Abstract group 648000.q: $C_5^3.C_{12}^2.S_3^2$

• Label: 648000.q
• Order: \( 2^{6} \cdot 3^{4} \cdot 5^{3} \)
• Exponent: \( 2^{3} \cdot 3^{2} \cdot 5 \)
• Nilpotent: no
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{2} \)
• $\card{Z(G)}$: 1
• $\card{\Aut(G)}$: \( 2^{8} \cdot 3^{5} \cdot 5^{3} \)
• $\card{\mathrm{Out}(G)}$: \( 2^{2} \cdot 3 \)
• Perm deg.: $24$
• Trans deg.: $45$
• Rank: $2$

Group information

Description:	$C_5^3.C_{12}^2.S_3^2$
Order:	\(648000\)\(\medspace = 2^{6} \cdot 3^{4} \cdot 5^{3} \)
Exponent:	\(360\)\(\medspace = 2^{3} \cdot 3^{2} \cdot 5 \)
Automorphism group:	$C_5^3.C_{12}^2.(C_{12}\times S_3^2)$, of order \(7776000\)\(\medspace = 2^{8} \cdot 3^{5} \cdot 5^{3} \)
Composition factors:	$C_2$ x 6, $C_3$ x 4, $C_5$ x 3
Derived length:	$4$

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

Group statistics

Order	1	2	3	4	5	6	8	9	10	12	15	20	24	30	45	60
Elements	1	2847	7226	43200	124	26070	36000	14400	29028	86400	32024	16800	126000	154680	57600	15600	648000
Conjugacy classes	1	5	6	8	6	15	4	1	22	36	42	2	10	48	4	13	223
Divisions	1	5	6	6	3	15	2	1	11	23	24	2	5	25	1	9	139
Autjugacy classes	1	5	4	8	3	9	4	1	11	20	14	2	6	15	1	7	111

Dimension	1	2	3	4	6	12	16	24	32	48	64	96	128	192	256	384
Irr. complex chars.	4	4	12	1	40	33	16	22	16	25	4	42	0	4	0	0	223
Irr. rational chars.	4	4	4	1	24	23	0	22	0	22	4	14	4	4	1	8	139

Minimal presentations

Permutation degree:	$24$
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	24	24	24
Arbitrary	not computed	not computed	not computed

Constructions

Presentation:	${\langle a, b, c, d, e, f \mid c^{12}=d^{60}=e^{5}=f^{15}=[a,f]=[e,f]=1, a^{2}= \!\cdots\! \rangle}$
Permutation group:	Degree $24$ $\langle(2,4)(3,6,11,8)(5,9,12,15,14,13,10,7)(17,19)(18,21)(20,23)(22,24), (1,2,3,5,8,12,6,10,11,14)(4,7,9,13,15)(16,17,18,20,21,24)(19,22,23)\rangle$
Transitive group:	45T1647				more information
Direct product:	not isomorphic to a non-trivial direct product
Semidirect product:	$C_5^3$ $\,\rtimes\,$ $(C_{12}^2.S_3^2)$				more information
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Possibly split product:	$C_{15}^3$ . $(C_4^2:D_6)$	$C_3^3$ . $(C_5:F_5^2:D_6)$	$(C_5:F_5^2:\He_3)$ . $D_6$	$(C_{15}^3.C_4^2.C_6)$ . $C_2$	all 24

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

Homology

Abelianization:	$C_{2}^{2} $
Schur multiplier:	$C_{2}^{2}$
Commutator length:	$1$

Subgroups

There are 3862592 subgroups in 5904 conjugacy classes, 27 normal, 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 $C_5^3.C_{12}^2.C_3^2$
Frattini:	a subgroup isomorphic to $C_3^2$
Fitting:	not computed
Radical:	not computed
Socle:	not computed
5-Sylow subgroup:	$P_{ 5 } \simeq$ $C_5^3$

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 200 (order at least 3240) are shown, as well as all normal subgroups of any index.

Order 648000: $C_5^3.C_{12}^2.S_3^2$

Order 324000: $C_{15}^3.C_4^2.C_6$, $C_5^3.C_6^2.C_3:S_4$, $C_5^3.C_{12}^2.C_3.C_6$

Order 216000: $C_{15}^3.C_4.C_2^3.C_2$, $C_5^3:(C_{12}^2:D_6)$

Order 162000: $C_5^3.C_{12}^2.C_3^2$, $C_5^3.C_6^2.S_3^2$

Order 108000: $C_{15}^3.C_8.C_2^2$, $C_{15}^3.C_4\wr C_2$, $C_{15}^3.C_4.D_4$, $C_{15}^3.C_2.C_2^4$, $C_5^3.(C_6\times C_{12}).C_6.C_2$, $C_3^3:C_2\times C_5^3:C_4^2$, $C_3^2\times C_5^3.D_{12}:C_4$, $C_5^3:(C_6^2.S_4)$, $C_5^3:(C_{12}^2:S_3)$, $C_5^3:(C_{12}^2:S_3)$, $C_5^3:(C_{12}^2:C_6)$

Order 81000: $C_3^3:(C_5^3:S_4)$, $C_3^3:(C_5^3:S_4)$, $C_{15}^3.A_4.C_2$

Order 72000: $C_5^3.(C_6\times C_{12}).C_2^3$ x 4, $C_5^3.C_4\wr C_2\times C_3:S_3$, $C_5:F_5^2:S_3^2$

Order 54000: $C_{15}^3.D_4.C_2$ x 4, $C_{15}^3.C_2^3.C_2$ x 2, $C_{15}^3.\OD_{16}$, $C_{15}^3.C_8.C_2$, $C_{15}^3.C_4^2$, $C_{15}^3.C_4.C_2^2$, $C_5\times C_{15}^2.C_6.C_2^3$, $C_3^3\times C_5^3:C_4^2$, $C_3^3:C_2\times C_5^3:(C_2\times C_4)$, $C_3^2\times C_5^3.D_{12}.C_2$, $C_3^2\times C_5^3.C_3:\OD_{16}$, $(C_5\times C_{15}^2).C_{24}.C_2$, $(C_5\times C_{15}^2).C_6.C_2^3$, $C_5^3:(C_{12}^2.C_3)$, $C_5:F_5^2:\He_3$, $C_{15}:D_5^2.S_3^2$

Order 43200: $C_{15}^2.C_6.C_2^4.C_2$

Order 40500: $C_3^3:(C_5^3:A_4)$, $C_5\times C_{15}^2.S_3^2$

Order 36000: $C_3\times C_5^3.D_{12}:C_4$ x 7, $C_3:S_3\times C_5^3:C_4^2$ x 5, $(C_5\times C_{15}^2).C_4\wr C_2$ x 5, $(C_5\times C_{15}^2).C_4.D_4$ x 5, $C_5^3.C_4:S_3^2.C_2$ x 4, $C_5^3.C_3:(S_3\times \OD_{16})$ x 4, $C_5^3:\OD_{16}\times C_3:S_3$, $C_5^3:(D_4:C_2)\times C_3:S_3$, $C_5^3.C_{12}.C_6.C_2^2$, $C_5^3.C_3:(D_{12}:C_4)$, $C_3^2\times C_5^3.C_4\wr C_2$, $C_5:F_5^2:D_9$, $C_{15}:D_5^2.S_4$, $C_{15}:D_5^2.S_4$, $C_5:F_5^2:C_3\times S_3$

Order 27000: $C_{15}^3.C_4.C_2$ x 3, $C_{15}^3.C_2^2.C_2$ x 3, $C_{15}^3.D_4$ x 2, $C_5\times C_{15}^2.D_6.C_2$ x 2, $C_{15}^3.Q_8$, $C_5^3.(C_{12}\times C_3:S_3)$, $C_5^3.(C_3\times C_{12}.S_3)$, $C_5^3.(C_3\times C_6^2:C_2)$, $C_5\times C_{15}^2.C_6.C_2^2$, $C_5\times C_3^3:C_2\times C_5:F_5$, $C_3^3\times C_5^3:(C_2\times C_4)$, $C_3^3:C_2\times C_5^3:C_2^2$, $C_3^3:C_2\times C_5:(C_5^2:C_4)$, $C_3^2\times C_5^3:(C_4\times S_3)$, $C_3^2\times C_5^3:(C_3:Q_8)$, $C_3^2\times C_5^3:(C_3:D_4)$, $C_3^2\times C_5^3:(C_3:C_8)$, $C_3\times C_{15}\times C_5^2:D_{12}$, $(C_5\times C_{15}^2).C_{24}$, $(C_5\times C_{15}^2).S_4$, $(C_5\times C_{15}^2):S_4$, $(C_5\times C_{15}^2):S_4$, $C_5^3:(C_6^2:C_6)$

Order 24000: $C_5^3.C_4\wr C_2\times S_3$ x 4, $C_5^3.C_6.C_2^4.C_2$, $C_5:F_5^2:D_6$

Order 21600: $C_{15}^2.D_{12}:C_2.C_2$, $C_{15}^2.C_{12}.D_4$, $C_{15}^2.C_{12}.C_2^3$, $C_{15}^2.C_6.C_2^4$, $C_{15}^2.C_4.C_6.C_2^2$, $C_3^3:C_2\times F_5^2$, $C_3^2\times C_5^2.D_{12}:C_4$

Order 20250: $C_{15}\wr C_3:C_2$, $C_{15}\wr S_3$, $C_{15}^3.C_6$

Order 18000: $C_5^3.C_6^2.C_2^2$ x 10, $C_3:S_3\times C_5^3:(C_2\times C_4)$ x 9, $C_5^3.C_3:Q_8:S_3$ x 8, $C_3\times C_5^3.D_{12}.C_2$ x 7, $C_3\times C_5^3.C_3:\OD_{16}$ x 7, $C_3^2\times C_5^3:C_4^2$ x 5, $C_5^3.D_6:S_3.C_2$ x 4, $C_5^3.D_6:D_6$ x 4, $C_5^3.C_3^2:D_4.C_2$ x 4, $C_5^3.C_3:(S_3\times Q_8)$ x 4, $C_5^3.C_3:(C_8\times S_3)$ x 4, $C_5^3.(C_4\times S_3^2)$ x 4, $C_5\times C_5^2.C_4:S_3^2$ x 4, $(C_5\times C_{15}^2).\OD_{16}$ x 4, $C_5^3.C_3:D_{12}.C_2$ x 2, $C_5^3.(C_{12}.D_6)$ x 2, $C_5^3:Q_8\times C_3:S_3$, $C_5^3:D_4\times C_3:S_3$, $C_5^3:C_8\times C_3:S_3$, $C_5^3:(C_2\times C_4)\times C_3:S_3$, $C_5^3.C_{12}.C_6.C_2$, $C_5^3.C_6.C_6.C_2^2$, $C_5^3.C_3^2:\OD_{16}$, $C_5^3.C_3^2:C_8.C_2$, $C_5\times D_5\wr C_2\times C_3:S_3$, $C_3^2\times C_5^3:\OD_{16}$, $C_3^2\times C_5^3:(D_4:C_2)$, $C_5:F_5^2:C_9$, $C_5:F_5^2:C_3^2$, $S_3\times C_5^3:S_4$

Order 14400: $F_5^2:S_3^2$ x 4, $C_5^2:C_4\wr C_2\times C_3:S_3$

Order 13500: $C_5\times C_5^2:(S_3\times C_3:S_3)$ x 2, $C_{15}^3.C_4$, $C_{15}^3.C_2^2$, $C_5\times C_5^2:(C_6\times C_3:S_3)$, $C_5\times C_5^2:(C_3^3:C_4)$, $C_5\times C_3^3:C_2\times C_5:D_5$, $C_3^3\times C_5^3:C_2^2$, $C_3^3\times C_5:(C_5^2:C_4)$, $C_3^2\times C_{15}\times C_5:F_5$, $C_3^2\times C_5^3:(C_3:C_4)$, $C_3\times C_{15}\times D_{15}:D_5$, $(C_5\times C_{15}^2).C_{12}$, $(C_5\times C_{15}^2).A_4$, $(C_5\times C_{15}^2):A_4$, $C_5\times C_{15}^2:D_6$

Order 12000: $S_3\times C_5^3:C_4^2$ x 5, $C_5^3.D_{12}:C_4$ x 5, $C_5^3.C_{12}.D_4$ x 5, $C_3\times C_5^3.C_4\wr C_2$ x 5, $C_5^3:\OD_{16}\times S_3$ x 4, $C_5^3:(D_4:C_2)\times S_3$ x 4, $C_5^3.C_3:C_4\wr C_2$ x 4, $C_5:F_5^2:S_3$ x 2, $C_5^3.D_6.C_2^3$, $C_5^3.C_{12}.C_2^3$, $C_5^2.D_{12}:F_5$, $C_5:F_5^2:C_6$

Order 10800: $C_{15}^2.C_6.C_2^3$ x 5, $C_{15}^2.D_{12}.C_2$ x 2, $C_{15}^2.D_4:S_3$ x 2, $C_{15}^2.C_{24}.C_2$ x 2, $C_{15}^2.C_6.C_4.C_2$ x 2, $S_3\times C_3:S_3\times D_5.D_5$, $C_{15}^2.(C_8\times S_3)$, $C_3^3\times F_5^2$, $C_3^3:C_2\times D_5\times F_5$, $C_3^3:C_2\times D_5:F_5$, $C_3^2\times C_5^2:(D_{12}:C_2)$, $C_3^2\times C_5^2:(C_3:\OD_{16})$

Order 10125: $C_{15}\wr C_3$

Order 9000: $C_5^3.(C_4\times C_3:S_3)$ x 20, $C_3^2\times C_5^3:(C_2\times C_4)$ x 10, $C_5^3.C_3^2:C_4.C_2$ x 9, $C_5\times C_3:S_3\times C_5:F_5$ x 9, $C_3:S_3\times C_5:(C_5^2:C_4)$ x 9, $C_5^3.C_3^2:D_4$ x 8, $C_{15}\times C_5^2:D_{12}$ x 7, $C_3\times C_5^3:(C_4\times S_3)$ x 7, $C_3\times C_5^3:(C_3:Q_8)$ x 7, $C_3\times C_5^3:(C_3:D_4)$ x 7, $C_3\times C_5^3:(C_3:C_8)$ x 7, $C_3:S_3\times C_5^3:C_2^2$ x 5, $C_5^3.C_3^2:Q_8$ x 4, $C_5^3.(S_3\times C_3:C_4)$ x 4, $C_5^3.(C_6\times S_3).C_2$ x 4, $C_5^3.(C_2.S_3^2)$ x 4, $C_5\times C_5^2:(D_6:S_3)$ x 4, $C_5\times C_5^2:(C_2\times S_3^2)$ x 4, $C_5^3.C_6.C_6.C_2$ x 3, $C_5^3.C_6^2.C_2$ x 2, $(C_5\times C_{15}^2).Q_8$ x 2, $C_5^3.C_{12}.C_6$, $C_5^3.C_6^2:C_2$, $C_5^3.C_3^2:C_8$, $C_5^3.C_3:D_{12}$, $C_5^2:(C_5:C_4)\times C_3:S_3$, $C_5\times D_5^2\times C_3:S_3$, $C_5\times C_5^2:(C_6^2:C_2)$, $C_5\times C_5^2:(C_3\times D_{12})$, $C_5\times C_5^2:(C_3:D_{12})$, $C_3^2\times C_5^3:Q_8$, $C_3^2\times C_5^3:D_4$, $C_3^2\times C_5^3:C_8$, $C_3\times C_{15}\times D_5\wr C_2$, $(C_5^2\times C_{15}).S_4$, $(C_5^2\times C_{15}):S_4$, $C_3\times C_5^3:S_4$, $S_3\times C_5^3:A_4$

Order 8640: $C_{15}.(C_6\times C_{12}).C_2^3$

Order 8100: $C_{15}^2.S_3^2$

Order 8000: $C_2\times C_5^3.C_4\wr C_2$

Order 7200: $C_3:S_3\times F_5^2$ x 9, $(C_3\times F_5)\wr C_2$ x 8, $C_{15}^2:C_4\wr C_2$ x 8, $C_{15}^2:(C_2\times \OD_{16})$ x 4, $D_5^2.(S_3\times D_6)$ x 4, $C_{15}^2.C_4\wr C_2$ x 2, $D_5^2.C_4\times C_3:S_3$, $D_5\wr C_2:C_2\times C_3:S_3$, $C_{15}^2.C_4.D_4$, $C_3^2\times C_5^2:C_4\wr C_2$

Order 6750: $C_5^3\times C_3^3:C_2$, $C_5\times C_5^2:(C_3^3:C_2)$, $C_5\times C_5^2:(C_3\times C_3:S_3)$, $C_3^2\times C_{15}\times C_5:D_5$, $C_3\times C_{15}\times C_5\times D_{15}$, $C_5\times C_{15}^2.S_3$, $C_5\times C_{15}^2:S_3$, $C_5\times C_{15}^2:S_3$, $C_{15}^2:C_{30}$

Order 6000: $C_5^3.D_{12}.C_2$ x 11, $C_5^3.D_4:S_3$ x 10, $S_3\times C_5^3:(C_2\times C_4)$ x 9, $C_5^3.C_6.C_4.C_2$ x 5, $C_5^3.C_3:\OD_{16}$ x 5, $C_3\times C_5^3:\OD_{16}$ x 5, $C_3\times C_5^3:C_4^2$ x 5, $C_3\times C_5^3:(D_4:C_2)$ x 5, $C_5^3:Q_8\times S_3$ x 4, $C_5^3:D_4\times S_3$ x 4, $C_5^3:C_8\times S_3$ x 4, $C_5^3:(C_2\times C_4)\times S_3$ x 4, $C_5^3.D_{12}:C_2$ x 4, $C_5^3.C_{24}.C_2$ x 4, $C_5^3.C_6.C_2^2.C_2$ x 4, $C_5\times D_5\wr C_2\times S_3$ x 4, $C_5^3.D_6.C_2^2$ x 2, $C_5^3.C_{24}:C_2$, $C_5^3.C_6.C_2^3$, $C_5^3.(C_8\times S_3)$, $C_5\times C_5^2:(S_3\times D_4)$, $C_2\times C_5^3:S_4$, $C_5:F_5^2:C_3$

Order 5400: $C_{15}^2.C_6.C_2^2$ x 7, $C_{15}^2.C_{12}.C_2$ x 5, $C_{15}^2.D_6.C_2$ x 4, $C_{15}^2.D_{12}$ x 2, $C_3^3:C_2\times D_5.D_5$ x 2, $C_3^2\times C_5^2:(C_4\times S_3)$ x 2, $C_{15}^2.C_{24}$, $C_{15}^2.C_3:Q_8$, $C_{15}^2.C_3:D_4$, $C_5\times S_3\times C_3:S_3\times D_5$, $C_5\times C_3^3:C_2\times F_5$, $C_5:(C_3^3:C_4)\times D_5$, $C_3^3\times D_5\times F_5$, $C_3^3\times D_5:F_5$, $C_3^3:C_2\times D_5^2$, $C_3^3:C_2\times C_5^2:C_4$, $C_3^3:C_2\times C_5:F_5$, $C_3^2\times C_5^2:D_{12}$, $C_3^2\times C_5^2:(C_3:Q_8)$, $C_3^2\times C_5^2:(C_3:D_4)$, $C_3^2\times C_5^2:(C_3:C_8)$, $C_3^2:D_{15}\times F_5$

Order 5184: $C_{12}^2.S_3^2$

Order 4800: $F_5^2:D_6$ x 5

Order 4500: $C_5^3:(C_3^2:C_4)$ x 10, $C_5^3:(C_2\times C_3:S_3)$ x 9, $C_5\times C_5^2:(C_3^2:C_4)$ x 9, $C_5\times C_3:S_3\times C_5:D_5$ x 9, $C_3^2\times C_5:(C_5^2:C_4)$ x 9, $C_3\times C_{15}\times C_5:F_5$ x 9, $C_5\times C_5^2:S_3^2$ x 8, $C_{15}\times D_{15}:D_5$ x 7, $C_3\times C_5^3:(C_3:C_4)$ x 7, $C_3^2\times C_5^3:C_2^2$ x 5, $C_5\times C_5^2:(C_2\times C_3:S_3)$ x 2, $C_5^3:(C_3\times C_3:C_4)$, $C_5\times C_5^2:(C_6\times S_3)$, $C_5\times C_5\times D_5\times C_3:S_3$, $C_3^2\times C_5^2:(C_5:C_4)$, $C_3\times C_{15}\times D_5^2$, $(C_5^2\times C_{15}).A_4$, $C_3\times C_5^3:A_4$, $C_5^3:S_3^2$

Order 4320: $C_{15}.D_4:(C_3^2:C_4)$, $C_{15}.C_6^2.C_2^3$, $C_{15}.(\OD_{16}\times C_3:S_3)$, $C_{15}.(C_3\times D_{12}:C_4)$, $C_4\times C_3^3:C_2\times F_5$, $C_3^2\times (C_{12}\times F_5):C_2$, $(C_3\times C_{15}).C_{12}.D_4$

Order 4050: $(C_3\times C_{15}^2):S_3$, $(C_3\times C_{15}^2):S_3$, $(C_3\times C_{15}^2):C_6$

Order 4000: $C_5:F_5\wr C_2$ x 4, $C_2\times C_5^3:\OD_{16}$, $C_2\times C_5^3:C_4^2$, $C_2\times C_5^3:(D_4:C_2)$

Order 3600: $C_{15}^2.C_4^2$ x 22, $C_3:S_3\times D_5\times F_5$ x 13, $C_3^2\times F_5^2$ x 9, $C_3:S_3\times D_5:F_5$ x 9, $C_{15}^2:\OD_{16}$ x 8, $C_3\times D_5^2.D_6$ x 8, $C_{15}^2:D_4:C_2$ x 8, $C_{15}^2.D_4.C_2$ x 5, $S_3^2\times D_5.D_5$ x 4, $D_5^2:S_3^2$ x 4, $C_{15}^2:\OD_{16}$ x 4, $D_5^2.S_3^2$ x 4, $D_{15}^2:C_4$ x 4, $C_{15}^2:(C_2\times D_4)$ x 4, $C_{15}^2:D_4:C_2$ x 4, $C_{15}^2:(C_2\times Q_8)$ x 4, $C_{15}^2:(C_2\times C_8)$ x 4, $D_5\wr C_2\times C_3:S_3$ x 2, $C_{15}^2.\OD_{16}$ x 2, $D_5:F_5\times C_3:S_3$, $C_5^2:Q_8\times C_3:S_3$, $C_5:D_5.C_4\times C_3:S_3$, $C_3^2\times D_5^2.C_4$, $C_3^2\times D_5\wr C_2:C_2$, $C_2\times D_5.D_5\times C_3:S_3$

Order 3375: $C_{15}^3$, $C_{15}^2.C_{15}$, $C_{15}^2:C_{15}$

Order 2000: $C_5:F_5^2$

Order 1500: $C_{15}:D_5^2$

Order 1125: $C_5\times C_{15}^2$

Order 500: $C_5:D_5^2$

Order 375: $C_5^2\times C_{15}$

Order 125: $C_5^3$

Order 54: $C_3^2:S_3$

Order 27: $C_3^3$

Order 9: $C_3^2$

Order 3: $C_3$

Order 1: $C_1$

Classes of subgroups up to automorphism

not computed

Normal subgroups (quotient in parentheses)

Order 648000: $C_5^3.C_{12}^2.S_3^2$ ($C_1$)

Order 324000: $C_{15}^3.C_4^2.C_6$ ($C_2$), $C_5^3.C_6^2.C_3:S_4$ ($C_2$), $C_5^3.C_{12}^2.C_3.C_6$ ($C_2$)

Order 162000: $C_5^3.C_{12}^2.C_3^2$ ($C_2^2$)

Order 108000: $C_3^3:C_2\times C_5^3:C_4^2$ ($S_3$), $C_5^3:(C_{12}^2:S_3)$ ($S_3$)

Order 54000: $C_3^3\times C_5^3:C_4^2$ ($D_6$), $C_5:F_5^2:\He_3$ ($D_6$)

Order 27000: $C_3^3:C_2\times C_5^3:C_2^2$ ($S_4$)

Order 18000: $C_3^2\times C_5^3:C_4^2$ ($S_3^2$)

Order 13500: $C_3^3\times C_5^3:C_2^2$ ($C_2\times S_4$)

Order 6750: $C_5^3\times C_3^3:C_2$ ($C_4^2:S_3$)

Order 6000: $C_3\times C_5^3:C_4^2$ ($C_3^2:D_6$)

Order 4500: $C_3^2\times C_5^3:C_2^2$ ($S_3\times S_4$)

Order 3375: $C_{15}^3$ ($C_4^2:D_6$)

Order 2000: $C_5:F_5^2$ ($C_3^3:D_6$)

Order 1500: $C_{15}:D_5^2$ ($C_6^2:D_6$)

Order 1125: $C_5\times C_{15}^2$ ($C_4^2:S_3^2$)

Order 500: $C_5:D_5^2$ ($C_6^2.S_3^2$)

Order 375: $C_5^2\times C_{15}$ ($C_{12}^2:D_6$)

Order 125: $C_5^3$ ($C_{12}^2.S_3^2$)

Order 54: $C_3^2:S_3$ ($C_5:F_5^2:S_3$)

Order 27: $C_3^3$ ($C_5:F_5^2:D_6$)

Order 9: $C_3^2$ ($C_5:F_5^2:S_3^2$)

Order 3: $C_3$ ($C_5^3:(C_{12}^2:D_6)$)

Order 1: $C_1$ ($C_5^3.C_{12}^2.S_3^2$)

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 3 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 $223 \times 223$ 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 $139 \times 139$ rational character table (warning: may be slow to load).