Abstract group 5062500.c: $C_5^6:(C_3^3:D_6)$

• Label: 5062500.c
• Order: \( 2^{2} \cdot 3^{4} \cdot 5^{6} \)
• Exponent: \( 2 \cdot 3^{2} \cdot 5 \)
• Nilpotent: no
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{2} \cdot 3 \)
• $\card{Z(G)}$: \( 1 \)
• $\card{\Aut(G)}$: \( 2^{5} \cdot 3^{4} \cdot 5^{6} \)
• $\card{\mathrm{Out}(G)}$: \( 2^{3} \)
• Perm deg.: $45$
• Trans deg.: $45$
• Rank: $2$

Group information

Description:	$C_5^6:(C_3^3:D_6)$
Order:	\(5062500\)\(\medspace = 2^{2} \cdot 3^{4} \cdot 5^{6} \)
Exponent:	\(90\)\(\medspace = 2 \cdot 3^{2} \cdot 5 \)
Automorphism group:	$C_5^6.\He_3.C_{12}.C_2^3$, of order \(40500000\)\(\medspace = 2^{5} \cdot 3^{4} \cdot 5^{6} \)
Composition factors:	$C_2$ x 2, $C_3$ x 4, $C_5$ x 6
Derived length:	$4$

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

Group statistics

Order	1	2	3	5	6	9	10	15	18	30
Elements	1	21475	143900	15624	2127500	562500	275400	543600	562500	810000	5062500
Conjugacy classes	1	3	10	84	26	2	44	64	2	24	260
Divisions	1	3	6	42	14	1	22	18	1	6	114
Autjugacy classes	1	3	6	16	14	1	9	9	1	3	63

Dimension	1	2	3	4	6	18	36	54	72	108	144	162	216	324	432	648
Irr. complex chars.	12	6	24	0	2	24	12	56	0	40	0	56	0	28	0	0	260
Irr. rational chars.	4	6	0	2	14	0	4	0	6	12	2	0	16	28	6	14	114

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	18	18	36
Arbitrary	not computed	not computed	not computed

Constructions

Presentation:	${\langle a, b, c, d, e, f, g, h, i \mid a^{6}=b^{6}=d^{15}=e^{5}=f^{5}=g^{5}= \!\cdots\! \rangle}$
Permutation group:	Degree $45$ $\langle(1,43,7,5,44,6)(2,42,8,4,45,10)(3,41,9)(11,26,25,36,17,31)(12,30,21,40,18,35) \!\cdots\! \rangle$
Transitive group:	45T2343				more information
Direct product:	not isomorphic to a non-trivial direct product
Semidirect product:	$C_5^6$ $\,\rtimes\,$ $(C_3^3:D_6)$	$(C_5^6:C_3\wr C_3)$ $\,\rtimes\,$ $C_2^2$			more information
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Possibly split product:	$(C_5^6.C_3^3)$ . $D_6$	$(C_5^6:C_3\wr S_3)$ . $C_2$	$(C_5^6:C_3\wr S_3)$ . $C_2$	$(C_5^6.C_3^3.C_2)$ . $S_3$	all 15

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

Homology

Abelianization:	$C_{2} \times C_{6} \simeq C_{2}^{2} \times C_{3}$
Schur multiplier:	not computed
Commutator length:	$1$

Subgroups

There are 19 normal subgroups, and all normal subgroups are characteristic.

Characteristic subgroups are shown in this color.

Special subgroups

Center:	$Z \simeq$ $C_1$	$G/Z \simeq$ $C_5^6:(C_3^3:D_6)$
Commutator:	$G' \simeq$ $C_5^6:\He_3$	$G/G' \simeq$ $C_2\times C_6$
Frattini:	$\Phi \simeq$ $C_1$	$G/\Phi \simeq$ $C_5^6:(C_3^3:D_6)$
Fitting:	$\operatorname{Fit} \simeq$ $C_5^6$	$G/\operatorname{Fit} \simeq$ $C_3^3:D_6$
Radical:	$R \simeq$ $C_5^6:(C_3^3:D_6)$	$G/R \simeq$ $C_1$
Socle:	$\operatorname{soc} \simeq$ $C_5^6$	$G/\operatorname{soc} \simeq$ $C_3^3:D_6$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $C_2^2$
3-Sylow subgroup:	$P_{ 3 } \simeq$ $C_3\wr C_3$
5-Sylow subgroup:	$P_{ 5 } \simeq$ $C_5^6$

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 3750 (order at least 1350) are shown, as well as all normal subgroups of any index.

Order 5062500: $C_5^6:(C_3^3:D_6)$

Order 2531250: $C_5^6:C_3\wr S_3$, $C_5^6:C_3\wr S_3$, $C_5^6:(C_3^3:C_6)$

Order 1687500: $C_5^6.C_3.C_6^2$, $C_5^6:\He_3:C_2^2$

Order 1265625: $C_5^6:C_3\wr C_3$

Order 843750: $C_5^6.C_3^3.C_2$, $C_5^6.C_3^3.C_2$, $C_5^6.C_3^3.C_2$, $C_5^6:\He_3:C_2$, $C_5^6:\He_3:C_2$, $C_5^6:(C_2\times \He_3)$, $C_5^6:(C_9:C_6)$

Order 562500: $C_5^6:(C_6\times S_3)$ x 3, $C_5^6.(C_6\times S_3)$, $C_5^6.(C_6\times S_3)$, $C_5^6.C_6^2$

Order 421875: $C_5^6.C_3^3$, $C_5^6:\He_3$, $C_5^6.C_9:C_3$

Order 281250: $C_5^6:(C_3\times S_3)$ x 3, $C_5^6:(C_3\times S_3)$ x 3, $C_5^6:(C_3\times C_6)$ x 3, $C_5^6.C_3.C_6$ x 2, $C_5^6.C_3.C_6$, $C_5^6.C_3.C_6$, $C_5^6.C_3.C_6$, $C_5^6.C_3.C_6$, $C_5\times C_5^4.C_{15}.C_6$, $C_5^6.C_3.C_6$, $C_5^6.C_3.C_6$, $C_5^6:C_{18}$

Order 187500: $C_5^6.C_6.C_2$ x 2, $C_5^6.D_6$ x 2, $C_5^6.C_6.C_2$, $C_5^6.C_6.C_2$

Order 140625: $C_5^6:C_3^2$ x 3, $C_5^6.C_3^2$ x 2, $C_5\times C_5^5:C_3^2$, $C_5^6:C_9$

Order 112500: $C_5^5:(C_6\times S_3)$ x 2

Order 93750: $C_5^6.C_6$ x 3, $C_5^6.C_6$ x 2, $C_5^6.C_6$ x 2, $C_5^6.S_3$ x 2, $C_5^6.C_6$ x 2, $C_5\times C_5^5:C_6$, $C_5^6.C_6$, $C_5^6.C_6$, $C_5^6.C_6$, $C_5^6.C_6$, $C_5^4.C_{15}.C_{10}$, $C_5\times C_5^5:S_3$

Order 67500: $C_5^4.C_3.C_6^2$

Order 62500: $C_5^5.D_{10}$

Order 56250: $C_5^5:(C_3\times C_6)$ x 2, $C_5^4:(S_3\times C_{15})$ x 2, $C_5^4:(C_3\times D_{15})$ x 2

Order 46875: $C_5^6:C_3$ x 2, $C_5^6:C_3$ x 2, $C_5^2\times C_5^4:C_3$, $C_5\times C_5^5:C_3$

Order 37500: $C_5^5:D_6$ x 4, $C_5^5.C_6.C_2$ x 2, $C_5^5.C_6.C_2$ x 2, $C_5^5:(C_2\times C_6)$ x 2

Order 33750: $C_5^4.C_3^3.C_2$ x 2, $C_5^4.C_3^3.C_2$

Order 31250: $C_5^6:C_2$, $C_5\times C_5^5:C_2$, $C_5^6:C_2$

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

Order 22500: $C_5^4:(C_6\times S_3)$ x 4, $C_5^4:(C_6\times S_3)$ x 2, $C_5^4.C_6^2$, $C_5^4.C_6^2$, $C_5^4.C_6^2$, $C_3\times C_5^4:D_6$

Order 18750: $C_5^5:C_6$ x 12, $C_5^5:C_6$ x 6, $C_5^4:D_{15}$ x 4, $C_5^5:S_3$ x 4, $C_5^5:C_6$ x 2, $C_5^5:C_6$ x 2, $C_5^5:C_6$ x 2, $C_5^5:C_6$ x 2, $C_5^4:C_{30}$ x 2, $C_5^5:C_6$ x 2

Order 16875: $C_5^4:C_3^3$

Order 15625: $C_5^6$

Order 12500: $C_5^5:C_2^2$ x 20, $C_5^5:C_2^2$ x 2

Order 11250: $C_5^4:(C_3\times C_6)$ x 18, $C_5^4:(C_3\times S_3)$ x 8, $C_5^4:(C_3\times C_6)$ x 2, $C_5^4:(C_3\times C_6)$ x 2, $C_5^4:(C_3\times C_6)$ x 2, $C_5^4:(C_3\times S_3)$ x 2, $C_5^4:(C_3\times S_3)$ x 2, $C_5^4:(C_3\times S_3)$, $C_3\times C_5^4:S_3$, $C_5^4:(C_3\times C_6)$, $C_2\times C_5^4:C_3^2$

Order 9375: $C_5^5:C_3$ x 12, $C_5^5:C_3$ x 6

Order 7500: $C_5^4:(C_2\times C_6)$ x 21, $C_5^3:C_6\times D_5$ x 12, $C_5^4:D_6$ x 4, $C_5^4:(C_2\times C_6)$ x 2, $C_5^4:(C_2\times C_6)$ x 2, $C_5^4:(C_2\times C_6)$ x 2, $C_5^4:D_6$ x 2, $C_3\times C_5^4:C_2^2$

Order 6250: $C_5^5:C_2$ x 42, $C_5^5:C_2$ x 20, $C_5^5:C_2$ x 20, $C_5^4\times D_5$ x 2, $C_5^5:C_2$ x 2

Order 5625: $C_5^4:C_3^2$ x 17, $C_5^4:C_3^2$ x 2, $C_{15}\times C_5\times C_5^2:C_3$

Order 3750: $C_5^4:C_6$ x 78, $C_5^4:C_6$ x 77, $C_5^4:C_6$ x 42, $C_5^3:C_{30}$ x 12, $C_5^3:C_{30}$ x 12, $C_5^4:S_3$ x 4, $C_5^4:S_3$ x 4, $C_5^4:S_3$ x 2, $C_5\times C_5^3:S_3$ x 2, $C_2\times C_5^4:C_3$ x 2, $C_3\times C_5^4:C_2$ x 2, $C_5^3:C_{30}$ x 2, $C_5^4:C_6$, $C_5\times C_5^3:C_6$, $C_5^4:C_6$, $C_5:D_5\times C_5^2:C_3$, $C_3\times C_5^4:C_2$

Order 3125: $C_5^5$ x 42

Order 2700: $C_5^2:(C_3\times C_6\times S_3)$

Order 2500: $C_5^4:C_2^2$ x 112, $C_5^4:C_2^2$ x 102, $C_5^4:C_2^2$

Order 2250: $C_3\times C_5^3:C_6$ x 2

Order 1875: $C_5^3:C_{15}$ x 76, $C_5^4:C_3$ x 75, $C_5^3\times C_{15}$

Order 1500: $C_5^3:D_6$ x 16, $C_{15}:D_5^2$ x 4, $C_2\times C_5^3:C_6$ x 4, $C_5^2:C_6\times D_5$ x 4

Order 1350: $C_{15}^2:C_6$, $C_{15}^2:C_6$, $C_{15}^2:C_6$

Order 324: $C_3^3:D_6$

Order 81: $C_3\wr C_3$

Order 4: $C_2^2$

Order 1: $C_1$

Classes of subgroups up to automorphism

All subgroups of index up to 3750 (order at least 1350) are shown, as well as all normal subgroups of any index.

Order 5062500: $C_5^6:(C_3^3:D_6)$

Order 2531250: $C_5^6:C_3\wr S_3$, $C_5^6:C_3\wr S_3$, $C_5^6:(C_3^3:C_6)$

Order 1687500: $C_5^6.C_3.C_6^2$, $C_5^6:\He_3:C_2^2$

Order 1265625: $C_5^6:C_3\wr C_3$

Order 843750: $C_5^6.C_3^3.C_2$, $C_5^6.C_3^3.C_2$, $C_5^6.C_3^3.C_2$, $C_5^6:\He_3:C_2$, $C_5^6:\He_3:C_2$, $C_5^6:(C_2\times \He_3)$, $C_5^6:(C_9:C_6)$

Order 562500: $C_5^6:(C_6\times S_3)$ x 3, $C_5^6.(C_6\times S_3)$, $C_5^6.(C_6\times S_3)$, $C_5^6.C_6^2$

Order 421875: $C_5^6.C_3^3$, $C_5^6:\He_3$, $C_5^6.C_9:C_3$

Order 281250: $C_5^6:(C_3\times S_3)$ x 3, $C_5^6:(C_3\times S_3)$ x 3, $C_5^6:(C_3\times C_6)$ x 3, $C_5^6.C_3.C_6$ x 2, $C_5^6.C_3.C_6$, $C_5^6.C_3.C_6$, $C_5^6.C_3.C_6$, $C_5^6.C_3.C_6$, $C_5\times C_5^4.C_{15}.C_6$, $C_5^6.C_3.C_6$, $C_5^6.C_3.C_6$, $C_5^6:C_{18}$

Order 187500: $C_5^6.C_6.C_2$ x 2, $C_5^6.D_6$ x 2, $C_5^6.C_6.C_2$, $C_5^6.C_6.C_2$

Order 140625: $C_5^6:C_3^2$ x 3, $C_5^6.C_3^2$ x 2, $C_5\times C_5^5:C_3^2$, $C_5^6:C_9$

Order 112500: $C_5^5:(C_6\times S_3)$

Order 93750: $C_5^6.C_6$ x 3, $C_5^6.C_6$ x 2, $C_5^6.C_6$ x 2, $C_5^6.S_3$ x 2, $C_5^6.C_6$ x 2, $C_5\times C_5^5:C_6$, $C_5^6.C_6$, $C_5^6.C_6$, $C_5^6.C_6$, $C_5^6.C_6$, $C_5^4.C_{15}.C_{10}$, $C_5\times C_5^5:S_3$

Order 67500: $C_5^4.C_3.C_6^2$

Order 62500: $C_5^5.D_{10}$

Order 56250: $C_5^5:(C_3\times C_6)$, $C_5^4:(S_3\times C_{15})$, $C_5^4:(C_3\times D_{15})$

Order 46875: $C_5^6:C_3$ x 2, $C_5^6:C_3$ x 2, $C_5^2\times C_5^4:C_3$, $C_5\times C_5^5:C_3$

Order 37500: $C_5^5:D_6$ x 2, $C_5^5.C_6.C_2$, $C_5^5.C_6.C_2$, $C_5^5:(C_2\times C_6)$

Order 33750: $C_5^4.C_3^3.C_2$ x 2, $C_5^4.C_3^3.C_2$

Order 31250: $C_5^6:C_2$, $C_5\times C_5^5:C_2$, $C_5^6:C_2$

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

Order 22500: $C_5^4:(C_6\times S_3)$ x 4, $C_5^4.C_6^2$, $C_5^4.C_6^2$, $C_5^4.C_6^2$, $C_3\times C_5^4:D_6$, $C_5^4:(C_6\times S_3)$

Order 18750: $C_5^5:C_6$ x 5, $C_5^5:C_6$ x 3, $C_5^4:D_{15}$ x 2, $C_5^5:S_3$ x 2, $C_5^5:C_6$, $C_5^5:C_6$, $C_5^5:C_6$, $C_5^5:C_6$, $C_5^4:C_{30}$, $C_5^5:C_6$

Order 16875: $C_5^4:C_3^3$

Order 15625: $C_5^6$

Order 12500: $C_5^5:C_2^2$ x 8, $C_5^5:C_2^2$

Order 11250: $C_5^4:(C_3\times C_6)$ x 15, $C_5^4:(C_3\times S_3)$ x 8, $C_5^4:(C_3\times C_6)$ x 2, $C_5^4:(C_3\times C_6)$ x 2, $C_5^4:(C_3\times C_6)$ x 2, $C_5^4:(C_3\times S_3)$, $C_3\times C_5^4:S_3$, $C_5^4:(C_3\times C_6)$, $C_2\times C_5^4:C_3^2$, $C_5^4:(C_3\times S_3)$, $C_5^4:(C_3\times S_3)$

Order 9375: $C_5^5:C_3$ x 5, $C_5^5:C_3$ x 3

Order 7500: $C_5^4:(C_2\times C_6)$ x 15, $C_5^3:C_6\times D_5$ x 4, $C_5^4:D_6$ x 2, $C_5^4:(C_2\times C_6)$ x 2, $C_5^4:(C_2\times C_6)$ x 2, $C_5^4:(C_2\times C_6)$ x 2, $C_5^4:D_6$ x 2, $C_3\times C_5^4:C_2^2$

Order 6250: $C_5^5:C_2$ x 16, $C_5^5:C_2$ x 8, $C_5^5:C_2$ x 8, $C_5^4\times D_5$, $C_5^5:C_2$

Order 5625: $C_5^4:C_3^2$ x 14, $C_5^4:C_3^2$ x 2, $C_{15}\times C_5\times C_5^2:C_3$

Order 3750: $C_5^4:C_6$ x 47, $C_5^4:C_6$ x 43, $C_5^4:C_6$ x 30, $C_5^4:S_3$ x 4, $C_5^3:C_{30}$ x 4, $C_5^3:C_{30}$ x 4, $C_5^4:S_3$ x 2, $C_2\times C_5^4:C_3$ x 2, $C_3\times C_5^4:C_2$ x 2, $C_5^3:C_{30}$ x 2, $C_5^4:C_6$, $C_5\times C_5^3:C_6$, $C_5^4:S_3$, $C_5\times C_5^3:S_3$, $C_5^4:C_6$, $C_5:D_5\times C_5^2:C_3$, $C_3\times C_5^4:C_2$

Order 3125: $C_5^5$ x 16

Order 2700: $C_5^2:(C_3\times C_6\times S_3)$

Order 2500: $C_5^4:C_2^2$ x 44, $C_5^4:C_2^2$ x 34, $C_5^4:C_2^2$

Order 2250: $C_3\times C_5^3:C_6$

Order 1875: $C_5^4:C_3$ x 45, $C_5^3:C_{15}$ x 41, $C_5^3\times C_{15}$

Order 1500: $C_5^3:D_6$ x 6, $C_{15}:D_5^2$ x 2, $C_2\times C_5^3:C_6$ x 2, $C_5^2:C_6\times D_5$ x 2

Order 1350: $C_{15}^2:C_6$, $C_{15}^2:C_6$, $C_{15}^2:C_6$

Order 324: $C_3^3:D_6$

Order 81: $C_3\wr C_3$

Order 4: $C_2^2$

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 5062500: $C_5^6:(C_3^3:D_6)$ ($C_1$)

Order 2531250: $C_5^6:C_3\wr S_3$ ($C_2$), $C_5^6:C_3\wr S_3$ ($C_2$), $C_5^6:(C_3^3:C_6)$ ($C_2$)

Order 1687500: $C_5^6:\He_3:C_2^2$ ($C_3$)

Order 1265625: $C_5^6:C_3\wr C_3$ ($C_2^2$)

Order 843750: $C_5^6.C_3^3.C_2$ ($S_3$), $C_5^6:\He_3:C_2$ ($C_6$), $C_5^6:\He_3:C_2$ ($C_6$), $C_5^6:(C_2\times \He_3)$ ($C_6$)

Order 421875: $C_5^6.C_3^3$ ($D_6$), $C_5^6:\He_3$ ($C_2\times C_6$)

Order 281250: $C_5^6:(C_3\times C_6)$ ($C_3\times S_3$)

Order 140625: $C_5^6:C_3^2$ ($C_6\times S_3$)

Order 93750: $C_5^6.C_6$ ($C_3^2:C_6$)

Order 46875: $C_5^6:C_3$ ($C_3^2:D_6$)

Order 31250: $C_5^6:C_2$ ($C_3\wr S_3$)

Order 15625: $C_5^6$ ($C_3^3:D_6$)

Order 1: $C_1$ ($C_5^6:(C_3^3:D_6)$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 5062500: $C_5^6:(C_3^3:D_6)$ ($C_1$)

Order 2531250: $C_5^6:C_3\wr S_3$ ($C_2$), $C_5^6:C_3\wr S_3$ ($C_2$), $C_5^6:(C_3^3:C_6)$ ($C_2$)

Order 1687500: $C_5^6:\He_3:C_2^2$ ($C_3$)

Order 1265625: $C_5^6:C_3\wr C_3$ ($C_2^2$)

Order 843750: $C_5^6.C_3^3.C_2$ ($S_3$), $C_5^6:\He_3:C_2$ ($C_6$), $C_5^6:\He_3:C_2$ ($C_6$), $C_5^6:(C_2\times \He_3)$ ($C_6$)

Order 421875: $C_5^6.C_3^3$ ($D_6$), $C_5^6:\He_3$ ($C_2\times C_6$)

Order 281250: $C_5^6:(C_3\times C_6)$ ($C_3\times S_3$)

Order 140625: $C_5^6:C_3^2$ ($C_6\times S_3$)

Order 93750: $C_5^6.C_6$ ($C_3^2:C_6$)

Order 46875: $C_5^6:C_3$ ($C_3^2:D_6$)

Order 31250: $C_5^6:C_2$ ($C_3\wr S_3$)

Order 15625: $C_5^6$ ($C_3^3:D_6$)

Order 1: $C_1$ ($C_5^6:(C_3^3:D_6)$)

Series

Derived series

$C_5^6:(C_3^3:D_6)$

$\rhd$

$C_5^6:\He_3$

$\rhd$

$C_5^6:C_3$

$\rhd$

$C_5^6$

$\rhd$

$C_1$

Chief series

$C_5^6:(C_3^3:D_6)$

$\rhd$

$C_5^6:(C_3^3:C_6)$

$\rhd$

$C_5^6:(C_2\times \He_3)$

$\rhd$

$C_5^6:\He_3$

$\rhd$

$C_5^6:C_3^2$

$\rhd$

$C_5^6:C_3$

$\rhd$

$C_5^6$

$\rhd$

$C_1$

Lower central series

$C_5^6:(C_3^3:D_6)$

$\rhd$

$C_5^6:\He_3$

Upper central series

$C_1$

Supergroups

This group is a maximal subgroup of 7 larger groups in the database.

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

Character theory

Complex character table

See the $260 \times 260$ character table (warning: may be slow to load). Alternatively, you may search for characters of this group with desired properties.

Rational character table

See the $114 \times 114$ rational character table.