Abstract group 10080.v: $S_3\times D_5\times \GL(3,2)$

• Label: 10080.v
• Order: \( 2^{5} \cdot 3^{2} \cdot 5 \cdot 7 \)
• Exponent: \( 2^{2} \cdot 3 \cdot 5 \cdot 7 \)
• Nilpotent: no
• Solvable: no
• $\card{G^{\mathrm{ab}}}$: \( 2^{2} \)
• $\card{Z(G)}$: \( 1 \)
• $\card{\Aut(G)}$: \( 2^{7} \cdot 3^{2} \cdot 5 \cdot 7 \)
• $\card{\mathrm{Out}(G)}$: \( 2^{2} \)
• Perm deg.: $15$
• Trans deg.: $105$
• Rank: $2$

Group information

Description:	$S_3\times D_5\times \GL(3,2)$
Order:	\(10080\)\(\medspace = 2^{5} \cdot 3^{2} \cdot 5 \cdot 7 \)
Exponent:	\(420\)\(\medspace = 2^{2} \cdot 3 \cdot 5 \cdot 7 \)
Automorphism group:	$\SO(3,7)\times S_3\times F_5$, of order \(40320\)\(\medspace = 2^{7} \cdot 3^{2} \cdot 5 \cdot 7 \)
Composition factors:	$C_2$ x 2, $C_3$, $C_5$, $\PSL(2,7)$
Derived length:	$2$

This group is nonabelian and nonsolvable.

Group statistics

Order	1	2	3	4	5	6	7	10	12	14	15	20	21	30	35	42	60	70	105
Elements	1	527	170	1008	4	2110	48	348	504	1104	680	672	96	840	192	480	336	576	384	10080
Conjugacy classes	1	7	3	4	2	7	2	6	2	6	6	4	2	4	4	2	2	4	4	72
Divisions	1	7	3	4	1	7	1	3	2	3	3	2	1	2	1	1	1	1	1	45
Autjugacy classes	1	7	3	4	1	7	1	3	2	3	3	2	1	2	1	1	1	1	1	45

Dimension	1	2	3	4	6	7	8	12	14	16	24	28	32	48	56	64
Irr. complex chars.	4	6	8	2	16	4	4	10	6	6	2	2	2	0	0	0	72
Irr. rational chars.	4	2	0	2	8	4	5	4	2	2	4	2	2	2	1	1	45

Minimal presentations

Permutation degree:	$15$
Transitive degree:	$105$
Rank:	$2$
Inequivalent generating pairs:	$342$

Minimal degrees of faithful linear representations

	Over $\mathbb{C}$	Over $\mathbb{R}$	Over $\mathbb{Q}$
Irreducible	12	24	48
Arbitrary	7	10	12

Constructions

Permutation group:	Degree $15$ $\langle(4,8)(5,7), (1,2,3), (1,2)(4,7,5,8,6), (1,2,3)(4,6,8,5,7)(9,15)(10,12), (1,3,2)(4,7)(5,6)(9,11,14,12)(13,15), (4,8,7,6,5)\rangle$
Direct product:	$S_3$ $\, \times\, $ $D_5$ $\, \times\, $ $\PSL(2,7)$
Semidirect product:	$C_5$ $\,\rtimes\,$ $(D_6\times \GL(3,2))$	$(C_5\times \PSL(2,7))$ $\,\rtimes\,$ $D_6$	$(D_{15}\times \GL(3,2))$ $\,\rtimes\,$ $C_2$	$(C_3\times \GL(3,2))$ $\,\rtimes\,$ $D_{10}$	all 12
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product

Elements of the group are displayed as permutations of degree 15.

Homology

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

Subgroups

There are 44180 subgroups in 606 conjugacy classes, 20 normal, and all normal subgroups are characteristic.

Characteristic subgroups are shown in this color.

Special subgroups

Center:	$Z \simeq$ $C_1$	$G/Z \simeq$ $S_3\times D_5\times \GL(3,2)$
Commutator:	$G' \simeq$ $C_{15}\times \GL(3,2)$	$G/G' \simeq$ $C_2^2$
Frattini:	$\Phi \simeq$ $C_1$	$G/\Phi \simeq$ $S_3\times D_5\times \GL(3,2)$
Fitting:	$\operatorname{Fit} \simeq$ $C_{15}$	$G/\operatorname{Fit} \simeq$ $C_2^2\times \GL(3,2)$
Radical:	$R \simeq$ $S_3\times D_5$	$G/R \simeq$ $\PSL(2,7)$
Socle:	$\operatorname{soc} \simeq$ $C_{15}\times \GL(3,2)$	$G/\operatorname{soc} \simeq$ $C_2^2$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $C_2^2\times D_4$
3-Sylow subgroup:	$P_{ 3 } \simeq$ $C_3^2$
5-Sylow subgroup:	$P_{ 5 } \simeq$ $C_5$
7-Sylow subgroup:	$P_{ 7 } \simeq$ $C_7$

Hi

Subgroup diagram and profile

Classes of subgroups up to conjugation

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Classes of subgroups up to automorphism

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Normal subgroups

Normal subgroups up to automorphism

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Classes of subgroups up to conjugation

Order 10080: $S_3\times D_5\times \GL(3,2)$

Order 5040: $D_{15}\times \GL(3,2)$, $C_5\times S_3\times \GL(3,2)$, $C_3\times D_5\times \GL(3,2)$

Order 3360: $D_{10}\times \PSL(2,7)$

Order 2520: $C_{15}\times \GL(3,2)$

Order 2016: $D_6\times \GL(3,2)$

Order 1680: $D_5\times \GL(3,2)$ x 2, $C_{10}\times \GL(3,2)$

Order 1440: $S_3\times D_5\times S_4$ x 2

Order 1260: $C_{35}:C_6\times S_3$

Order 1008: $S_3\times \GL(3,2)$ x 2, $C_6\times \GL(3,2)$

Order 840: $C_5\times \PSL(2,7)$

Order 720: $S_3\times D_5\times A_4$ x 2, $D_5\times C_3:S_4$ x 2, $D_{15}:S_4$ x 2, $S_4\times D_{15}$ x 2, $S_3\times C_5:S_4$ x 2, $C_5\times S_3\times S_4$ x 2, $C_3\times D_5\times S_4$ x 2

Order 672: $C_2^2\times \GL(3,2)$

Order 630: $C_{105}:C_6$, $C_{21}:C_{30}$, $C_{105}:C_6$

Order 504: $C_3\times \GL(3,2)$

Order 480: $D_{10}\times S_4$ x 2, $D_{12}:D_{10}$

Order 420: $C_{21}:D_{10}$, $C_{70}:C_6$

Order 360: $A_4\times D_{15}$ x 2, $C_5\times S_3\times A_4$ x 2, $C_3\times D_5\times A_4$ x 2, $C_{15}:S_4$ x 2, $C_{15}:S_4$ x 2, $C_{15}:S_4$ x 2, $C_{15}\times S_4$ x 2, $D_5\times S_3^2$

Order 336: $C_2\times \GL(3,2)$ x 3

Order 315: $C_{105}:C_3$

Order 288: $D_6\times S_4$ x 2

Order 252: $C_{42}:C_6$

Order 240: $D_5\times S_4$ x 10, $D_6\times D_{10}$ x 2, $A_4\times D_{10}$ x 2, $C_{10}:S_4$ x 2, $C_{10}\times S_4$ x 2, $D_6:D_{10}$ x 2, $D_6:D_{10}$ x 2, $D_6:D_{10}$ x 2, $D_4\times D_{15}$, $C_{20}:D_6$, $C_{12}:D_{10}$, $C_{12}:D_{10}$, $S_3\times D_{20}$, $D_5\times D_{12}$, $C_{12}:D_{10}$

Order 210: $C_{35}:C_6$ x 3, $S_3\times C_{35}$, $D_5\times C_{21}$, $C_7:C_{30}$, $C_7\times D_{15}$

Order 180: $A_4\times C_{15}$ x 2, $S_3\times D_{15}$ x 2, $C_{15}:D_6$ x 2, $C_{15}:D_6$, $C_5\times S_3^2$, $C_{15}:D_6$

Order 168: $\PSL(2,7)$

Order 160: $D_4\times D_{10}$

Order 144: $S_3\times S_4$ x 8, $A_4\times D_6$ x 2, $C_6:S_4$ x 2, $C_6\times S_4$ x 2

Order 140: $C_7\times D_{10}$

Order 126: $C_{21}:C_6$ x 2, $C_{21}:C_6$

Order 120: $S_3\times D_{10}$ x 8, $D_5\times A_4$ x 6, $C_5:S_4$ x 6, $C_5\times S_4$ x 6, $C_2\times D_{30}$ x 2, $C_{10}\times D_6$ x 2, $C_6\times D_{10}$ x 2, $C_{10}\times A_4$ x 2, $C_{15}:D_4$ x 2, $C_{15}:D_4$ x 2, $C_{15}:D_4$ x 2, $C_5:D_{12}$ x 2, $C_3:D_{20}$ x 2, $C_{15}:D_4$ x 2, $C_6.D_{10}$, $C_6.D_{10}$, $D_4\times C_{15}$, $D_{60}$, $C_4\times D_{15}$, $C_5\times D_{12}$, $S_3\times C_{20}$, $C_3\times D_{20}$, $D_5\times C_{12}$, $D_{15}:C_4$

Order 105: $C_7:C_{15}$ x 2, $C_{105}$

Order 96: $C_2^2\times S_4$ x 2, $D_4\times D_6$

Order 90: $C_3\times D_{15}$ x 2, $S_3\times C_{15}$ x 2, $C_3:D_{15}$, $C_{15}:S_3$, $C_3^2\times D_5$

Order 84: $C_{14}:C_6$, $S_3\times C_{14}$

Order 80: $D_4\times D_5$ x 8, $C_2^2\times D_{10}$ x 2, $C_{10}:D_4$ x 2, $D_4\times C_{10}$, $C_2\times D_{20}$, $C_4\times D_{10}$

Order 72: $S_3\times A_4$ x 4, $C_3:S_4$ x 4, $C_3\times S_4$ x 4, $C_6\times A_4$ x 2, $S_3\times D_6$

Order 70: $C_7\times D_5$ x 2, $C_{70}$

Order 63: $C_{21}:C_3$

Order 60: $S_3\times D_5$ x 11, $C_5\times A_4$ x 4, $D_{30}$ x 4, $S_3\times C_{10}$ x 4, $C_3\times D_{10}$ x 4, $C_2\times C_{30}$ x 2, $C_{60}$, $C_{15}:C_4$, $C_5:C_{12}$, $C_3:C_{20}$

Order 48: $C_2\times S_4$ x 14, $S_3\times D_4$ x 8, $C_2^2\times D_6$ x 2, $C_2^2\times A_4$ x 2, $C_6:D_4$ x 2, $C_6\times D_4$, $C_2\times D_{12}$, $C_4\times D_6$

Order 45: $C_3\times C_{15}$

Order 42: $C_7:C_6$ x 4, $S_3\times C_7$ x 2, $C_{42}$

Order 40: $C_2\times D_{10}$ x 11, $C_5:D_4$ x 8, $D_{20}$ x 4, $C_4\times D_5$ x 4, $C_5\times D_4$ x 4, $C_2^2\times C_{10}$ x 2, $C_2\times C_{20}$, $C_{10}:C_4$

Order 36: $S_3^2$ x 4, $C_6\times S_3$ x 2, $C_3\times A_4$ x 2, $C_6:S_3$

Order 35: $C_{35}$

Order 32: $C_2^2\times D_4$

Order 30: $D_{15}$ x 5, $C_3\times D_5$ x 5, $C_5\times S_3$ x 5, $C_{30}$ x 2

Order 28: $C_2\times C_{14}$

Order 24: $C_2\times D_6$ x 12, $S_4$ x 12, $C_3:D_4$ x 8, $C_2\times A_4$ x 8, $D_{12}$ x 4, $C_4\times S_3$ x 4, $C_3\times D_4$ x 4, $C_2^2\times C_6$ x 2, $C_2\times C_{12}$, $C_6:C_4$

Order 21: $C_7:C_3$ x 2, $C_{21}$

Order 20: $D_{10}$ x 12, $C_2\times C_{10}$ x 5, $C_{20}$ x 2, $C_5:C_4$ x 2

Order 18: $C_3\times S_3$ x 4, $C_3:S_3$ x 2, $C_3\times C_6$

Order 16: $C_2\times D_4$ x 12, $C_2^4$ x 2, $C_2^2\times C_4$

Order 15: $C_{15}$ x 3

Order 14: $C_{14}$ x 3

Order 12: $D_6$ x 19, $C_2\times C_6$ x 6, $A_4$ x 4, $C_{12}$ x 2, $C_3:C_4$ x 2

Order 10: $D_5$ x 4, $C_{10}$ x 3

Order 9: $C_3^2$

Order 8: $D_4$ x 16, $C_2^3$ x 13, $C_2\times C_4$ x 6

Order 7: $C_7$

Order 6: $S_3$ x 10, $C_6$ x 7

Order 5: $C_5$

Order 4: $C_2^2$ x 17, $C_4$ x 4

Order 3: $C_3$ x 3

Order 2: $C_2$ x 7

Order 1: $C_1$

Classes of subgroups up to automorphism

Order 10080: $S_3\times D_5\times \GL(3,2)$

Order 5040: $D_{15}\times \GL(3,2)$, $C_5\times S_3\times \GL(3,2)$, $C_3\times D_5\times \GL(3,2)$

Order 3360: $D_{10}\times \PSL(2,7)$

Order 2520: $C_{15}\times \GL(3,2)$

Order 2016: $D_6\times \GL(3,2)$

Order 1680: $D_5\times \GL(3,2)$ x 2, $C_{10}\times \GL(3,2)$

Order 1440: $S_3\times D_5\times S_4$

Order 1260: $C_{35}:C_6\times S_3$

Order 1008: $S_3\times \GL(3,2)$ x 2, $C_6\times \GL(3,2)$

Order 840: $C_5\times \PSL(2,7)$

Order 720: $S_3\times D_5\times A_4$, $D_5\times C_3:S_4$, $D_{15}:S_4$, $S_4\times D_{15}$, $S_3\times C_5:S_4$, $C_5\times S_3\times S_4$, $C_3\times D_5\times S_4$

Order 672: $C_2^2\times \GL(3,2)$

Order 630: $C_{105}:C_6$, $C_{21}:C_{30}$, $C_{105}:C_6$

Order 504: $C_3\times \GL(3,2)$

Order 480: $D_{10}\times S_4$, $D_{12}:D_{10}$

Order 420: $C_{21}:D_{10}$, $C_{70}:C_6$

Order 360: $A_4\times D_{15}$, $C_5\times S_3\times A_4$, $C_3\times D_5\times A_4$, $C_{15}:S_4$, $C_{15}:S_4$, $C_{15}:S_4$, $C_{15}\times S_4$, $D_5\times S_3^2$

Order 336: $C_2\times \GL(3,2)$ x 3

Order 315: $C_{105}:C_3$

Order 288: $D_6\times S_4$

Order 252: $C_{42}:C_6$

Order 240: $D_5\times S_4$ x 5, $D_6\times D_{10}$, $A_4\times D_{10}$, $C_{10}:S_4$, $C_{10}\times S_4$, $D_4\times D_{15}$, $C_{20}:D_6$, $C_{12}:D_{10}$, $D_6:D_{10}$, $D_6:D_{10}$, $D_6:D_{10}$, $C_{12}:D_{10}$, $S_3\times D_{20}$, $D_5\times D_{12}$, $C_{12}:D_{10}$

Order 210: $C_{35}:C_6$ x 3, $S_3\times C_{35}$, $D_5\times C_{21}$, $C_7:C_{30}$, $C_7\times D_{15}$

Order 180: $S_3\times D_{15}$ x 2, $C_{15}:D_6$ x 2, $A_4\times C_{15}$, $C_{15}:D_6$, $C_5\times S_3^2$, $C_{15}:D_6$

Order 168: $\PSL(2,7)$

Order 160: $D_4\times D_{10}$

Order 144: $S_3\times S_4$ x 4, $A_4\times D_6$, $C_6:S_4$, $C_6\times S_4$

Order 140: $C_7\times D_{10}$

Order 126: $C_{21}:C_6$ x 2, $C_{21}:C_6$

Order 120: $S_3\times D_{10}$ x 5, $D_5\times A_4$ x 3, $C_5:S_4$ x 3, $C_5\times S_4$ x 3, $C_6.D_{10}$, $C_6.D_{10}$, $C_2\times D_{30}$, $C_{10}\times D_6$, $C_6\times D_{10}$, $C_{10}\times A_4$, $D_4\times C_{15}$, $C_{15}:D_4$, $D_{60}$, $C_4\times D_{15}$, $C_{15}:D_4$, $C_5\times D_{12}$, $S_3\times C_{20}$, $C_{15}:D_4$, $C_3\times D_{20}$, $D_5\times C_{12}$, $C_5:D_{12}$, $C_3:D_{20}$, $C_{15}:D_4$, $D_{15}:C_4$

Order 105: $C_7:C_{15}$ x 2, $C_{105}$

Order 96: $C_2^2\times S_4$, $D_4\times D_6$

Order 90: $C_3\times D_{15}$ x 2, $S_3\times C_{15}$ x 2, $C_3:D_{15}$, $C_{15}:S_3$, $C_3^2\times D_5$

Order 84: $C_{14}:C_6$, $S_3\times C_{14}$

Order 80: $D_4\times D_5$ x 6, $C_2^2\times D_{10}$, $D_4\times C_{10}$, $C_{10}:D_4$, $C_2\times D_{20}$, $C_4\times D_{10}$

Order 72: $S_3\times A_4$ x 2, $C_3:S_4$ x 2, $C_3\times S_4$ x 2, $C_6\times A_4$, $S_3\times D_6$

Order 70: $C_7\times D_5$ x 2, $C_{70}$

Order 63: $C_{21}:C_3$

Order 60: $S_3\times D_5$ x 10, $D_{30}$ x 3, $S_3\times C_{10}$ x 3, $C_3\times D_{10}$ x 3, $C_5\times A_4$ x 2, $C_{60}$, $C_{15}:C_4$, $C_5:C_{12}$, $C_2\times C_{30}$, $C_3:C_{20}$

Order 48: $C_2\times S_4$ x 7, $S_3\times D_4$ x 6, $C_2^2\times D_6$, $C_2^2\times A_4$, $C_6\times D_4$, $C_6:D_4$, $C_2\times D_{12}$, $C_4\times D_6$

Order 45: $C_3\times C_{15}$

Order 42: $C_7:C_6$ x 4, $S_3\times C_7$ x 2, $C_{42}$

Order 40: $C_2\times D_{10}$ x 6, $C_5:D_4$ x 4, $C_4\times D_5$ x 4, $D_{20}$ x 3, $C_5\times D_4$ x 3, $C_2\times C_{20}$, $C_{10}:C_4$, $C_2^2\times C_{10}$

Order 36: $S_3^2$ x 4, $C_6\times S_3$ x 2, $C_6:S_3$, $C_3\times A_4$

Order 35: $C_{35}$

Order 32: $C_2^2\times D_4$

Order 30: $D_{15}$ x 5, $C_3\times D_5$ x 5, $C_5\times S_3$ x 5, $C_{30}$ x 2

Order 28: $C_2\times C_{14}$

Order 24: $C_2\times D_6$ x 7, $S_4$ x 6, $C_3:D_4$ x 4, $C_4\times S_3$ x 4, $C_2\times A_4$ x 4, $D_{12}$ x 3, $C_3\times D_4$ x 3, $C_2\times C_{12}$, $C_6:C_4$, $C_2^2\times C_6$

Order 21: $C_7:C_3$ x 2, $C_{21}$

Order 20: $D_{10}$ x 9, $C_2\times C_{10}$ x 3, $C_{20}$ x 2, $C_5:C_4$ x 2

Order 18: $C_3\times S_3$ x 4, $C_3:S_3$ x 2, $C_3\times C_6$

Order 16: $C_2\times D_4$ x 9, $C_2^4$, $C_2^2\times C_4$

Order 15: $C_{15}$ x 3

Order 14: $C_{14}$ x 3

Order 12: $D_6$ x 16, $C_2\times C_6$ x 4, $A_4$ x 2, $C_{12}$ x 2, $C_3:C_4$ x 2

Order 10: $D_5$ x 4, $C_{10}$ x 3

Order 9: $C_3^2$

Order 8: $D_4$ x 10, $C_2^3$ x 7, $C_2\times C_4$ x 6

Order 7: $C_7$

Order 6: $S_3$ x 10, $C_6$ x 7

Order 5: $C_5$

Order 4: $C_2^2$ x 12, $C_4$ x 4

Order 3: $C_3$ x 3

Order 2: $C_2$ x 7

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 10080: $S_3\times D_5\times \GL(3,2)$ ($C_1$)

Order 5040: $D_{15}\times \GL(3,2)$ ($C_2$), $C_5\times S_3\times \GL(3,2)$ ($C_2$), $C_3\times D_5\times \GL(3,2)$ ($C_2$)

Order 2520: $C_{15}\times \GL(3,2)$ ($C_2^2$)

Order 1680: $D_5\times \GL(3,2)$ ($S_3$)

Order 1008: $S_3\times \GL(3,2)$ ($D_5$)

Order 840: $C_5\times \PSL(2,7)$ ($D_6$)

Order 504: $C_3\times \GL(3,2)$ ($D_{10}$)

Order 168: $\PSL(2,7)$ ($S_3\times D_5$)

Order 60: $S_3\times D_5$ ($\PSL(2,7)$)

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

Order 15: $C_{15}$ ($C_2^2\times \GL(3,2)$)

Order 10: $D_5$ ($S_3\times \GL(3,2)$)

Order 6: $S_3$ ($D_5\times \GL(3,2)$)

Order 5: $C_5$ ($D_6\times \GL(3,2)$)

Order 3: $C_3$ ($D_{10}\times \PSL(2,7)$)

Order 1: $C_1$ ($S_3\times D_5\times \GL(3,2)$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 10080: $S_3\times D_5\times \GL(3,2)$ ($C_1$)

Order 5040: $D_{15}\times \GL(3,2)$ ($C_2$), $C_5\times S_3\times \GL(3,2)$ ($C_2$), $C_3\times D_5\times \GL(3,2)$ ($C_2$)

Order 2520: $C_{15}\times \GL(3,2)$ ($C_2^2$)

Order 1680: $D_5\times \GL(3,2)$ ($S_3$)

Order 1008: $S_3\times \GL(3,2)$ ($D_5$)

Order 840: $C_5\times \PSL(2,7)$ ($D_6$)

Order 504: $C_3\times \GL(3,2)$ ($D_{10}$)

Order 168: $\PSL(2,7)$ ($S_3\times D_5$)

Order 60: $S_3\times D_5$ ($\PSL(2,7)$)

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

Order 15: $C_{15}$ ($C_2^2\times \GL(3,2)$)

Order 10: $D_5$ ($S_3\times \GL(3,2)$)

Order 6: $S_3$ ($D_5\times \GL(3,2)$)

Order 5: $C_5$ ($D_6\times \GL(3,2)$)

Order 3: $C_3$ ($D_{10}\times \PSL(2,7)$)

Order 1: $C_1$ ($S_3\times D_5\times \GL(3,2)$)

Series

Derived series

$S_3\times D_5\times \GL(3,2)$

$\rhd$

$C_{15}\times \GL(3,2)$

$\rhd$

$\PSL(2,7)$

Chief series

$S_3\times D_5\times \GL(3,2)$

$\rhd$

$S_3\times D_5$

$\rhd$

$C_5\times S_3$

$\rhd$

$C_{15}$

$\rhd$

$C_5$

$\rhd$

$C_1$

Lower central series

$S_3\times D_5\times \GL(3,2)$

$\rhd$

$C_{15}\times \GL(3,2)$

Upper central series

$C_1$

Supergroups

This group is a maximal subgroup of 3 larger groups in the database.

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

Character theory

Complex character table

See the $72 \times 72$ character table. Alternatively, you may search for characters of this group with desired properties.

Rational character table

See the $45 \times 45$ rational character table.