

Order 117: $C_{39}:C_3$ x 2, $C_3\times C_{39}$

Order 108: $C_3^2:D_6$

Order 78: $C_{78}$ x 2, $D_{39}$ x 2, $C_{13}:C_6$ x 2, $C_{13}:C_6$

Order 54: $C_3^2:C_6$, $C_2\times \He_3$

Order 52: $D_{26}$

Order 39: $C_{39}$ x 2, $C_{13}:C_3$ x 2

Order 36: $C_6:S_3$, $C_6\times S_3$

Order 27: $\He_3$

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

Order 18: $C_3\times C_6$ x 3, $C_3:S_3$, $C_3\times S_3$

Order 13: $C_{13}$

Order 12: $D_6$ x 2, $C_2\times C_6$

Order 9: $C_3^2$ x 3

Order 6: $C_6$ x 5, $S_3$ x 2

Order 4: $C_2^2$

Order 3: $C_3$ x 4

Order 2: $C_2$ x 2

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 1404: $(C_3\times C_{78}):C_6$ ($C_1$)

Order 702: $(C_3\times C_{39}):C_6$ ($C_2$) x 2, $C_{26}:\He_3$ ($C_2$)

Order 468: $C_3:D_{78}$ ($C_3$)

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

Order 234: $C_3:D_{39}$ ($C_6$) x 2, $C_3\times C_{78}$ ($C_6$), $C_{39}:C_6$ ($S_3$)

Order 117: $C_3\times C_{39}$ ($C_2\times C_6$), $C_{39}:C_3$ ($D_6$)

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

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

Order 26: $C_{26}$ ($C_3^2:C_6$)

Order 18: $C_3\times C_6$ ($C_{13}:C_6$)

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

Order 9: $C_3^2$ ($C_{26}:C_6$)

Order 6: $C_6$ ($C_{39}:C_6$)

Order 3: $C_3$ ($C_{78}:C_6$)

Order 2: $C_2$ ($(C_3\times C_{39}):C_6$)

Order 1: $C_1$ ($(C_3\times C_{78}):C_6$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 1404: $(C_3\times C_{78}):C_6$ ($C_1$)

Order 702: $C_{26}:\He_3$ ($C_2$), $(C_3\times C_{39}):C_6$ ($C_2$)

Order 468: $C_3:D_{78}$ ($C_3$)

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

Order 234: $C_3\times C_{78}$ ($C_6$), $C_3:D_{39}$ ($C_6$), $C_{39}:C_6$ ($S_3$)

Order 117: $C_3\times C_{39}$ ($C_2\times C_6$), $C_{39}:C_3$ ($D_6$)

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

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

Order 26: $C_{26}$ ($C_3^2:C_6$)

Order 18: $C_3\times C_6$ ($C_{13}:C_6$)

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

Order 9: $C_3^2$ ($C_{26}:C_6$)

Order 6: $C_6$ ($C_{39}:C_6$)

Order 3: $C_3$ ($C_{78}:C_6$)

Order 2: $C_2$ ($(C_3\times C_{39}):C_6$)

Order 1: $C_1$ ($(C_3\times C_{78}):C_6$)

Series

Derived series

$(C_3\times C_{78}):C_6$

$\rhd$

$C_3\times C_{39}$

$\rhd$

$C_1$

Chief series

$(C_3\times C_{78}):C_6$

$\rhd$

$C_{26}:\He_3$

$\rhd$

$C_3\times C_{78}$

$\rhd$

$C_3\times C_{39}$

$\rhd$

$C_{39}$

$\rhd$

$C_{13}$

$\rhd$

$C_1$

Lower central series

$(C_3\times C_{78}):C_6$

$\rhd$

$C_3\times C_{39}$

Upper central series

$C_1$

$\lhd$

$C_2$

Character theory

Complex character table

See the $56 \times 56$ character table. Alternatively, you may search for characters of this group with desired properties.

Rational character table

See the $20 \times 20$ rational character table.