Abstract group 624.138: $D_{26}:C_{12}$

• Label: 624.138
• Order: \( 2^{4} \cdot 3 \cdot 13 \)
• Exponent: \( 2^{2} \cdot 3 \cdot 13 \)
• Nilpotent: no
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{4} \cdot 3 \)
• $\card{Z(G)}$: \( 2^{3} \)
• $\card{\Aut(G)}$: \( 2^{7} \cdot 3 \cdot 13 \)
• $\card{\mathrm{Out}(G)}$: \( 2^{6} \)
• Perm deg.: $19$
• Trans deg.: $104$
• Rank: $3$

Group information

Description:	$D_{26}:C_{12}$
Order:	\(624\)\(\medspace = 2^{4} \cdot 3 \cdot 13 \)
Exponent:	\(156\)\(\medspace = 2^{2} \cdot 3 \cdot 13 \)
Automorphism group:	$C_2^2\wr C_2\times F_{13}$, of order \(4992\)\(\medspace = 2^{7} \cdot 3 \cdot 13 \)
Composition factors:	$C_2$ x 4, $C_3$, $C_{13}$
Derived length:	$2$

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

Group statistics

Order	1	2	3	4	6	12	13	26	52
Elements	1	55	26	56	182	208	12	36	48	624
Conjugacy classes	1	7	2	8	14	16	2	6	8	64
Divisions	1	7	1	4	7	4	1	3	2	30
Autjugacy classes	1	3	2	2	6	4	1	2	1	22

Dimension	1	2	4	6	12	24
Irr. complex chars.	48	0	0	16	0	0	64
Irr. rational chars.	8	12	4	0	4	2	30

Minimal presentations

Permutation degree:	$19$
Transitive degree:	$104$
Rank:	$3$
Inequivalent generating triples:	$15288$

Minimal degrees of faithful linear representations

	Over $\mathbb{C}$	Over $\mathbb{R}$	Over $\mathbb{Q}$
Irreducible	none	none	none
Arbitrary	7	8	14

Constructions

Presentation:	$\langle a, b, c \mid a^{4}=b^{6}=c^{26}=[a,b]=[a,c]=1, c^{b}=c^{23} \rangle$
Permutation group:	Degree $19$ $\langle(2,3)(4,6)(5,9)(7,8)(10,12)(11,13)(14,15)(16,17), (14,15)(16,17)(18,19) \!\cdots\! \rangle$
Matrix group:	$\left\langle \left(\begin{array}{rrr} 6 & 5 & 1 \\ 3 & 1 & 10 \\ 5 & 2 & 2 \end{array}\right), \left(\begin{array}{rrr} 2 & 7 & 7 \\ 6 & 3 & 7 \\ 12 & 1 & 10 \end{array}\right), \left(\begin{array}{rrr} 6 & 9 & 4 \\ 10 & 6 & 3 \\ 0 & 12 & 10 \end{array}\right), \left(\begin{array}{rrr} 4 & 10 & 10 \\ 0 & 1 & 0 \\ 3 & 10 & 11 \end{array}\right), \left(\begin{array}{rrr} 0 & 3 & 3 \\ 10 & 6 & 3 \\ 7 & 6 & 9 \end{array}\right), \left(\begin{array}{rrr} 12 & 0 & 0 \\ 0 & 12 & 0 \\ 0 & 0 & 12 \end{array}\right) \right\rangle \subseteq \GL_{3}(\F_{13})$
Direct product:	$C_2$ $\, \times\, $ $C_4$ $\, \times\, $ $(C_{13}:C_6)$
Semidirect product:	$D_{26}$ $\,\rtimes\,$ $C_{12}$ (4)	$(C_4\times D_{26})$ $\,\rtimes\,$ $C_3$	$(C_4\times D_{13})$ $\,\rtimes\,$ $C_6$ (4)	$(C_2\times C_{52})$ $\,\rtimes\,$ $C_6$	all 16
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Non-split product:	$(D_{26}:C_6)$ . $C_2$	$(C_2\times D_{26})$ . $C_6$	$D_{26}$ . $(C_2\times C_6)$ (2)	$C_2$ . $(D_{26}:C_6)$	all 10

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

Homology

Abelianization:	$C_{2}^{2} \times C_{12} \simeq C_{2}^{2} \times C_{4} \times C_{3}$
Schur multiplier:	$C_{2}^{3}$
Commutator length:	$1$

Subgroups

There are 660 subgroups in 108 conjugacy classes, 62 normal (18 characteristic).

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

Special subgroups

Center:	$Z \simeq$ $C_2\times C_4$	$G/Z \simeq$ $C_{13}:C_6$
Commutator:	$G' \simeq$ $C_{13}$	$G/G' \simeq$ $C_2^2\times C_{12}$
Frattini:	$\Phi \simeq$ $C_2$	$G/\Phi \simeq$ $D_{26}:C_6$
Fitting:	$\operatorname{Fit} \simeq$ $C_2\times C_{52}$	$G/\operatorname{Fit} \simeq$ $C_6$
Radical:	$R \simeq$ $D_{26}:C_{12}$	$G/R \simeq$ $C_1$
Socle:	$\operatorname{soc} \simeq$ $C_2\times C_{26}$	$G/\operatorname{soc} \simeq$ $C_2\times C_6$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $C_2^2\times C_4$
3-Sylow subgroup:	$P_{ 3 } \simeq$ $C_3$
13-Sylow subgroup:	$P_{ 13 } \simeq$ $C_{13}$

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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Subgroup information

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Classes of subgroups up to conjugation

Order 624: $D_{26}:C_{12}$

Order 312: $C_{52}:C_6$ x 4, $D_{26}:C_6$, $C_{26}:C_{12}$, $C_{26}:C_{12}$

Order 208: $C_4\times D_{26}$

Order 156: $C_{26}:C_6$ x 6, $C_{13}:C_{12}$ x 2, $C_{13}:C_{12}$ x 2, $C_{26}:C_6$

Order 104: $C_4\times D_{13}$ x 4, $C_2\times C_{52}$, $C_{26}:C_4$, $C_2\times D_{26}$

Order 78: $C_{13}:C_6$ x 4, $C_{13}:C_6$ x 3

Order 52: $D_{26}$ x 6, $C_{52}$ x 2, $C_{13}:C_4$ x 2, $C_2\times C_{26}$

Order 48: $C_2^2\times C_{12}$

Order 39: $C_{13}:C_3$

Order 26: $D_{13}$ x 4, $C_{26}$ x 3

Order 24: $C_2\times C_{12}$ x 6, $C_2^2\times C_6$

Order 16: $C_2^2\times C_4$

Order 13: $C_{13}$

Order 12: $C_2\times C_6$ x 7, $C_{12}$ x 4

Order 8: $C_2\times C_4$ x 6, $C_2^3$

Order 6: $C_6$ x 7

Order 4: $C_2^2$ x 7, $C_4$ x 4

Order 3: $C_3$

Order 2: $C_2$ x 7

Order 1: $C_1$

Classes of subgroups up to automorphism

Order 624: $D_{26}:C_{12}$

Order 312: $C_{52}:C_6$, $D_{26}:C_6$, $C_{26}:C_{12}$, $C_{26}:C_{12}$

Order 208: $C_4\times D_{26}$

Order 156: $C_{26}:C_6$ x 2, $C_{13}:C_{12}$, $C_{26}:C_6$, $C_{13}:C_{12}$

Order 104: $C_2\times C_{52}$, $C_{26}:C_4$, $C_4\times D_{13}$, $C_2\times D_{26}$

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

Order 52: $D_{26}$ x 2, $C_2\times C_{26}$, $C_{52}$, $C_{13}:C_4$

Order 48: $C_2^2\times C_{12}$

Order 39: $C_{13}:C_3$

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

Order 24: $C_2\times C_{12}$ x 3, $C_2^2\times C_6$

Order 16: $C_2^2\times C_4$

Order 13: $C_{13}$

Order 12: $C_2\times C_6$ x 3, $C_{12}$ x 2

Order 8: $C_2\times C_4$ x 3, $C_2^3$

Order 6: $C_6$ x 3

Order 4: $C_2^2$ x 3, $C_4$ x 2

Order 3: $C_3$

Order 2: $C_2$ x 3

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 624: $D_{26}:C_{12}$ ($C_1$)

Order 312: $C_{52}:C_6$ ($C_2$) x 4, $D_{26}:C_6$ ($C_2$), $C_{26}:C_{12}$ ($C_2$), $C_{26}:C_{12}$ ($C_2$)

Order 208: $C_4\times D_{26}$ ($C_3$)

Order 156: $C_{26}:C_6$ ($C_4$) x 4, $C_{26}:C_6$ ($C_2^2$) x 2, $C_{13}:C_{12}$ ($C_2^2$) x 2, $C_{13}:C_{12}$ ($C_2^2$) x 2, $C_{26}:C_6$ ($C_2^2$)

Order 104: $C_4\times D_{13}$ ($C_6$) x 4, $C_2\times C_{52}$ ($C_6$), $C_{26}:C_4$ ($C_6$), $C_2\times D_{26}$ ($C_6$)

Order 78: $C_{13}:C_6$ ($C_2\times C_4$) x 4, $C_{13}:C_6$ ($C_2\times C_4$) x 2, $C_{13}:C_6$ ($C_2^3$)

Order 52: $D_{26}$ ($C_{12}$) x 4, $D_{26}$ ($C_2\times C_6$) x 2, $C_{52}$ ($C_2\times C_6$) x 2, $C_{13}:C_4$ ($C_2\times C_6$) x 2, $C_2\times C_{26}$ ($C_2\times C_6$)

Order 39: $C_{13}:C_3$ ($C_2^2\times C_4$)

Order 26: $D_{13}$ ($C_2\times C_{12}$) x 4, $C_{26}$ ($C_2\times C_{12}$) x 2, $C_{26}$ ($C_2^2\times C_6$)

Order 13: $C_{13}$ ($C_2^2\times C_{12}$)

Order 8: $C_2\times C_4$ ($C_{13}:C_6$)

Order 4: $C_4$ ($C_{26}:C_6$) x 2, $C_2^2$ ($C_{26}:C_6$)

Order 2: $C_2$ ($C_{52}:C_6$) x 2, $C_2$ ($D_{26}:C_6$)

Order 1: $C_1$ ($D_{26}:C_{12}$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 624: $D_{26}:C_{12}$ ($C_1$)

Order 312: $C_{52}:C_6$ ($C_2$), $D_{26}:C_6$ ($C_2$), $C_{26}:C_{12}$ ($C_2$), $C_{26}:C_{12}$ ($C_2$)

Order 208: $C_4\times D_{26}$ ($C_3$)

Order 156: $C_{26}:C_6$ ($C_2^2$), $C_{26}:C_6$ ($C_4$), $C_{13}:C_{12}$ ($C_2^2$), $C_{26}:C_6$ ($C_2^2$), $C_{13}:C_{12}$ ($C_2^2$)

Order 104: $C_2\times C_{52}$ ($C_6$), $C_{26}:C_4$ ($C_6$), $C_4\times D_{13}$ ($C_6$), $C_2\times D_{26}$ ($C_6$)

Order 78: $C_{13}:C_6$ ($C_2^3$), $C_{13}:C_6$ ($C_2\times C_4$), $C_{13}:C_6$ ($C_2\times C_4$)

Order 52: $C_2\times C_{26}$ ($C_2\times C_6$), $D_{26}$ ($C_2\times C_6$), $D_{26}$ ($C_{12}$), $C_{52}$ ($C_2\times C_6$), $C_{13}:C_4$ ($C_2\times C_6$)

Order 39: $C_{13}:C_3$ ($C_2^2\times C_4$)

Order 26: $C_{26}$ ($C_2\times C_{12}$), $C_{26}$ ($C_2^2\times C_6$), $D_{13}$ ($C_2\times C_{12}$)

Order 13: $C_{13}$ ($C_2^2\times C_{12}$)

Order 8: $C_2\times C_4$ ($C_{13}:C_6$)

Order 4: $C_2^2$ ($C_{26}:C_6$), $C_4$ ($C_{26}:C_6$)

Order 2: $C_2$ ($C_{52}:C_6$), $C_2$ ($D_{26}:C_6$)

Order 1: $C_1$ ($D_{26}:C_{12}$)

Series

Derived series

$D_{26}:C_{12}$

$\rhd$

$C_{13}$

$\rhd$

$C_1$

Chief series

$D_{26}:C_{12}$

$\rhd$

$C_{26}:C_{12}$

$\rhd$

$C_{26}:C_6$

$\rhd$

$C_{13}:C_6$

$\rhd$

$C_{26}$

$\rhd$

$C_{13}$

$\rhd$

$C_1$

Lower central series

$D_{26}:C_{12}$

$\rhd$

$C_{13}$

Upper central series

$C_1$

$\lhd$

$C_2\times C_4$

Supergroups

This group is a maximal subgroup of 41 larger groups in the database.

This group is a maximal quotient of 27 larger groups in the database.

Character theory

Complex character table

See the $64 \times 64$ character table. Alternatively, you may search for characters of this group with desired properties.

Rational character table

See the $30 \times 30$ rational character table.