Abstract group 105468750.f: $C_5^6.C_{15}^2:D_{15}$

• Label: 105468750.f
• Order: \( 2 \cdot 3^{3} \cdot 5^{9} \)
• Exponent: \( 2 \cdot 3 \cdot 5 \)
• Nilpotent: no
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2 \cdot 3 \)
• $\card{Z(G)}$: 1
• $\card{\Aut(G)}$: \( 2^{8} \cdot 3^{3} \cdot 5^{9} \)
• $\card{\mathrm{Out}(G)}$: \( 2^{7} \)
• Perm deg.: $45$
• Trans deg.: $45$
• Rank: $2$

Group information

Description:	$C_5^6.C_{15}^2:D_{15}$
Order:	\(105468750\)\(\medspace = 2 \cdot 3^{3} \cdot 5^{9} \)
Exponent:	\(30\)\(\medspace = 2 \cdot 3 \cdot 5 \)
Automorphism group:	Group of order \(13500000000\)\(\medspace = 2^{8} \cdot 3^{3} \cdot 5^{9} \)
Composition factors:	$C_2$, $C_3$ x 3, $C_5$ x 9
Derived length:	$3$

This group is nonabelian and solvable. Whether it is monomial or rational has not been computed.

Group statistics

Statistics about orders of elements in this group have not been computed.

Minimal presentations

Permutation degree:	$45$
Transitive degree:	$45$
Rank:	$2$
Inequivalent generating pairs:	not computed

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

Constructions

Presentation:	${\langle a, b, c, d, e, f, g, h, i, j \mid b^{15}=c^{15}=d^{5}=e^{5}=f^{5}= \!\cdots\! \rangle}$
Permutation group:	Degree $45$ $\langle(1,2)(3,5)(6,44,9,41,7,43,10,45,8,42)(11,33,24,40,18,26,13,35,21,38,16,28,15,32,23,36,19,30,12,34,25,39,17,27,14,31,22,37,20,29) \!\cdots\! \rangle$
Transitive group:	45T3794				more information
Direct product:	not computed
Semidirect product:	not computed
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Possibly split product:	$(C_5^9.C_3^2)$ . $S_3$	$(C_5^8.\He_3)$ . $D_5$	$C_5^7$ . $(C_{15}^2:S_3)$	$(C_5^8.C_3^2)$ . $D_{15}$	all 18

Elements of the group are displayed as permutations of degree 45.

Homology

Abelianization:	$C_{6} \simeq C_{2} \times C_{3}$
Schur multiplier:	not computed
Commutator length:	not computed

Subgroups

There are 22 normal subgroups, and all normal subgroups are characteristic.

Characteristic subgroups are shown in this color.

Special subgroups

Center:	a subgroup isomorphic to $C_1$
Commutator:	a subgroup isomorphic to $C_5^2.C_5^7:C_3^2$
Frattini:	a subgroup isomorphic to $C_1$
Fitting:	not computed
Radical:	not computed
Socle:	not computed
2-Sylow subgroup:	$P_{ 2 } \simeq$ $C_2$
3-Sylow subgroup:	$P_{ 3 } \simeq$ $\He_3$
5-Sylow subgroup:	$P_{ 5 } \simeq$ $C_5^9$

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 250 (order at least 421875) are shown, as well as all normal subgroups of any index.

Order 105468750: $C_5^6.C_{15}^2:D_{15}$

Order 52734375: $C_5^8.C_{15}.C_3^2$

Order 35156250: $C_5^8.C_{15}.C_6$, $C_5^9.C_3.S_3$

Order 21093750: $C_5^6.C_{15}^2:S_3$

Order 17578125: $C_5^2.C_5^7:C_3^2$ x 2, $C_5^9.C_3^2$

Order 11718750: $C_5^8.C_{15}.C_2$ x 2, $C_5^9.C_6$

Order 10546875: $C_5^8.\He_3$

Order 7031250: $C_5^7.C_{15}.C_6$ x 6, $C_5^8.C_3.C_6$, $C_5^8.C_3.S_3$

Order 5859375: $C_5^9.C_3$ x 4

Order 4218750: $C_5^7:\He_3:C_2$

Order 3906250: $C_5^9.C_2$

Order 3515625: $C_5^8.C_3^2$ x 9, $C_5^8.C_3^2$ x 2

Order 2343750: $C_5^8.C_6$ x 37, $C_5^7.C_{15}.C_2$ x 12, $C_5^8.S_3$ x 2

Order 2109375: $C_5^7:\He_3$

Order 1953125: $C_5^9$

Order 1406250: $C_5^7:(C_3\times S_3)$ x 6, $C_5^7:C_3^2:C_2$ x 2, $C_5^6.C_{15}.C_6$, $C_5^7:(C_3\times S_3)$

Order 1171875: $C_5^8.C_3$ x 188

Order 843750: $C_5^6:\He_3:C_2$

Order 781250: $C_5^8.C_2$ x 337

Order 703125: $C_5^7:C_3^2$ x 13, $C_5^7.C_3^2$

Order 468750: $C_5^7.C_6$ x 220, $C_5^6.C_{15}.C_2$ x 44, $C_5^7.S_3$ x 12

Order 421875: $C_5^6:\He_3$

Order 390625: $C_5^8$

Order 234375: $C_5^7.C_3$

Order 140625: $C_5^6:C_3^2$

Order 78125: $C_5^7$

Order 46875: $C_5^6:C_3$

Order 15625: $C_5^6$

Order 6750: $C_{15}^2:D_{15}$

Order 125: $C_5^3$

Order 27: $\He_3$

Order 25: $C_5^2$

Order 5: $C_5$

Order 2: $C_2$

Order 1: $C_1$

Classes of subgroups up to automorphism

not computed

Normal subgroups (quotient in parentheses)

Order 105468750: $C_5^6.C_{15}^2:D_{15}$ ($C_1$)

Order 52734375: $C_5^8.C_{15}.C_3^2$ ($C_2$)

Order 35156250: $C_5^9.C_3.S_3$ ($C_3$)

Order 17578125: $C_5^9.C_3^2$ ($S_3$), $C_5^2.C_5^7:C_3^2$ ($C_6$)

Order 10546875: $C_5^8.\He_3$ ($D_5$)

Order 5859375: $C_5^9.C_3$ ($C_3\times S_3$)

Order 3515625: $C_5^8.C_3^2$ ($D_{15}$), $C_5^8.C_3^2$ ($C_3\times D_5$)

Order 1953125: $C_5^9$ ($C_3^2:C_6$)

Order 1171875: $C_5^8.C_3$ ($C_3\times D_{15}$)

Order 703125: $C_5^7:C_3^2$ ($C_5^2:S_3$)

Order 390625: $C_5^8$ ($C_3^2:D_{15}$)

Order 234375: $C_5^7.C_3$ ($C_3\times C_5^2:S_3$)

Order 140625: $C_5^6:C_3^2$ ($C_5^2:D_{15}$)

Order 78125: $C_5^7$ ($C_{15}^2:S_3$)

Order 46875: $C_5^6:C_3$ ($C_3\times C_5^2:D_{15}$)

Order 15625: $C_5^6$ ($C_{15}^2:D_{15}$)

Order 125: $C_5^3$ ($C_5^6:\He_3:C_2$)

Order 25: $C_5^2$ ($C_5^7:\He_3:C_2$)

Order 5: $C_5$ ($C_5^6.C_{15}^2:S_3$)

Order 1: $C_1$ ($C_5^6.C_{15}^2:D_{15}$)

Normal subgroups up to automorphism (quotient in parentheses)

not computed

Series

Derived series	not computed
Chief series	not computed
Lower central series	not computed
Upper central series	not computed

Supergroups

This group is a maximal subgroup of 6 larger groups in the database.

This group is a maximal quotient of 0 larger groups in the database.

Character theory

The character tables for this group have not been computed.