Abstract group 27744.ba: $C_{136}.D_{102}$

• Label: 27744.ba
• Order: \( 2^{5} \cdot 3 \cdot 17^{2} \)
• Exponent: \( 2^{3} \cdot 3 \cdot 17 \)
• Nilpotent: no
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{4} \cdot 17 \)
• $\card{Z(G)}$: \( 2^{3} \cdot 17 \)
• $\card{\Aut(G)}$: \( 2^{14} \cdot 3 \cdot 17 \)
• $\card{\mathrm{Out}(G)}$: \( 2^{12} \)
• Perm deg.: $53$
• Trans deg.: $816$
• Rank: $3$

Group information

Description:	$C_{136}.D_{102}$
Order:	\(27744\)\(\medspace = 2^{5} \cdot 3 \cdot 17^{2} \)
Exponent:	\(408\)\(\medspace = 2^{3} \cdot 3 \cdot 17 \)
Automorphism group:	$C_{51}:((C_2\times C_{16}^2).C_2^5)$, of order \(835584\)\(\medspace = 2^{14} \cdot 3 \cdot 17 \)
Composition factors:	$C_2$ x 5, $C_3$, $C_{17}$ x 2
Derived length:	$2$

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

Group statistics

Order	1	2	3	4	6	8	12	17	24	34	51	68	102	136	204	408
Elements	1	207	2	208	6	416	8	288	16	4128	576	4416	1728	8832	2304	4608	27744
Conjugacy classes	1	4	1	5	3	10	4	152	8	472	288	624	864	1248	1152	2304	7140
Divisions	1	4	1	4	2	4	2	10	2	30	10	22	28	22	20	20	182
Autjugacy classes	1	3	1	3	2	3	2	3	2	7	3	7	6	7	6	6	62

Minimal presentations

Permutation degree:	$53$
Transitive degree:	$816$
Rank:	$3$
Inequivalent generating triples:	not computed

Minimal degrees of linear representations for this group have not been computed

Constructions

Presentation:	$\langle a, b, c \mid a^{2}=b^{102}=c^{136}=[b,c]=1, b^{a}=b^{53}c^{52}, c^{a}=b^{48}c^{49} \rangle$
Permutation group:	Degree $53$ $\langle(2,4)(3,7)(11,13)(12,14)(35,36)(37,38)(39,40)(41,42)(43,44)(45,46)(47,48) \!\cdots\! \rangle$
Matrix group:	$\left\langle \left(\begin{array}{rr} 21 & 0 \\ 0 & 236 \end{array}\right), \left(\begin{array}{rr} 32 & 0 \\ 0 & 294 \end{array}\right), \left(\begin{array}{rr} 0 & 1 \\ 1 & 0 \end{array}\right) \right\rangle \subseteq \GL_{2}(\F_{409})$
Direct product:	not computed
Semidirect product:	not isomorphic to a non-trivial semidirect product
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Possibly split product:	$C_{408}$ . $D_{34}$ (2)	$D_{204}$ . $C_{68}$	$C_{136}$ . $D_{102}$ (2)	$C_{68}$ . $(S_3\times C_{68})$	all 83

Elements of the group are displayed as matrices in $\GL_{2}(\F_{409})$.

Homology

Abelianization:	$C_{2}^{2} \times C_{68} \simeq C_{2}^{2} \times C_{4} \times C_{17}$
Schur multiplier:	$C_{2}^{2}$
Commutator length:	not computed

Subgroups

There are 3752 subgroups in 424 conjugacy classes, 110 normal (82 characteristic).

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

Special subgroups

Center:	$Z \simeq$ $C_{136}$	$G/Z \simeq$ $D_{102}$
Commutator:	$G' \simeq$ $C_{102}$	$G/G' \simeq$ $C_2^2\times C_{68}$
Frattini:	$\Phi \simeq$ $C_4$	$G/\Phi \simeq$ $C_{17}^2:(C_2^2\times S_3)$
Fitting:	$\operatorname{Fit} \simeq$ $C_{34}\times C_{408}$	$G/\operatorname{Fit} \simeq$ $C_2$
Radical:	$R \simeq$ $C_{136}.D_{102}$	$G/R \simeq$ $C_1$
Socle:	$\operatorname{soc} \simeq$ $C_{17}\times C_{102}$	$G/\operatorname{soc} \simeq$ $C_2^2\times C_4$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $\OD_{16}:C_2$
3-Sylow subgroup:	$P_{ 3 } \simeq$ $C_3$
17-Sylow subgroup:	$P_{ 17 } \simeq$ $C_{17}^2$

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

Order 27744: $C_{136}.D_{102}$

Order 13872: $D_{51}\times C_{136}$ x 2, $C_{408}:C_{34}$ x 2, $C_{204}.C_{68}$, $C_{68}.D_{102}$, $C_{34}\times C_{408}$

Order 9248: $C_{136}.D_{34}$

Order 6936: $C_{51}:C_{136}$ x 2, $C_{68}\times D_{51}$ x 2, $C_{17}\times C_{408}$ x 2, $D_{102}:C_{34}$ x 2, $C_{34}\times C_{204}$, $C_{204}.C_{34}$, $C_{17}\times D_{204}$

Order 4624: $C_{136}:C_{34}$ x 2, $D_{17}\times C_{136}$ x 2, $C_{68}.C_{68}$, $D_{68}:C_{34}$, $C_{34}\times C_{136}$

Order 3468: $C_{17}\times C_{204}$ x 2, $C_{51}:C_{68}$ x 2, $C_{17}\times D_{102}$ x 2, $C_{34}\times C_{102}$

Order 2312: $D_{17}\times C_{68}$ x 2, $C_{17}:C_{136}$ x 2, $C_{17}\times C_{136}$ x 2, $C_{34}\wr C_2$ x 2, $C_{17}\times D_{68}$, $C_{34}\times C_{68}$, $C_{17}^2:Q_8$

Order 1734: $C_{17}\times D_{51}$ x 2, $C_{17}\times C_{102}$ x 2

Order 1632: $C_8.D_{102}$, $D_{12}.C_{68}$

Order 1156: $C_{17}\times C_{68}$ x 2, $C_{17}:C_{68}$ x 2, $C_{17}\times D_{34}$ x 2, $C_{34}^2$

Order 867: $C_{17}\times C_{51}$

Order 816: $C_2\times C_{408}$ x 10, $C_{408}:C_2$ x 2, $C_8\times D_{51}$ x 2, $C_{24}:C_{34}$ x 2, $S_3\times C_{136}$ x 2, $C_{204}.C_4$, $C_{12}.C_{68}$, $D_{204}:C_2$, $D_{12}:C_{34}$

Order 578: $C_{17}\times C_{34}$ x 2, $C_{17}\times D_{17}$ x 2

Order 544: $\OD_{16}:C_{34}$, $C_8.D_{34}$

Order 408: $C_{408}$ x 20, $C_2\times C_{204}$ x 10, $C_{51}:C_8$ x 2, $C_{51}:D_4$ x 2, $C_4\times D_{51}$ x 2, $D_6:C_{34}$ x 2, $S_3\times C_{68}$ x 2, $C_3:C_{136}$ x 2, $D_{204}$, $C_{51}:Q_8$, $C_{17}\times D_{12}$, $C_{51}:Q_8$

Order 289: $C_{17}^2$

Order 272: $C_2\times C_{136}$ x 12, $\OD_{16}\times C_{17}$ x 3, $C_{136}:C_2$ x 2, $C_8\times D_{17}$ x 2, $D_4:C_{34}$, $D_{68}:C_2$, $C_{17}:\OD_{16}$

Order 204: $C_{204}$ x 20, $C_2\times C_{102}$ x 10, $C_{51}:C_4$ x 2, $D_{102}$ x 2, $S_3\times C_{34}$ x 2, $C_3:C_{68}$ x 2

Order 136: $C_{136}$ x 22, $C_2\times C_{68}$ x 12, $D_4\times C_{17}$ x 3, $C_{17}:D_4$ x 2, $C_4\times D_{17}$ x 2, $C_{17}:C_8$ x 2, $D_{68}$, $C_{17}:Q_8$, $Q_8\times C_{17}$

Order 102: $C_{102}$ x 28, $D_{51}$ x 2, $S_3\times C_{17}$ x 2

Order 96: $C_8.D_6$

Order 68: $C_{68}$ x 22, $C_2\times C_{34}$ x 12, $D_{34}$ x 2, $C_{17}:C_4$ x 2

Order 51: $C_{51}$ x 10

Order 48: $C_{24}:C_2$ x 2, $S_3\times C_8$ x 2, $D_{12}:C_2$, $C_2\times C_{24}$, $C_3:\OD_{16}$

Order 34: $C_{34}$ x 30, $D_{17}$ x 2

Order 32: $\OD_{16}:C_2$

Order 24: $C_3:D_4$ x 2, $C_4\times S_3$ x 2, $C_{24}$ x 2, $C_3:C_8$ x 2, $C_2\times C_{12}$, $D_{12}$, $C_3:Q_8$

Order 17: $C_{17}$ x 10

Order 16: $\OD_{16}$ x 3, $C_2\times C_8$ x 3, $D_4:C_2$

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

Order 8: $C_8$ x 4, $D_4$ x 3, $C_2\times C_4$ x 3, $Q_8$

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

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

Order 3: $C_3$

Order 2: $C_2$ x 4

Order 1: $C_1$

Classes of subgroups up to automorphism

Order 27744: $C_{136}.D_{102}$

Order 13872: $D_{51}\times C_{136}$

Order 9248: $C_{136}.D_{34}$

Order 867: $C_{17}\times C_{51}$

Order 289: $C_{17}^2$

Order 96: $C_8.D_6$

Order 32: $\OD_{16}:C_2$

Order 3: $C_3$

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 27744: $C_{136}.D_{102}$ ($C_1$)

Order 13872: $D_{51}\times C_{136}$ ($C_2$) x 2, $C_{408}:C_{34}$ ($C_2$) x 2, $C_{204}.C_{68}$ ($C_2$), $C_{68}.D_{102}$ ($C_2$), $C_{34}\times C_{408}$ ($C_2$)

Order 6936: $C_{51}:C_{136}$ ($C_2^2$) x 2, $C_{68}\times D_{51}$ ($C_2^2$) x 2, $C_{17}\times C_{408}$ ($C_2^2$) x 2, $D_{102}:C_{34}$ ($C_4$) x 2, $C_{34}\times C_{204}$ ($C_2^2$), $C_{204}.C_{34}$ ($C_4$), $C_{17}\times D_{204}$ ($C_4$)

Order 4624: $C_{34}\times C_{136}$ ($S_3$)

Order 3468: $C_{51}:C_{68}$ ($C_2\times C_4$) x 2, $C_{17}\times D_{102}$ ($C_2\times C_4$) x 2, $C_{17}\times C_{204}$ ($C_2^3$), $C_{17}\times C_{204}$ ($C_2\times C_4$), $C_{34}\times C_{102}$ ($C_2\times C_4$)

Order 2312: $C_{17}\times C_{136}$ ($D_6$) x 2, $C_{34}\times C_{68}$ ($D_6$)

Order 1734: $C_{17}\times C_{102}$ ($C_2^2\times C_4$)

Order 1632: $C_8.D_{102}$ ($C_{17}$)

Order 1156: $C_{17}\times C_{68}$ ($C_4\times S_3$), $C_{17}\times C_{68}$ ($C_2\times D_6$), $C_{34}^2$ ($C_4\times S_3$)

Order 867: $C_{17}\times C_{51}$ ($\OD_{16}:C_2$)

Order 816: $C_{408}:C_2$ ($C_{34}$) x 2, $C_8\times D_{51}$ ($C_{34}$) x 2, $C_2\times C_{408}$ ($C_{34}$), $C_2\times C_{408}$ ($D_{17}$), $C_{204}.C_4$ ($C_{34}$), $D_{204}:C_2$ ($C_{34}$)

Order 578: $C_{17}\times C_{34}$ ($C_4\times D_6$)

Order 408: $C_{408}$ ($C_2\times C_{34}$) x 2, $C_{408}$ ($D_{34}$) x 2, $C_{51}:C_8$ ($C_2\times C_{34}$) x 2, $C_{51}:D_4$ ($C_{68}$) x 2, $C_4\times D_{51}$ ($C_2\times C_{34}$) x 2, $C_2\times C_{204}$ ($C_2\times C_{34}$), $C_2\times C_{204}$ ($D_{34}$), $D_{204}$ ($C_{68}$), $C_{51}:Q_8$ ($C_{68}$)

Order 289: $C_{17}^2$ ($C_8.D_6$)

Order 272: $C_2\times C_{136}$ ($D_{51}$), $C_2\times C_{136}$ ($S_3\times C_{17}$)

Order 204: $C_{51}:C_4$ ($C_2\times C_{68}$) x 2, $D_{102}$ ($C_2\times C_{68}$) x 2, $C_{204}$ ($C_2\times C_{68}$), $C_{204}$ ($C_4\times D_{17}$), $C_{204}$ ($C_2^2\times C_{34}$), $C_{204}$ ($C_2\times D_{34}$), $C_2\times C_{102}$ ($C_2\times C_{68}$), $C_2\times C_{102}$ ($C_4\times D_{17}$)

Order 136: $C_{136}$ ($D_{102}$) x 2, $C_{136}$ ($S_3\times C_{34}$) x 2, $C_2\times C_{68}$ ($D_{102}$), $C_2\times C_{68}$ ($S_3\times C_{34}$)

Order 102: $C_{102}$ ($C_2^2\times C_{68}$), $C_{102}$ ($C_4\times D_{34}$)

Order 68: $C_2\times C_{34}$ ($C_4\times D_{51}$), $C_2\times C_{34}$ ($S_3\times C_{68}$), $C_{68}$ ($C_2\times D_{102}$), $C_{68}$ ($D_6\times C_{34}$), $C_{68}$ ($C_4\times D_{51}$), $C_{68}$ ($S_3\times C_{68}$)

Order 51: $C_{51}$ ($\OD_{16}:C_{34}$), $C_{51}$ ($C_8.D_{34}$)

Order 48: $C_2\times C_{24}$ ($C_{17}\times D_{17}$)

Order 34: $C_{34}$ ($C_4\times D_{102}$), $C_{34}$ ($D_6\times C_{68}$)

Order 24: $C_{24}$ ($C_{17}\times D_{34}$) x 2, $C_2\times C_{12}$ ($C_{17}\times D_{34}$)

Order 17: $C_{17}$ ($C_8.D_{102}$), $C_{17}$ ($D_{12}.C_{68}$)

Order 16: $C_2\times C_8$ ($C_{17}\times D_{51}$)

Order 12: $C_2\times C_6$ ($D_{17}\times C_{68}$), $C_{12}$ ($C_{17}^2:C_2^3$), $C_{12}$ ($D_{17}\times C_{68}$)

Order 8: $C_8$ ($C_{17}\times D_{102}$) x 2, $C_2\times C_4$ ($C_{17}\times D_{102}$)

Order 6: $C_6$ ($C_{17}^2:(C_2^2\times C_4)$)

Order 4: $C_2^2$ ($C_{68}\times D_{51}$), $C_4$ ($C_{17}^2:(C_2^2\times S_3)$), $C_4$ ($C_{68}\times D_{51}$)

Order 3: $C_3$ ($C_{136}.D_{34}$)

Order 2: $C_2$ ($C_{17}^2:(C_2\times C_4\times S_3)$)

Order 1: $C_1$ ($C_{136}.D_{102}$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 27744: $C_{136}.D_{102}$ ($C_1$)

Order 13872: $D_{51}\times C_{136}$ ($C_2$)

Order 867: $C_{17}\times C_{51}$ ($\OD_{16}:C_2$)

Order 289: $C_{17}^2$ ($C_8.D_6$)

Order 3: $C_3$ ($C_{136}.D_{34}$)

Order 1: $C_1$ ($C_{136}.D_{102}$)

Series

Derived series

$C_{136}.D_{102}$

$\rhd$

$C_{102}$

$\rhd$

$C_1$

Chief series

$C_{136}.D_{102}$

$\rhd$

$C_{34}\times C_{408}$

$\rhd$

$C_{34}\times C_{204}$

$\rhd$

$C_{17}\times C_{204}$

$\rhd$

$C_{17}\times C_{102}$

$\rhd$

$C_{102}$

$\rhd$

$C_{51}$

$\rhd$

$C_{17}$

$\rhd$

$C_1$

Lower central series

$C_{136}.D_{102}$

$\rhd$

$C_{102}$

$\rhd$

$C_{51}$

Upper central series

$C_1$

$\lhd$

$C_{136}$

$\lhd$

$C_2\times C_{136}$

Supergroups

This group is a maximal subgroup of 2 larger groups in the database.

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

Character theory

Complex character table

The $7140 \times 7140$ character table is not available for this group.

Rational character table

The $182 \times 182$ rational character table is not available for this group.