Abstract group 1392.160: $D_6\times C_{116}$

• Label: 1392.160
• Order: \( 2^{4} \cdot 3 \cdot 29 \)
• Exponent: \( 2^{2} \cdot 3 \cdot 29 \)
• Nilpotent: no
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{4} \cdot 29 \)
• $\card{Z(G)}$: \( 2^{3} \cdot 29 \)
• $\card{\Aut(G)}$: \( 2^{8} \cdot 3 \cdot 7 \)
• $\card{\mathrm{Out}(G)}$: \( 2^{7} \cdot 7 \)
• Perm deg.: $38$
• Trans deg.: $696$
• Rank: $3$

Group information

Description:	$D_6\times C_{116}$
Order:	\(1392\)\(\medspace = 2^{4} \cdot 3 \cdot 29 \)
Exponent:	\(348\)\(\medspace = 2^{2} \cdot 3 \cdot 29 \)
Automorphism group:	$C_{21}:(C_2^4.C_2^4)$, of order \(5376\)\(\medspace = 2^{8} \cdot 3 \cdot 7 \)
Composition factors:	$C_2$ x 4, $C_3$, $C_{29}$
Derived length:	$2$

This group is nonabelian, supersolvable (hence solvable and monomial), hyperelementary for $p = 2$, metabelian, and an A-group.

Group statistics

Order	1	2	3	4	6	12	29	58	87	116	174	348
Elements	1	15	2	16	6	8	28	420	56	448	168	224	1392
Conjugacy classes	1	7	1	8	3	4	28	196	28	224	84	112	696
Divisions	1	7	1	4	3	2	1	7	1	4	3	2	36
Autjugacy classes	1	3	1	2	2	1	1	3	1	2	2	1	20

Dimension	1	2	4	28	56	112
Irr. complex chars.	464	232	0	0	0	0	696
Irr. rational chars.	8	8	2	8	8	2	36

Minimal presentations

Permutation degree:	$38$
Transitive degree:	$696$
Rank:	$3$
Inequivalent generating triples:	$146328$

Minimal degrees of faithful linear representations

	Over $\mathbb{C}$	Over $\mathbb{R}$	Over $\mathbb{Q}$
Irreducible	none	not computed	none
Arbitrary	not computed	not computed	not computed

Constructions

Presentation:	$\langle a, b, c \mid a^{2}=b^{4}=c^{174}=[a,b]=[b,c]=1, c^{a}=c^{59} \rangle$
Permutation group:	Degree $38$ $\langle(1,2)(3,4)(8,9), (5,6), (1,3,2,4), (10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38), (1,2)(3,4), (7,8,9)\rangle$
Matrix group:	$\left\langle \left(\begin{array}{rr} 46 & 0 \\ 0 & 46 \end{array}\right), \left(\begin{array}{rr} 1 & 3 \\ 0 & 1 \end{array}\right), \left(\begin{array}{rr} 28 & 0 \\ 0 & 28 \end{array}\right), \left(\begin{array}{rr} 46 & 29 \\ 0 & 17 \end{array}\right), \left(\begin{array}{rr} 1 & 29 \\ 0 & 1 \end{array}\right), \left(\begin{array}{rr} 41 & 0 \\ 0 & 41 \end{array}\right) \right\rangle \subseteq \GL_{2}(\Z/87\Z)$
Direct product:	$C_2$ $\, \times\, $ $C_4$ $\, \times\, $ $C_{29}$ $\, \times\, $ $S_3$
Semidirect product:	$C_{348}$ $\,\rtimes\,$ $C_2^2$ (2)	$(C_6:C_4)$ $\,\rtimes\,$ $C_{58}$	$(C_2\times C_{348})$ $\,\rtimes\,$ $C_2$	$(C_6:C_{116})$ $\,\rtimes\,$ $C_2$	all 12
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Non-split product:	$C_{174}$ . $C_2^3$	$(D_6\times C_{58})$ . $C_2$	$(C_2\times C_{58})$ . $D_6$	$C_{58}$ . $(C_2\times D_6)$	all 12
Aut. group:	$\Aut(C_2\times C_{590})$	$\Aut(S_3\times C_{295})$

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

Homology

Abelianization:	$C_{2}^{2} \times C_{116} \simeq C_{2}^{2} \times C_{4} \times C_{29}$
Schur multiplier:	$C_{2}^{3}$
Commutator length:	$1$

Subgroups

There are 184 subgroups in 108 conjugacy classes, 70 normal (22 characteristic).

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

Special subgroups

Center:	$Z \simeq$ $C_2\times C_{116}$	$G/Z \simeq$ $S_3$
Commutator:	$G' \simeq$ $C_3$	$G/G' \simeq$ $C_2^2\times C_{116}$
Frattini:	$\Phi \simeq$ $C_2$	$G/\Phi \simeq$ $D_6\times C_{58}$
Fitting:	$\operatorname{Fit} \simeq$ $C_2\times C_{348}$	$G/\operatorname{Fit} \simeq$ $C_2$
Radical:	$R \simeq$ $D_6\times C_{116}$	$G/R \simeq$ $C_1$
Socle:	$\operatorname{soc} \simeq$ $C_2\times C_{174}$	$G/\operatorname{soc} \simeq$ $C_2^2$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $C_2^2\times C_4$
3-Sylow subgroup:	$P_{ 3 } \simeq$ $C_3$
29-Sylow subgroup:	$P_{ 29 } \simeq$ $C_{29}$

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 1392: $D_6\times C_{116}$

Order 696: $S_3\times C_{116}$ x 4, $D_6\times C_{58}$, $C_2\times C_{348}$, $C_6:C_{116}$

Order 464: $C_2^2\times C_{116}$

Order 348: $S_3\times C_{58}$ x 6, $C_{348}$ x 2, $C_3:C_{116}$ x 2, $C_2\times C_{174}$

Order 232: $C_2\times C_{116}$ x 6, $C_2^2\times C_{58}$

Order 174: $S_3\times C_{29}$ x 4, $C_{174}$ x 3

Order 116: $C_2\times C_{58}$ x 7, $C_{116}$ x 4

Order 87: $C_{87}$

Order 58: $C_{58}$ x 7

Order 48: $C_4\times D_6$

Order 29: $C_{29}$

Order 24: $C_4\times S_3$ x 4, $C_2\times C_{12}$, $C_6:C_4$, $C_2\times D_6$

Order 16: $C_2^2\times C_4$

Order 12: $D_6$ x 6, $C_{12}$ x 2, $C_3:C_4$ x 2, $C_2\times C_6$

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

Order 6: $S_3$ x 4, $C_6$ x 3

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 1392: $D_6\times C_{116}$

Order 696: $D_6\times C_{58}$, $C_2\times C_{348}$, $C_6:C_{116}$, $S_3\times C_{116}$

Order 464: $C_2^2\times C_{116}$

Order 348: $S_3\times C_{58}$ x 2, $C_{348}$, $C_2\times C_{174}$, $C_3:C_{116}$

Order 232: $C_2\times C_{116}$ x 3, $C_2^2\times C_{58}$

Order 174: $C_{174}$ x 2, $S_3\times C_{29}$

Order 116: $C_2\times C_{58}$ x 3, $C_{116}$ x 2

Order 87: $C_{87}$

Order 58: $C_{58}$ x 3

Order 48: $C_4\times D_6$

Order 29: $C_{29}$

Order 24: $C_2\times C_{12}$, $C_6:C_4$, $C_4\times S_3$, $C_2\times D_6$

Order 16: $C_2^2\times C_4$

Order 12: $D_6$ x 2, $C_2\times C_6$, $C_{12}$, $C_3:C_4$

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

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

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 1392: $D_6\times C_{116}$ ($C_1$)

Order 696: $S_3\times C_{116}$ ($C_2$) x 4, $D_6\times C_{58}$ ($C_2$), $C_2\times C_{348}$ ($C_2$), $C_6:C_{116}$ ($C_2$)

Order 348: $S_3\times C_{58}$ ($C_4$) x 4, $C_{348}$ ($C_2^2$) x 2, $S_3\times C_{58}$ ($C_2^2$) x 2, $C_3:C_{116}$ ($C_2^2$) x 2, $C_2\times C_{174}$ ($C_2^2$)

Order 232: $C_2\times C_{116}$ ($S_3$)

Order 174: $S_3\times C_{29}$ ($C_2\times C_4$) x 4, $C_{174}$ ($C_2\times C_4$) x 2, $C_{174}$ ($C_2^3$)

Order 116: $C_{116}$ ($D_6$) x 2, $C_2\times C_{58}$ ($D_6$)

Order 87: $C_{87}$ ($C_2^2\times C_4$)

Order 58: $C_{58}$ ($C_4\times S_3$) x 2, $C_{58}$ ($C_2\times D_6$)

Order 48: $C_4\times D_6$ ($C_{29}$)

Order 29: $C_{29}$ ($C_4\times D_6$)

Order 24: $C_4\times S_3$ ($C_{58}$) x 4, $C_2\times C_{12}$ ($C_{58}$), $C_6:C_4$ ($C_{58}$), $C_2\times D_6$ ($C_{58}$)

Order 12: $D_6$ ($C_{116}$) x 4, $D_6$ ($C_2\times C_{58}$) x 2, $C_{12}$ ($C_2\times C_{58}$) x 2, $C_3:C_4$ ($C_2\times C_{58}$) x 2, $C_2\times C_6$ ($C_2\times C_{58}$)

Order 8: $C_2\times C_4$ ($S_3\times C_{29}$)

Order 6: $S_3$ ($C_2\times C_{116}$) x 4, $C_6$ ($C_2\times C_{116}$) x 2, $C_6$ ($C_2^2\times C_{58}$)

Order 4: $C_4$ ($S_3\times C_{58}$) x 2, $C_2^2$ ($S_3\times C_{58}$)

Order 3: $C_3$ ($C_2^2\times C_{116}$)

Order 2: $C_2$ ($S_3\times C_{116}$) x 2, $C_2$ ($D_6\times C_{58}$)

Order 1: $C_1$ ($D_6\times C_{116}$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 1392: $D_6\times C_{116}$ ($C_1$)

Order 696: $D_6\times C_{58}$ ($C_2$), $C_2\times C_{348}$ ($C_2$), $C_6:C_{116}$ ($C_2$), $S_3\times C_{116}$ ($C_2$)

Order 348: $C_{348}$ ($C_2^2$), $C_2\times C_{174}$ ($C_2^2$), $S_3\times C_{58}$ ($C_2^2$), $S_3\times C_{58}$ ($C_4$), $C_3:C_{116}$ ($C_2^2$)

Order 232: $C_2\times C_{116}$ ($S_3$)

Order 174: $C_{174}$ ($C_2^3$), $C_{174}$ ($C_2\times C_4$), $S_3\times C_{29}$ ($C_2\times C_4$)

Order 116: $C_2\times C_{58}$ ($D_6$), $C_{116}$ ($D_6$)

Order 87: $C_{87}$ ($C_2^2\times C_4$)

Order 58: $C_{58}$ ($C_4\times S_3$), $C_{58}$ ($C_2\times D_6$)

Order 48: $C_4\times D_6$ ($C_{29}$)

Order 29: $C_{29}$ ($C_4\times D_6$)

Order 24: $C_2\times C_{12}$ ($C_{58}$), $C_6:C_4$ ($C_{58}$), $C_4\times S_3$ ($C_{58}$), $C_2\times D_6$ ($C_{58}$)

Order 12: $C_2\times C_6$ ($C_2\times C_{58}$), $D_6$ ($C_2\times C_{58}$), $D_6$ ($C_{116}$), $C_{12}$ ($C_2\times C_{58}$), $C_3:C_4$ ($C_2\times C_{58}$)

Order 8: $C_2\times C_4$ ($S_3\times C_{29}$)

Order 6: $C_6$ ($C_2\times C_{116}$), $C_6$ ($C_2^2\times C_{58}$), $S_3$ ($C_2\times C_{116}$)

Order 4: $C_2^2$ ($S_3\times C_{58}$), $C_4$ ($S_3\times C_{58}$)

Order 3: $C_3$ ($C_2^2\times C_{116}$)

Order 2: $C_2$ ($D_6\times C_{58}$), $C_2$ ($S_3\times C_{116}$)

Order 1: $C_1$ ($D_6\times C_{116}$)

Series

Derived series

$D_6\times C_{116}$

$\rhd$

$C_3$

$\rhd$

$C_1$

Chief series

$D_6\times C_{116}$

$\rhd$

$S_3\times C_{116}$

$\rhd$

$S_3\times C_{58}$

$\rhd$

$C_{174}$

$\rhd$

$C_{87}$

$\rhd$

$C_{29}$

$\rhd$

$C_1$

Lower central series

$D_6\times C_{116}$

$\rhd$

$C_3$

Upper central series

$C_1$

$\lhd$

$C_2\times C_{116}$

Character theory

Complex character table

The $696 \times 696$ character table is not available for this group.

Rational character table

See the $36 \times 36$ rational character table.