Abstract group 2109375.b: $C_5^7:\He_3$

• Label: 2109375.b
• Order: \( 3^{3} \cdot 5^{7} \)
• Exponent: \( 3 \cdot 5 \)
• Nilpotent: no
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 3^{2} \cdot 5 \)
• $\card{Z(G)}$: \( 5 \)
• $\card{\Aut(G)}$: \( 2^{9} \cdot 3^{4} \cdot 5^{6} \)
• $\card{\mathrm{Out}(G)}$: \( 2^{9} \cdot 3 \)
• Perm deg.: $45$
• Trans deg.: $45$
• Rank: $2$

Group information

Description:	$C_5^7:\He_3$
Order:	\(2109375\)\(\medspace = 3^{3} \cdot 5^{7} \)
Exponent:	\(15\)\(\medspace = 3 \cdot 5 \)
Automorphism group:	$C_5^6.\He_3.Q_8.C_6.C_4^2.C_2$, of order \(648000000\)\(\medspace = 2^{9} \cdot 3^{4} \cdot 5^{6} \)
Composition factors:	$C_3$ x 3, $C_5$ x 7
Derived length:	$3$

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

Group statistics

Order	1	3	5	15
Elements	1	46250	78124	1985000	2109375
Conjugacy classes	1	10	3004	360	3375
Divisions	1	5	751	45	802
Autjugacy classes	1	2	17	4	24

Minimal presentations

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

Minimal degrees of faithful linear representations

	Over $\mathbb{C}$	Over $\mathbb{R}$	Over $\mathbb{Q}$
Irreducible	9	not computed	not computed
Arbitrary	not computed	not computed	not computed

Constructions

Presentation:	${\langle a, b, c, d, e, f, g, h \mid a^{15}=b^{3}=c^{15}=d^{5}=e^{5}=f^{5}= \!\cdots\! \rangle}$
Permutation group:	Degree $45$ $\langle(1,25,30,2,21,26,3,22,27,4,23,28,5,24,29)(6,15,31,9,13,34,7,11,32,10,14,35,8,12,33) \!\cdots\! \rangle$
Transitive group:	45T1989				more information
Direct product:	$C_5$ $\, \times\, $ $(C_5^6:\He_3)$
Semidirect product:	$C_5^7$ $\,\rtimes\,$ $\He_3$				more information
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Possibly split product:	$(C_5^7:C_3^2)$ . $C_3$	$(C_5^7.C_3)$ . $C_3^2$	$(C_5^6:\He_3)$ . $C_5$	$C_5^6$ . $(C_5\times \He_3)$	all 7

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

Homology

Abelianization:	$C_{3} \times C_{15} \simeq C_{3}^{2} \times C_{5}$
Schur multiplier:	not computed
Commutator length:	$1$

Subgroups

There are 16 normal subgroups (8 characteristic).

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

Special subgroups

Center:	$Z \simeq$ $C_5$	$G/Z \simeq$ $C_5^6:\He_3$
Commutator:	$G' \simeq$ $C_5^6:C_3$	$G/G' \simeq$ $C_3\times C_{15}$
Frattini:	$\Phi \simeq$ $C_1$	$G/\Phi \simeq$ $C_5^7:\He_3$
Fitting:	$\operatorname{Fit} \simeq$ $C_5^7$	$G/\operatorname{Fit} \simeq$ $\He_3$
Radical:	$R \simeq$ $C_5^7:\He_3$	$G/R \simeq$ $C_1$
Socle:	$\operatorname{soc} \simeq$ $C_5^7$	$G/\operatorname{soc} \simeq$ $\He_3$
3-Sylow subgroup:	$P_{ 3 } \simeq$ $\He_3$
5-Sylow subgroup:	$P_{ 5 } \simeq$ $C_5^7$

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 75 (order at least 28125) are shown, as well as all normal subgroups of any index.

Order 2109375: $C_5^7:\He_3$

Order 703125: $C_5^7:C_3^2$ x 4

Order 421875: $C_5^6:\He_3$

Order 234375: $C_5^7.C_3$ x 4, $C_5^7.C_3$

Order 140625: $C_5^6:C_3^2$ x 4

Order 78125: $C_5^7$

Order 46875: $C_5^6:C_3$ x 44, $C_5^6:C_3$

Order 28125: $C_5^5:C_3^2$ x 4

Order 15625: $C_5^6$

Order 135: $C_5\times \He_3$

Order 27: $\He_3$

Order 5: $C_5$

Order 1: $C_1$

Classes of subgroups up to automorphism

All subgroups of index up to 75 (order at least 28125) are shown, as well as all normal subgroups of any index.

Order 2109375: $C_5^7:\He_3$

Order 703125: $C_5^7:C_3^2$

Order 421875: $C_5^6:\He_3$

Order 234375: $C_5^7.C_3$, $C_5^7.C_3$

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

Order 78125: $C_5^7$

Order 46875: $C_5^6:C_3$ x 3, $C_5^6:C_3$

Order 28125: $C_5^5:C_3^2$

Order 15625: $C_5^6$

Order 135: $C_5\times \He_3$

Order 27: $\He_3$

Order 5: $C_5$

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 2109375: $C_5^7:\He_3$ ($C_1$)

Order 703125: $C_5^7:C_3^2$ ($C_3$) x 4

Order 421875: $C_5^6:\He_3$ ($C_5$)

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

Order 140625: $C_5^6:C_3^2$ ($C_{15}$) x 4

Order 78125: $C_5^7$ ($\He_3$)

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

Order 15625: $C_5^6$ ($C_5\times \He_3$)

Order 5: $C_5$ ($C_5^6:\He_3$)

Order 1: $C_1$ ($C_5^7:\He_3$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 2109375: $C_5^7:\He_3$ ($C_1$)

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

Order 421875: $C_5^6:\He_3$ ($C_5$)

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

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

Order 78125: $C_5^7$ ($\He_3$)

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

Order 15625: $C_5^6$ ($C_5\times \He_3$)

Order 5: $C_5$ ($C_5^6:\He_3$)

Order 1: $C_1$ ($C_5^7:\He_3$)

Series

Derived series

$C_5^7:\He_3$

$\rhd$

$C_5^6:C_3$

$\rhd$

$C_5^6$

$\rhd$

$C_1$

Chief series

$C_5^7:\He_3$

$\rhd$

$C_5^7:C_3^2$

$\rhd$

$C_5^6:C_3^2$

$\rhd$

$C_5^6:C_3$

$\rhd$

$C_5^6$

$\rhd$

$C_1$

Lower central series

$C_5^7:\He_3$

$\rhd$

$C_5^6:C_3$

$\rhd$

$C_5^6$

Upper central series

$C_1$

$\lhd$

$C_5$

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

Complex character table

The $3375 \times 3375$ character table is not available for this group.

Rational character table

The $802 \times 802$ rational character table is not available for this group.